cp's OEIS Frontend

A244754 a(n) = Sum_{k=0..n} C(n,k) * (1 + 2^k)^(n-k).

%I A244754 #19 Jan 25 2015 06:59:05
%S A244754 1,3,11,51,311,2583,30011,495771,11740271,399511023,19531952051,
%T A244754 1369534859091,137461591250951,19708614005005383,4029559971566918891,
%U A244754 1172950335844577723211,485515762655939377001951,285459356061242116657495263,238215406681004045293498284131
%N A244754 a(n) = Sum_{k=0..n} C(n,k) * (1 + 2^k)^(n-k).
%H A244754 Paul D. Hanna, <a href="/A244754/b244754.txt">Table of n, a(n) for n = 0..100</a>
%F A244754 E.g.f.: Sum_{n>=0} exp((1+2^n)*x) * x^n/n!.
%F A244754 O.g.f.: Sum_{n>=0} x^n/(1 - (1+2^n)*x)^(n+1).
%F A244754 a(n) ~ c * 2^(n^2/4 + n + 1/2) / sqrt(Pi*n), where c = JacobiTheta3(0, 1/2) = EllipticTheta[3, 0, 1/2] = 2.1289368272118771586... if n is even, and c = JacobiTheta2(0, 1/2) = EllipticTheta[2, 0, 1/2] = 2.12893125051302755859... if n is odd. - _Vaclav Kotesovec_, Jan 25 2015
%e A244754 E.g.f.: A(x) = 1 + 3*x + 11*x^2/2! + 51*x^3/3! + 311*x^4/4! + 2583*x^5/5! +...
%e A244754 where
%e A244754 A(x) = exp(x)*(1 + 2*x + 6*x^2/2! + 26*x^3/3! + 162*x^4/4! + 1442*x^5/5! +...+ A047863(n)*x^n/n! +...).
%e A244754 ILLUSTRATION OF INITIAL TERMS:
%e A244754 a(1) = (1+2^0)^1 + (1+2^1)^0 = 3;
%e A244754 a(2) = (1+2^0)^2 + 2*(1+2^1)^1 + (1+2^2)^0 = 11;
%e A244754 a(3) = (1+2^0)^3 + 3*(1+2^1)^2 + 3*(1+2^2)^1 + (1+2^3)^0 = 51;
%e A244754 a(4) = (1+2^0)^4 + 4*(1+2^1)^3 + 6*(1+2^2)^2 + 4*(1+2^3)^1 + (1+2^4)^0 = 311; ...
%t A244754 Table[Sum[Binomial[n,k] * (1 + 2^k)^(n-k),{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jan 25 2015 *)
%o A244754 (PARI) {a(n) = sum(k=0,n,binomial(n,k) * (1 + 2^k)^(n-k) )}
%o A244754 for(n=0,25,print1(a(n),", "))
%o A244754 (PARI) /* E.g.f.: Sum_{n>=0} exp((1+2^n)*x)*x^n/n!: */
%o A244754 {a(n)=n!*polcoeff(sum(k=0, n, exp((1+2^k)*x +x*O(x^n))*x^k/k!), n)}
%o A244754 for(n=0,25,print1(a(n),", "))
%o A244754 (PARI) /* O.g.f. Sum_{n>=0} x^n/(1 - (1+2^n)*x)^(n+1): */
%o A244754 {a(n)=polcoeff(sum(k=0, n, x^k/(1-(1+2^k)*x +x*O(x^n))^(k+1)), n)}
%o A244754 for(n=0,25,print1(a(n),", "))
%Y A244754 Cf. A047863, A244755, A243918.
%K A244754 nonn
%O A244754 0,2
%A A244754 _Paul D. Hanna_, Jul 05 2014