A090362 G.f. satisfies A^6 = BINOMIAL(A)^5 and also equals A090358^5.

1, 5, 40, 460, 7220, 148276, 3831760, 120333680, 4460572870, 190679906990, 9230084185456, 498734395394840, 29740372199558420, 1939241402832412180, 137222625361036807760, 10470376552560151801616, 856818090423771231257245

Offset: 0

Author

Paul D. Hanna, Nov 26 2003

Keywords

• nonn
• easy

Comments

See comments in A090358.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..320

Crossrefs

Cf. A090358, A019538, A094418.

Cf. A090352, A090355, A090357.

Programs

Mathematica

nmax = 16; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x]^6 - A[x/(1 - x)]^5/(1 - x)^5 + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 02 2019 *)

PARI

{a(n)=local(A); if(n<0,0,A=1+x+x*O(x^n); for(k=1,n,B=subst(A,x,x/(1-x))/(1-x)+x*O(x^n); A=A-A^6+B^5);polcoeff(A,n,x))}

Formula

G.f.: A(x)^6 = A(x/(1-x))^5/(1-x)^5.

a(n) ~ (n-1)! / (6 * (log(6/5))^(n+1)). - Vaclav Kotesovec, Nov 19 2014

From Peter Bala, May 26 2015: (Start)

O.g.f.: A(x) = exp( Sum_{n >= 1} b(n)*x^n/n ), where b(n) = Sum_{k = 1..n} k!*Stirling2(n,k)*5^k = A094418(n).

BINOMIAL(A(x)) = exp( Sum_{n >= 1} c(n)*x^n/n ) where c(n) = (-1)^n*Sum_{k = 1..n} k!*Stirling2(n,k)*(-6)^k. A(x) = B(x)^5 and BINOMIAL(A(x)) = B(x)^6 where B(x) = 1 + x + 6*x^2 + 66*x^3 + 1071*x^4 + ... is the o.g.f. for A090358. See also A019538. (End)

G.f.: Product_{k>=1} 1/(1 - k*x)^((1/6) * (5/6)^k). - Seiichi Manyama, May 26 2025