A183146 - OEIS

%I #7 Mar 12 2022 13:26:01

%S 1,1,4,16,80,407,2221,12380,71196,417016,2484839,15001779,91603298,

%T 564661194,3509278042,21964437947,138330334357,875977578584,

%U 5574225259696,35626247068500,228592067446715,1471959684881231

%N G.f.: Sum_{n>=0} [Sum_{k=0..n} C(n,k)^2*x^k]^3 * x^n.

%C Compare g.f. to a g.f. of the Whitney numbers in A051286:

%C Sum_{n>=0} [Sum_{k=0..n} C(n,k)^2*x^k] * x^n.

%e G.f.: A(x) = 1 + x + 4*x^2 + 16*x^3 + 80*x^4 + 407*x^5 + 2221*x^6 +...

%e which equals the sum of the series:

%e A(x) = 1 + (1 + x)^3*x + (1 + 4*x + x^2)^3*x^2

%e + (1 + 9*x + 9*x^2 + x^3)^3*x^3

%e + (1 + 16*x + 36*x^2 + 16*x^3 + x^4)^3*x^4

%e + (1 + 25*x + 100*x^2 + 100*x^3 + 25*x^4 + x^5)^3*x^5

%e + (1 + 36*x + 225*x^2 + 400*x^3 + 225*x^4 + 36*x^5 + x^6)^3*x^6 +...

%o (PARI) {a(n)=polcoeff(sum(m=0,n,sum(k=0,m,binomial(m,k)^2*x^k)^3*x^m)+x*O(x^n),n)}

%Y Cf. A180717, A051286.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 26 2010