cp's OEIS Frontend

A251661 a(n) = Sum_{k=0..n} C(n,k) * (2^k + 3^k)^(n-k).

%I A251661 #16 May 23 2017 15:16:35
%S A251661 1,3,15,123,1671,37863,1447515,93965763,10456301871,2001375249423,
%T A251661 663553617119475,381265346343864843,381607689867265672551,
%U A251661 664239239404717367975223,2018751151993358704057734795,10680818706115450217386068210963,98710608829560784063971722066895711
%N A251661 a(n) = Sum_{k=0..n} C(n,k) * (2^k + 3^k)^(n-k).
%C A251661 a(n) == 0 (mod 3) for n>0. - _Paul D. Hanna_, May 02 2015
%H A251661 G. C. Greubel, <a href="/A251661/b251661.txt">Table of n, a(n) for n = 0..90</a>
%F A251661 E.g.f.: Sum_{n>=0} exp((2^n + 3^n)*x) * x^n/n!.
%F A251661 O.g.f.: Sum_{n>=0} x^n/(1 - (2^n + 3^n)*x)^(n+1).
%F A251661 a(n) ~ c * 3^(n^2/4) * 2^(n+1/2) / sqrt(Pi*n), where c = JacobiTheta3(0, 1/3) = EllipticTheta[3, 0, 1/3] = 1.69145968168171534134842... if n is even, and c = JacobiTheta2(0, 1/3) = EllipticTheta[2, 0, 1/3] = 1.69061120307521423305296... if n is odd. - _Vaclav Kotesovec_, Jan 25 2015
%e A251661 E.g.f.: A(x) = 1 + 3*x + 15*x^2/2! + 123*x^3/3! + 1671*x^4/4! + 37863*x^5/5! +...
%e A251661 where
%e A251661 A(x) = exp(2*x) + exp(5*x)*x + exp(13*x)*x^2/2! + exp(35*x)*x^3/3! + exp(97*x)*x^4/4! + exp(275*x)*x^5/5! + exp(793*x)*x^6/6! +...+ exp((2^n+3^n)*x)*x^n/n! +...
%e A251661 ILLUSTRATION OF INITIAL TERMS:
%e A251661 a(0) = 1*(2^0+3^0)^0 = 1;
%e A251661 a(1) = 1*(2^0+3^0)^1 + 1*(2^1+3^1)^0 = 3;
%e A251661 a(2) = 1*(2^0+3^0)^2 + 2*(2^1+3^1)^1 + 1*(2^2+3^2)^0 = 15;
%e A251661 a(3) = 1*(2^0+3^0)^3 + 3*(2^1+3^1)^2 + 3*(2^2+3^2)^1 + 1*(2^3+3^3)^0 = 123;
%e A251661 a(4) = 1*(2^0+3^0)^4 + 4*(2^1+3^1)^3 + 6*(2^2+3^2)^2 + 4*(2^3+3^3)^1 + 1*(2^4+3^4)^0 = 1671; ...
%t A251661 Table[Sum[Binomial[n,k] * (2^k + 3^k)^(n-k),{k,0,n}],{n,0,15}] (* _Vaclav Kotesovec_, Jan 25 2015 *)
%o A251661 (PARI) {a(n) = sum(k=0, n, binomial(n, k) * (2^k + 3^k)^(n-k) )}
%o A251661 for(n=0, 20, print1(a(n), ", "))
%o A251661 (PARI) /* E.g.f.: Sum_{n>=0} exp((2^n + 3^n)*x)*x^n/n!: */
%o A251661 {a(n)=n!*polcoeff(sum(k=0, n, exp((2^k + 3^k)*x +x*O(x^n))*x^k/k!), n)}
%o A251661 for(n=0, 20, print1(a(n), ", "))
%o A251661 (PARI) /* O.g.f. Sum_{n>=0} x^n/(1 - (2^n + 3^n)*x)^(n+1): */
%o A251661 {a(n)=polcoeff(sum(k=0, n, x^k/(1-(2^k + 3^k)*x +x*O(x^n))^(k+1)), n)}
%o A251661 for(n=0, 20, print1(a(n), ", "))
%Y A251661 Cf. A244754, A251671.
%K A251661 nonn
%O A251661 0,2
%A A251661 _Paul D. Hanna_, Jan 21 2015