A097851 G.f.: (1+x^8+x^9+x^10+x^18)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)).

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 13, 20, 22, 31, 36, 47, 54, 71, 80, 102, 117, 144, 164, 201, 227, 272, 309, 365, 411, 483, 540, 627, 702, 806, 898, 1026, 1137, 1289, 1427, 1606, 1770, 1985, 2179, 2429, 2663, 2952, 3225, 3565, 3882, 4272, 4644, 5090, 5518, 6032, 6522

Offset

0,5

Comments

Poincare series for invariant polynomial functions on the space of binary forms of degree 8.

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Andries Brouwer, Poincaré Series (See n=8).

J.-I. Igusa, Modular forms and projective invariants, Amer. J. Math., 89 (1967), 817-855; see p. 847.

Peter Littelmann and Claudio Procesi, On the Poincaré series of the invariants of binary forms, Journal of Algebra 133.2 (1990): 490-499. See last page.

Index entries for linear recurrences with constant coefficients, signature (-1,1,3,3,0,-3,-4,-3,-1,1,2,3,3,2,1,-1,-3,-4,-3,0,3,3,1,-1,-1).

Formula

a(n) = floor((6*n^5+135*n^4+2488*n^3+25488*n^2+129792*n+438291)/725760 - (n mod 2)*(n^2+9*n+50)/128 + ((2*n^2+1) mod 3)*n/27 + ((n^3+n^2+n+2) mod 5)/5). - Hoang Xuan Thanh, Apr 20 2026

Mathematica

CoefficientList[Series[(1+x^8+x^9+x^10+x^18)/((1-x^2)(1-x^3)(1-x^4)(1-x^5)(1-x^6)(1-x^7)), {x, 0, 60}], x] (* or *) LinearRecurrence[{-1, 1, 3, 3, 0, -3, -4, -3, -1, 1, 2, 3, 3, 2, 1, -1, -3, -4, -3, 0, 3, 3, 1, -1, -1}, {1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 13, 20, 22, 31, 36, 47, 54, 71, 80, 102, 117, 144, 164, 201}, 60] (* Harvey P. Dale, Jul 19 2025 *)

Prog

(PARI) Vec((1+x^8+x^9+x^10+x^18)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)) + O(x^50)) \\ Jinyuan Wang, Mar 10 2020

Crossrefs

For these Poincare series for d = 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 20, 24 see A097852, A293933, A097851, A293934, A293935, A293936, A293937, A293938, A293939, A293940, A293941, A293942, A293943 respectively.

This Poincare series is mentioned in A079293.

Sequence in context: A027591 A027596 A007213 * A266778 A107235 A266779

Adjacent sequences: A097848 A097849 A097850 * A097852 A097853 A097854

Keyword

nonn,easy

Author

N. J. A. Sloane, Sep 01 2004

Status

approved