A330209 Nonzero coefficients of the polynomials (x + d/dx)^n x^2, in row-major order.

1, 2, 1, 2, 5, 1, 12, 9, 1, 12, 39, 14, 1, 90, 95, 20, 1, 90, 375, 195, 27, 1, 840, 1155, 357, 35, 1, 840, 4305, 2940, 602, 44, 1, 9450, 16065, 6552, 954, 54, 1, 9450, 57645, 48825, 13230, 1440, 65, 1, 124740, 252945, 128205, 24750, 2090, 77, 1, 124740, 883575, 893970, 301455, 43560, 2937, 90, 1

Offset

0,2

Comments

The similar-looking polynomials (x-d/dx)^n * 1 are the probabilist's Hermite polynomials, A096713.

Links

Table of n, a(n) for n=0..60.

Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 19.

Example

x^2 ; 2x + x^3; 2 + 5x^2 + x^4; 12x + 9x^3 + x^5.

Crossrefs

Cf. A096713.

Sequence in context: A291694 A135506 A295516 * A068822 A351517 A337224

Adjacent sequences: A330206 A330207 A330208 * A330210 A330211 A330212

Keyword

nonn,tabf

Author

Chris Jones, Dec 05 2019

Status

approved