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 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. 3
 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 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

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Last modified April 11 08:59 EDT 2021. Contains 342886 sequences. (Running on oeis4.)