A209440 G.f.: 1 = Sum_{n>=0} a(n)*x^n * (1-x)^((n+1)^2).

1, 1, 4, 30, 340, 5235, 102756, 2464898, 70120020, 2313120225, 86962820000, 3674969314090, 172615622432040, 8928295918586815, 504561763088722500, 30946605756915149850, 2048137516834986743700, 145535818715694311408181, 11054204297079333714850260

Offset

0,3

Comments

Compare to a g.f. of the Catalan numbers: 1 = Sum_{n>=0} A000108(n)*x^n*(1-x)^(n+1).

Links

G. C. Greubel, Table of n, a(n) for n = 0..350

Formula

a(n) = Sum_{k=0..n-1} (-1)^(n+1-k) * a(k) * binomial((k+1)^2,n-k) for n>=1, with a(0)=1.

Example

G.f.: 1 = 1*(1-x) + 1*x*(1-x)^4 + 4*x^2*(1-x)^9 + 30*x^3*(1-x)^16 + 340*x^4*(1-x)^25 +...

Maple

a:= proc(n) option remember; `if`(n=0, 1, -add(a(j)
      *(-1)^(n-j)*binomial((j+1)^2, n-j), j=0..n-1))
    end:
seq(a(n), n=0..19);  # Alois P. Heinz, Jul 08 2022

Mathematica

a[0] := 1; a[n_] := a[n] = Sum[(-1)^(n + 1 - k)*a[k]*Binomial[(k + 1)^2, n - k], {k, 0, n - 1}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jan 02 2018 *)

Prog

(PARI) {a(n)=if(n==0, 1, -polcoeff(sum(m=0, n-1, a(m)*x^m*(1-x+x*O(x^n))^((m+1)^2)), n))}
(PARI) {a(n)=if(n==0, 1, sum(k=0, n-1, (-1)^(n+1-k)*a(k)*binomial((k+1)^2, n-k)))}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A000108, A107878, A177450, A180716.

Column k=2 of A355614.

Sequence in context: A292220 A099712 A370931 * A052316 A089918 A371041

Adjacent sequences: A209437 A209438 A209439 * A209441 A209442 A209443

Keyword

nonn

Author

Paul D. Hanna, Apr 07 2012

Status

approved