Disclaimer: The characters belong to their respective owners, and the song is the translation of an old Japanese song, "Sakura". From what I've seen/read of Inuyasha, nothing has been mentioned of Sesshomaru's mother or what happened to her, and so here's my take on things.
Lullaby
The youkai looked down at the small being in his arms, its eyes scrunched closed as it yawned and curled its tiny fingers into fists. He grazed a claw along the pup's cheek, tracing one long red marking. The moon-shaped mark on his forehead seemed to dominate his face, nestled between two tufts of silver-white bangs.
"Sesshomaru." She had insisted on the name, befitting of his future, she'd said. Destruction of the circle of life.(1) He watched his infant son's features relax into sleep; his breathing slow and peaceful.
He closed his eyes and leaned back into his mother's embrace, her soft lips brushing the tip of his ear as she hummed softly to him. The humming soon took on the words of his favorite lullaby, enveloping him in warmth and safety.
"Cherry blossoms, cherry blossoms, as far as I can see across the March sky..." He opened his eyes to look up at the sky, squinting as though he could envision the pink blossoms dancing along to her voice.
"The fragrance is carried like the mist or the clouds..." She continued, and he breathed in deeply, like he could smell them. "Now, let us go to see them now..."
Her voice fell into a whisper, and she murmured to him things of the future. Of his power--of their power. The simple song and the secret promises were of the same intimacy. But they were whispered, these secrets, because they were not so innocent as a lullaby.
That day, he learned all about dangerous secrets.
"Sesshomaru." His father's voice, low and calm, smooth with confidence. He wanted to be just like his father. He slipped out of his mother's arms and took his father's proffered hand. The demon lord knelt down to him, holding his son's gaze with a solemn air that was almost tangible. He then rose, releasing Sesshomaru's hand, and swept past the child and over to his mother. The pup turned to watch as his father reached down, strong, swift fingers clutching his mate by the throat, lifting her to her feet and then up off the ground. His mother's eyes were wide, and her lips were parted--but no soothing lullabies or secret whispers passed through them this time, only a strangled cry of alarm.
"Those who underestimate me don't live for very long." His father said flatly, iced-over anger glittering in the depths of his eyes. "Don't try to plead, or waste my time with petty declarations--I know you never loved me. To try and turn my own son against me--in that you have made a grievous mistake." He curled his claws into her throat, shredding it through from five different points.
Sesshomaru stood numb with shock. He fought the urge to step back as his mother's eyes fell on him, a moment before they went dull and she went lax in his father's grip, her blood dying the top of her kimono dark red. His father let her fall to the ground, then turned and approached Sesshomaru, who looked up at him.
"She was using you, just like she used me. She was dangerous to us both--an enemy." He told his son, kneeling down again. "It's time you started following in your father's footsteps, Sesshomaru. A demon lord must look out for himself, he must know his enemies and destroy them all, and he must never show weakness." He said, raising his clean hand to wipe away the tears that had strayed down Sesshomaru's cheeks. The pup bit his lower lip, his fingers clenching into fists at his sides as he fought down the sadness that threatened to unleash itself. He blinked his eyes clear, nodding as he forced his fists to uncurl. It was easy to let himself be hurt, but he was to be a powerful demon lord. A demon lord could not show hurt.
"Yes, father."
His father, whom he had wanted to be just like. His father, who had given his love to a mortal woman--that which he had not been able to give to Sesshomaru's mother. It still hurt to remember the way in which he had so easily killed her, without a moment's hesitation. Not that he hadn't had reason to--after she had been plotting to use Sesshomaru against him, in order to gain power for herself. And of course, Sesshomaru himself had behaved in exactly the same way as his father, countless times--bringing death was nothing to be concerned about, not for a demon lord.
"Is Sesshomaru-sama sad?" He was pulled from the fog of his memories as Rin shoved her face into his, innocent eyes holding his in curiosity. "Cheer up!" She exclaimed, and kissed him square on the nose. He quirked an eyebrow in response to the childish gesture, then pulled her down into his lap. She curled up against him, smiling sweetly as she reached out to pet his hair. He closed his eyes and leaned forward, resting his chin on the top of her head.
He had become just like his father. Ruthless killer, powerful demon lord, untouchable by everyone--except for the one mortal girl that had stolen his black heart. He felt the color rise in his cheeks as he conjured up words he had thought he'd forgotten long ago, his voice quiet as they passed over his lips. They were only for her to hear.
"Cherry blossoms...cherry blossoms, as far as I can see across the March sky..." He hesitated when Rin stilled in his arms, but her tight hug was encouragement enough, and he felt his embarrassment start to fade. He could sing to her--his child, who's innocence could easily be corrupted. But what purpose would it serve, to corrupt a little human girl? She was not the heir to his title; she could never become a demon lord like him or his father before. He had no desire to twist her like he had been twisted, shaped in his father's image. It was necessary for him--it was not for her.
She was his child, and her love for him was clear and untainted. All he wanted was to sing to her like his mother had sang to him, but without any secrets or manipulations--just out of love for the child in his arms.
"The fragrance is carried like the mist or the clouds...Now, let us go to see them now..." The rest of the song poured itself from his memory, the words flowing easily from his tongue. "Cherry blossoms, cherry blossoms, as far as I can see across the hills and valleys, like the mist or the clouds... The fragrance is carried into the morning sun...cherry blossoms, cherry blossoms, they are in full bloom..."
He opened his eyes to find her watching the sky, as though she could see the blossoms dancing in the distance.
End.
1 - Taken from the kanji for Sesshomaru. "The kanji of 'Sesshō' means 'to kill or destroy.' 'Maru' is usually a suffix that is often added at the end of a Japanese name for young males in samurai families. They would use this name until they were recognized as adults, whereupon they would adopt a new name. However, it is said that 'maru' can also translate as the word for 'circle', which is a common Japanese motiff for what in the West is called 'the circle of life.' Thus it is widely assumed that the name Sesshomaru translates literally as 'destruction of the circle of life', or 'end of the circle of life.'" - From wikipedia