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 A207434 L.g.f.: log( Sum_{n>=0} Product_{k=1..n} ((1+x)^k - 1) ) = Sum_{n>=1} a(n)*x^n/n. 2
 1, 3, 16, 103, 796, 7104, 71807, 810239, 10095145, 137686648, 2040943180, 32679948256, 562281127266, 10347659040127, 202849692259846, 4220573966037231, 92900793975348826, 2156973952747274733, 52686155932369860221, 1350605860832381895768, 36256679580764579284889 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS G. C. Greubel, Table of n, a(n) for n = 1..250 Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019. FORMULA L.g.f.: log( Sum_{n>=0} 1/(1+x)^(n^2) * Product_{k=1..n} ((1+x)^(2*k-1) - 1) ). EXAMPLE L.g.f.: L(x) = x + 3*x^2/2 + 16*x^3/3 + 103*x^4/4 + 796*x^5/5 + 7104*x^6/6 + ... where exponentiation yields the g.f. of A179525: exp(L(x)) = 1 + x + 2*x^2 + 7*x^3 + 33*x^4 + 197*x^5 + 1419*x^6 + 11966*x^7 + ... such that, by definition, exp(L(x)) = 1 + ((1+x)-1) + ((1+x)-1)*((1+x)^2-1) + ((1+x)-1)*((1+x)^2-1)*((1+x)^3-1) + ... MATHEMATICA Rest@With[{m = 25}, CoefficientList[Series[Log[Sum[Product[(1+x)^k -1, {k, j}], {j, 0, m+2}]], {x, 0, m}], x]*Range[0, m]] (* G. C. Greubel, Feb 05 2020 *) PROG (PARI) {a(n)=n*polcoeff(log(sum(m=0, n, prod(k=1, m, (1+x)^k-1, 1+x*O(x^n)))), n)} for(n=1, 31, print1(a(n), ", ")) CROSSREFS Cf. A179525 (exp). Sequence in context: A278429 A341320 A365752 * A074542 A366050 A368934 Adjacent sequences: A207431 A207432 A207433 * A207435 A207436 A207437 KEYWORD nonn AUTHOR Paul D. Hanna, Feb 19 2012 STATUS approved

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Last modified September 10 16:20 EDT 2024. Contains 375790 sequences. (Running on oeis4.)