A107878 Column 2 of triangle A107876.

1, 1, 3, 15, 106, 975, 11100, 151148, 2401365, 43681578, 896371205, 20504034645, 517705752096, 14310162565395, 430020328711305, 13963933247986995, 487456219774434795, 18209055555140970945, 724952705958984299025, 30650849492427960893946, 1371796147488157950190065

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..390

Formula

G.f.: 1 = Sum_{k>=0} a(k)*x^k*(1-x)^((k+1)*(k+2)/2).

From Benedict W. J. Irwin, Nov 29 2016: (Start)

Conjecture: a(n) is given by a series of nested sums,

a(1) = Sum_{i=1..1} 1,

a(2) = Sum_{i=1..1} Sum_{j=1..i+2} 1,

a(3) = Sum_{i=1..1} Sum_{j=1..i+2} Sum_{k=1..j+3} 1,

a(4) = Sum_{i=1..1} Sum_{j=1..i+2} Sum_{k=1..j+3} Sum_{l=1..k+4} 1. (End)

Example

G.f. = 1 + x + 3*x^2 + 15*x^3 + 106*x^4 + 975*x^5 + 11100*x^6 + 151148*x^7 + ...
1 = 1*(1-x)^1 + 1*x*(1-x)^3 + 3*x^2*(1-x)^6 + 15*x^3*(1-x)^10 + 106*x^4*(1-x)^15 + 975*x^5*(1-x)^21 + 11100*x^6*(1-x)^21 +...

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(a(j-1)*
      (-1)^(n-j)*binomial(j*(j+1)/2, n-j+1), j=1..n))
    end:
seq(a(n), n=0..22);  # Alois P. Heinz, Jul 10 2022

Mathematica

a[ n_, k_: 0, j_: 1] := If[ n < 1, Boole[n >= 0], a[ n, k, j] = Sum[ a[ n - 1, i, j + 1], {i, k + j}]]; (* Michael Somos, Nov 26 2016 *)

Prog

(PARI) {a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k*(1-x+x*O(x^n))^((k+1)*(k+2)/2)), n)}

Crossrefs

Cf. A000217, A107876, A107877, A107879, A209440 (variant).

Sequence in context: A353587 A128276 A295124 * A218688 A120016 A349874

Adjacent sequences: A107875 A107876 A107877 * A107879 A107880 A107881

Keyword

nonn

Author

Paul D. Hanna, Jun 04 2005

Status

approved