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 A305115 O.g.f. A(x) satisfies: [x^n] exp( n * x*A(x) ) * (n^2 + 1 - A(x)) = 0 for n >= 0. 7
 1, 1, 14, 450, 31144, 4041775, 890769366, 309205147860, 159530833094816, 116905524905145753, 117339344873068964150, 156605173710780053035502, 271173392660354548224099528, 596723380510396302812115056135, 1639486267597614501043345413095854, 5538914776834654404464150449671117000, 22706307619073102796968257487359193429120 (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. LINKS Paul D. Hanna, Table of n, a(n) for n = 0..300 FORMULA a(n) ~ c * n!^3, where c = 2.49393609789981559563078907122202821077556480458411... - Vaclav Kotesovec, Oct 06 2020 EXAMPLE O.g.f.: A(x) = 1 + x + 14*x^2 + 450*x^3 + 31144*x^4 + 4041775*x^5 + 890769366*x^6 + 309205147860*x^7 + 159530833094816*x^8 + 116905524905145753*x^9 + ... ILLUSTRATION OF DEFINITION. The table of coefficients of x^k/k! in exp( n * x*A(x) ) * (n^2 + 1 - A(x)) begins: n=0: [0, -1, -28, -2700, -747456, -485013000, -641353943520, ...]; n=1: [1, 0, -27, -2702, -747963, -485118684, -641396951615, ...]; n=2: [4, 7, 0, -2092, -678784, -462055752, -623679177536, ...]; n=3: [9, 26, 101, 0, -460275, -391250658, -569892209247, ...]; n=4: [16, 63, 348, 4828, 0, -246538056, -461135488928, ...]; n=5: [25, 124, 837, 14150, 810509, 0, -277891671695, ...]; n=6: [36, 215, 1688, 30348, 2099712, 378224376, 0, ...]; n=7: [49, 342, 3045, 56548, 4020741, 920163738, 393372598609, 0, ...]; ... in which the coefficients of x^n in row n form a diagonal of zeros. RELATED SERIES. exp(x*A(x)) = 1 + x + 3*x^2/2! + 91*x^3/3! + 11161*x^4/4! + 3793881*x^5/5! + 2933070331*x^6/6! + 4510118566003*x^7/7! + 12503335235913201*x^8/8! + ... PROG (PARI) {a(n) = my(A=[1], m); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp( (m-1)*x*(Ser(A)) ) * ((m-1)^2+1 - Ser(A)) )[m] ); A[n+1]} for(n=0, 20, print1(a(n), ", ")) CROSSREFS Cf. A305110, A305114, A305116, A317337. Sequence in context: A187358 A103916 A201546 * A282245 A319096 A297548 Adjacent sequences:  A305112 A305113 A305114 * A305116 A305117 A305118 KEYWORD nonn AUTHOR Paul D. Hanna, May 26 2018 STATUS approved

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Last modified November 29 01:15 EST 2021. Contains 349415 sequences. (Running on oeis4.)