A257605 Expansion of g.f.: exp( Sum_{n>=1} A251661(n)*x^n/n ) where A251661(n) = Sum_{k=0..n} C(n, k) * (2^k + 3^k)^(n-k).

1, 3, 12, 68, 606, 9438, 271154, 14272266, 1350900204, 226478780848, 67039275113982, 34862320055916606, 31905434621918041764, 51191148619374796495296, 144350180842362122992451022, 712487785268333349746955065478, 6171550949441004942637166827656834

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..90

Formula

G.f.: exp( Sum_{n>=1} A251661(n)*x^n/n ).

Example

G.f.: A(x) = 1 + 3*x + 12*x^2 + 68*x^3 + 606*x^4 + 9438*x^5 + 271154*x^6 + ...
The logarithm of g.f. A(x) begins:
log(A(x)) = 3*x + 15*x^2/2 + 123*x^3/3 + 1671*x^4/4 + 37863*x^5/5 + ... + A251661(n)*x^n/n + ...
which may be written as:
log(A(x)) = (2^1 + 1) * x +
(2^2 + 2*(2 + 3)^1 + 1) * x^2/2 +
(2^3 + 3*(2 + 3)^2 + 3*(2^2 + 3^2)^1 + 1) * x^3/3 +
(2^4 + 4*(2 + 3)^3 + 6*(2^2 + 3^2)^2 + 4*(2^3 + 3^3)^1 + 1) * x^4/4 +
(2^5 + 5*(2 + 3)^4 + 10*(2^2 + 3^2)^3 + 10*(2^3 + 3^3)^2 + 5*(2^4 + 3^4)^1 + 1) * x^5/5 + ...

Mathematica

A251661[n_]:= A251661[n]= Sum[Binomial[n, k]*(2^k +3^k)^(n-k), {k, 0, n}];
With[{m = 50}, CoefficientList[Series[Exp[Sum[A251661[j]*x^j/j, {j, 2*m}]], {x, 0, m}], x]] (* G. C. Greubel, Mar 24 2022 *)

Prog

(PARI) {A251661(n) = sum(k=0, n, binomial(n, k) * (2^k + 3^k)^(n-k) )}
{a(n) = local(A = exp( sum(m=1, n, A251661(m)*x^m/m) +x*O(x^n)) ); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(Sage)
m=40
@CachedFunction
def A251661(n): return sum( binomial(n, k)*(2^k + 3^k)^(n-k) for k in (0..n) )
def p(x): return exp( sum(A251661(j)*x^j/j for j in (1..2*m)) )
def A257605(n): return ( p(x) ).series(x, n+1).list()[n]
[A257605(n) for n in (0..m)] # G. C. Greubel, Mar 24 2022

Crossrefs

Cf. A251661.

Sequence in context: A004127 A058115 A101313 * A265886 A295762 A375811

Adjacent sequences: A257602 A257603 A257604 * A257606 A257607 A257608

Keyword

nonn

Author

Paul D. Hanna, May 02 2015

Status

approved