A156631 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

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.

LINKS

Table of n, a(n) for n=0..10.

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

EXAMPLE

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)

PROG

(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

CROSSREFS

Cf. A156630, A060690, A155200.

Sequence in context: A013043 A296741 A167406 * A088065 A053718 A053773

Adjacent sequences: A156628 A156629 A156630 * A156632 A156633 A156634

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 12 2009

STATUS

approved