0,2

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.

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. [From Paul D. Hanna (pauldhanna(AT)juno.com), Mar 10 2009]

G.f.: A(x) = 1 + 4*x + 64*x^2 + 3072*x^3 + 466944*x^4 + 283115520*x^5 +...

Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), 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 paranthesis) form this sequence. (End)

(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)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Mar 11 2009]

Cf. A156630, A060690, A155200.

Sequence in context: A002454 A013043 A167406 this_sequence A088065 A053718 A053773

Adjacent sequences: A156628 A156629 A156630 this_sequence A156632 A156633 A156634

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Feb 12 2009