A156631 G.f.: A(x) = Sum_{n>=0} ( Sum_{k>=1} (2^n*2^k*x)^k/k )^n / n!, a power series in x with integer coefficients.

1, 4, 64, 3072, 466944, 283115520, 814634500096, 10734635101192192, 601470215201514061824, 138785509787119430915850240, 130376354694095237162362352959488

Offset: 0

Author

Paul D. Hanna, Feb 12 2009

Keywords

• nonn

Comments

Compare to these dual g.f.s:

Sum_{n>=0} ( Sum_{k>=1} (2^n*x)^k/k )^n/n! (A060690);

Sum_{n>=0} ( Sum_{k>=1} (2^k*x)^k/k )^n/n! (A155200);

which, when expanded as power series in x, have only integer coefficients.

Examples

			G.f.: A(x) = 1 + 4*x + 64*x^2 + 3072*x^3 + 466944*x^4 + 283115520*x^5 + ...
From _Paul D. Hanna_, Mar 10 2009: (Start)
Let B(x) be the g.f. of A155200:
B(x) = 1 + 2*x + 10*x^2 + 188*x^3 + 16774*x^4 + 6745436*x^5 + ...
then a(n) is the coefficient of x^n in B(x)^(2^n):
B(x)^(2^0): [(1),2,10,188,16774,6745436,11466849412,...];
B(x)^(2^1): [1,(4),24,416,34400,13561728,22961051392,...];
B(x)^(2^2): [1,8,(64),1024,72704,27418624,46032420864,...];
B(x)^(2^3): [1,16,192,(3072),165888,56131584,92513894400,...];
B(x)^(2^4): [1,32,640,12288,(466944),118751232,186897137664,...];
B(x)^(2^5): [1,64,2304,65536,2129920,(283115520),382143037440,...];
B(x)^(2^6): [1,128,8704,425984,17956864,1140850688,(814634500096),...];
the terms along the diagonal (in parentheses) form this sequence. (End)
		

Crossrefs

Cf. A156630, A060690, A155200.

Programs

PARI

{a(n)=polcoeff(sum(j=0,n,sum(k=1, n, (2^(j+k)*x)^k/k+x*O(x^n))^j/j!),n)}

PARI

/* a(n) = [x^n] B(x)^(2^n) where B(x) is g.f. of A155200: */ {a(n)=polcoeff(exp( 2^n*sum(k=1,n, 2^(k^2)*x^k/k)+x*O(x^n)), n)} \\ Paul D. Hanna, Mar 11 2009

Formula

a(n) = [x^n] B(x)^(2^n) where B(x) = exp(Sum_{n>=1} 2^(n^2)*x^n/n) is the g.f. of A155200. - Paul D. Hanna, Mar 10 2009