A302108 G.f.: Sum_{n>=0} ( (1+x)^n - (1-x)^n )^n / 2^n.

1, 1, 4, 27, 256, 3152, 47680, 854802, 17711872, 416372620, 10947581056, 318304921165, 10140097538560, 351219420860694, 13141237470041536, 528208859187285899, 22698715714385041920, 1038485165851106374784, 50395972495225521776384, 2585595617532863164095240, 139835798146777767415142912, 7950987913261988583226011167

Offset: 0

Author

Paul D. Hanna, Apr 01 2018

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 27*x^3 + 256*x^4 + 3152*x^5 + 47680*x^6 + 854802*x^7 + 17711872*x^8 + 416372620*x^9 + 10947581056*x^10 + ...
such that
A(x) = 1  +  ((1+x) - (1-x))/2  +  ((1+x)^2 - (1-x)^2)^2/2^2  +  ((1+x)^3 - (1-x)^3)^3/2^3  +  ((1+x)^4 - (1-x)^4)^4/2^4  +  ((1+x)^5 - (1-x)^5)^5/2^5  +  ((1+x)^6 - (1-x)^6)^6/2^6  +  ((1+x)^7 - (1-x)^7)^7/2^7  + ...
Equivalently,
A(x) = 1  +  x  +  (2*x)^2  +  (3*x + x^3)^3  +  (4*x + 4*x^3)^4  +  (5*x + 10*x^3 + x^5)^5  +  (6*x + 20*x^3 + 6*x^5)^6  +  (7*x + 35*x^3 + 21*x^5 + x^7)^7  +  (8*x + 56*x^3 + 56*x^5 + 8*x^7)^8  +  (9*x + 84*x^3 + 126*x^5 + 36*x^7 + x^9)^9  +  (10*x + 120*x^3 + 252*x^5 + 120*x^7 + 10*x^9)^10  + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A301309, A301308.

Programs

PARI

{a(n) = my(A=1); A = sum(m=0, n, ((1+x)^m - (1-x)^m +x*O(x^n))^m/2^m ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=0} [ Sum_{k=0..[n/2]} binomial(n,2*k+1) * x^(2*k+1) ]^n.

G.f.: Sum_{n>=0} (1+x)^(n^2) * Sum_{k=0..n} (-1)^k * C(n,k) * ((1-x)/(1+x))^(n*k) / 2^n.

a(n) ~ c * 2^(2*n) * n^n / (3^n * exp(n) * log(2)^(2*n)), where c = 0.873241746441310441203224293323899407211809744132... - Vaclav Kotesovec, Oct 06 2020