A257605 - OEIS

%I #12 Mar 25 2022 15:51:40

%S 1,3,12,68,606,9438,271154,14272266,1350900204,226478780848,

%T 67039275113982,34862320055916606,31905434621918041764,

%U 51191148619374796495296,144350180842362122992451022,712487785268333349746955065478,6171550949441004942637166827656834

%N 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).

%H G. C. Greubel, <a href="/A257605/b257605.txt">Table of n, a(n) for n = 0..90</a>

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

%e G.f.: A(x) = 1 + 3*x + 12*x^2 + 68*x^3 + 606*x^4 + 9438*x^5 + 271154*x^6 + ...

%e The logarithm of g.f. A(x) begins:

%e 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 + ...

%e which may be written as:

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

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

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

%e (2^4 + 4*(2 + 3)^3 + 6*(2^2 + 3^2)^2 + 4*(2^3 + 3^3)^1 + 1) * x^4/4 +

%e (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 + ...

%t A251661[n_]:= A251661[n]= Sum[Binomial[n,k]*(2^k +3^k)^(n-k), {k,0,n}];

%t 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 *)

%o (PARI) {A251661(n) = sum(k=0, n, binomial(n, k) * (2^k + 3^k)^(n-k) )}

%o {a(n) = local(A = exp( sum(m=1,n,A251661(m)*x^m/m) +x*O(x^n)) );polcoeff(A,n)}

%o for(n=0, 20, print1(a(n), ", "))

%o (Sage)

%o m=40

%o @CachedFunction

%o def A251661(n): return sum( binomial(n, k)*(2^k + 3^k)^(n-k) for k in (0..n) )

%o def p(x): return exp( sum(A251661(j)*x^j/j for j in (1..2*m)) )

%o def A257605(n): return ( p(x) ).series(x, n+1).list()[n]

%o [A257605(n) for n in (0..m)] # _G. C. Greubel_, Mar 24 2022

%Y Cf. A251661.

%K nonn

%O 0,2

%A _Paul D. Hanna_, May 02 2015