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 A207557 G.f.: Sum_{n>=0} 1/(1+x)^(n^2-n) * Product_{k=1..n} ((1+x)^(2*k-1) - 1). 3
 1, 1, 3, 12, 64, 420, 3276, 29581, 303389, 3483053, 44245695, 616103046, 9330961666, 152700926414, 2685132170466, 50488787588936, 1010864433071206, 21470488933116138, 482176661100286182, 11415700804801064258, 284169548252819022230, 7419733139418740010570 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Compare g.f. to: Sum_{n>=0} 1/(1+x)^(n^2) * Product_{k=1..n} ((1+x)^(2*k-1) - 1), which is the g.f. of A179525. LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..135 Hsien-Kuei Hwang, and Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], p. 36, 2019. FORMULA Given A(x) is the g.f. of this sequence, note that: 1 + x*A(x) = Sum_{n>=0} 1/(1+x)^(n^2+n) * Product_{k=1..n} ((1+x)^(2*k-1) - 1). a(n) ~ 2*sqrt(6) * 12^(n+1) * n^(n+1) / (exp(n+Pi^2/24) * Pi^(2*n+3)). - Vaclav Kotesovec, May 07 2014 EXAMPLE G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 64*x^4 + 420*x^5 + 3276*x^6 +... such that, by definition, A(x) = 1 + ((1+x)-1) + ((1+x)-1)*((1+x)^3-1)/(1+x)^2 + ((1+x)-1)*((1+x)^3-1)*((1+x)^5-1)/(1+x)^6 + ((1+x)-1)*((1+x)^3-1)*((1+x)^5-1)*((1+x)^7-1)/(1+x)^20 +... PROG (PARI) {a(n)=polcoeff(sum(m=0, n, prod(k=1, m, (1+x)^(2*k-1)-1)/(1+x+x*O(x^n))^(m^2-m) ), n)} for(n=0, 25, print1(a(n), ", ")) CROSSREFS Cf. A179525, A207556. Sequence in context: A307724 A029851 A201720 * A235129 A222033 A302195 Adjacent sequences:  A207554 A207555 A207556 * A207558 A207559 A207560 KEYWORD nonn AUTHOR Paul D. Hanna, Feb 18 2012 STATUS approved

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Last modified June 2 11:35 EDT 2020. Contains 334771 sequences. (Running on oeis4.)