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 A304401 O.g.f. A(x) satisfies: [x^n] exp( n^3 * x*A(x) ) * (2 - A(x)) = 0 for n > 0. 1
 1, 1, 32, 3618, 845824, 332389375, 196888240512, 164288952970296, 184344892426059776, 268830705445490506509, 496348897291481486672000, 1136486246811467501138927540, 3173564392477075053313688696832, 10660730426979559461604460186833401, 42595326050479099018430338636152049280, 200526023793925980859314834103239034380000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Note: the factorial series, F(x) = Sum_{n>=0} n! * x^n, satisfies: (1) [x^n] exp( n * x*F(x) ) * (2 - F(x)) = 0 for n > 0, (2) [x^n] exp( x*F(x) ) * (n + 1 - F(x)) = 0 for n > 0. It is remarkable that this sequence should consist entirely of integers. Note: a(n) is divisible by n^3 for n >= 1. LINKS EXAMPLE O.g.f.: A(x) = 1 + x + 32*x^2 + 3618*x^3 + 845824*x^4 + 332389375*x^5 + 196888240512*x^6 + 164288952970296*x^7 + 184344892426059776*x^8 + ... ILLUSTRATION OF SEFINITION. The table of coefficients of x^k/k! in exp( n^3 * x*A(x) ) * (2 - A(x)) begins: n=0: [1, -1, -64, -21708, -20299776, -39886725000, ...]; n=1: [1, 0, -63, -21710, -20300931, -39887501724, ...]; n=2: [1, 7, 0, -21052, -20280064, -39880261512, ...]; n=3: [1, 26, 665, 0, -19381155, -39710564418, ...]; n=4: [1, 63, 4032, 252340, 0, -37416032136, ...]; n=5: [1, 124, 15561, 1977542, 245086349, 0, ...]; n=6: [1, 215, 46592, 10194660, 2254128384, 485581472376, 0, ...]; ... in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n^3 * x*A(x) ) * (2 - A(x)) = 0 for n > 0. Terms along the secondary diagonal in the above table are divisible by the differences of cubes: [1, 7/7, 665/19, 252340/37, 245086349/61, 485581472376/91, ...] = [1, 1, 35, 6820, 4017809, 5336060136, ...]. RELATED SERIES. exp( x*A(x) ) = 1 + x + 3*x^2/2! + 199*x^3/3! + 87625*x^4/4! + 101938881*x^5/5! + 239933646571*x^6/6! + 993998976594583*x^7/7! + 6632090620377452049*x^8/8! + ... Note that the factorial series F(x) = 1 + x + 2!*x^2 + 3!*x^3 + 4!*x^4 + 5!*x^5 + ... + n!*x^n + ... satisfies [x^n] exp( n*x*F(x) ) * (2 - F(x)) = 0 for n > 0. PROG (PARI) {a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)^3 * x * Ser(A) ) * (2 - Ser(A)) )[m] ); A[n+1]} for(n=0, 20, print1(a(n), ", ")) CROSSREFS Cf. A304400. Sequence in context: A221659 A222053 A264126 * A328275 A291829 A248075 Adjacent sequences:  A304398 A304399 A304400 * A304402 A304403 A304404 KEYWORD nonn AUTHOR Paul D. Hanna, May 25 2018 STATUS approved

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Last modified September 18 14:52 EDT 2021. Contains 347527 sequences. (Running on oeis4.)