|
%I #18 Jul 10 2022 08:37:10
%S 1,1,4,30,340,5235,102756,2464898,70120020,2313120225,86962820000,
%T 3674969314090,172615622432040,8928295918586815,504561763088722500,
%U 30946605756915149850,2048137516834986743700,145535818715694311408181,11054204297079333714850260
%N G.f.: 1 = Sum_{n>=0} a(n)*x^n * (1-x)^((n+1)^2).
%C Compare to a g.f. of the Catalan numbers: 1 = Sum_{n>=0} A000108(n)*x^n*(1-x)^(n+1).
%H G. C. Greubel, <a href="/A209440/b209440.txt">Table of n, a(n) for n = 0..350</a>
%F 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.
%e 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 +...
%p a:= proc(n) option remember; `if`(n=0, 1, -add(a(j)
%p *(-1)^(n-j)*binomial((j+1)^2, n-j), j=0..n-1))
%p end:
%p seq(a(n), n=0..19); # _Alois P. Heinz_, Jul 08 2022
%t 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 *)
%o (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))}
%o (PARI) {a(n)=if(n==0,1,sum(k=0,n-1,(-1)^(n+1-k)*a(k)*binomial((k+1)^2,n-k)))}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A000108, A107878, A177450, A180716.
%Y Column k=2 of A355614.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 07 2012
|
