A108959 Triangle arising in connection with deformations of type D Kleinian singularities.

1, 2, 1, 3, 4, 2, 4, 10, 14, 7, 5, 20, 54, 76, 38, 6, 35, 154, 419, 590, 295, 7, 56, 364, 1616, 4400, 6196, 3098, 8, 84, 756, 4962, 22048, 60036, 84542, 42271, 9, 120, 1428, 12984, 85300, 379052, 1032154, 1453468, 726734, 10, 165, 2508, 30162, 274516, 1803638, 8014990, 21824737, 30733358, 15366679

Offset

0,2

Links

Table of n, a(n) for n=0..54.

Paul Boddington, No-cycle algebras and representation theory, Ph.D. thesis, University of Warwick, 2004.

Formula

For k>=0 define p_k(x) = x(x+1)(x+3)...(x+k(k-1)/2) and consider the linear map taking each p_k(x) to k*p_k(x)/x. Then the images of x, x^2, x^3, ... are given by the rows. E.g., x^3 goes to 3x^2 + 4x + 2.

Example

Triangle begins:
  1;
  2,  1;
  3,  4,  2;
  4, 10, 14,  7;
  5, 20, 54, 76, 38;
  ...

Prog

(PARI) tabl(nn) = my(v = vector(nn)); for (n=1, nn, my(p=prod(i=1, n, x+i*(i-1)/2), q=n*p/x); v[n] = q - sum(i=1, n-1, polcoeff(p, i)*v[i])); vector(nn, k, Vec(v[k])); \\ Michel Marcus, Mar 18 2023

Crossrefs

This sequence is an improved version of A097418. Coefficients of 1 give A000366.

Cf. A128813 (the p_k polynomials).

Sequence in context: A245334 A102756 A086614 * A208750 A107893 A131987

Adjacent sequences: A108956 A108957 A108958 * A108960 A108961 A108962

Keyword

easy,nonn,tabl

Author

Paul Boddington, Jul 22 2005

Extensions

More terms from Michel Marcus, Mar 18 2023

Status

approved