A244754 a(n) = Sum_{k=0..n} C(n,k) * (1 + 2^k)^(n-k).
1, 3, 11, 51, 311, 2583, 30011, 495771, 11740271, 399511023, 19531952051, 1369534859091, 137461591250951, 19708614005005383, 4029559971566918891, 1172950335844577723211, 485515762655939377001951, 285459356061242116657495263, 238215406681004045293498284131
Offset: 0
Keywords
Examples
E.g.f.: A(x) = 1 + 3*x + 11*x^2/2! + 51*x^3/3! + 311*x^4/4! + 2583*x^5/5! +... where 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! +...). ILLUSTRATION OF INITIAL TERMS: a(1) = (1+2^0)^1 + (1+2^1)^0 = 3; a(2) = (1+2^0)^2 + 2*(1+2^1)^1 + (1+2^2)^0 = 11; a(3) = (1+2^0)^3 + 3*(1+2^1)^2 + 3*(1+2^2)^1 + (1+2^3)^0 = 51; 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; ...
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..100
Programs
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Mathematica
Table[Sum[Binomial[n,k] * (1 + 2^k)^(n-k),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jan 25 2015 *) -
PARI
{a(n) = sum(k=0,n,binomial(n,k) * (1 + 2^k)^(n-k) )} for(n=0,25,print1(a(n),", ")) -
PARI
/* E.g.f.: Sum_{n>=0} exp((1+2^n)*x)*x^n/n!: */ {a(n)=n!*polcoeff(sum(k=0, n, exp((1+2^k)*x +x*O(x^n))*x^k/k!), n)} for(n=0,25,print1(a(n),", ")) -
PARI
/* O.g.f. Sum_{n>=0} x^n/(1 - (1+2^n)*x)^(n+1): */ {a(n)=polcoeff(sum(k=0, n, x^k/(1-(1+2^k)*x +x*O(x^n))^(k+1)), n)} for(n=0,25,print1(a(n),", "))
Formula
E.g.f.: Sum_{n>=0} exp((1+2^n)*x) * x^n/n!.
O.g.f.: Sum_{n>=0} x^n/(1 - (1+2^n)*x)^(n+1).
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