cp's OEIS Frontend

A304400 O.g.f. A(x) satisfies: [x^n] exp( n^2 * x*A(x) ) * (2 - A(x)) = 0 for n > 0.

%I A304400 #22 Oct 06 2020 03:35:49
%S A304400 1,1,8,153,4736,205125,11606832,826208992,72258829312,7635270104361,
%T A304400 961709587281200,142709474491679777,24684776053129473408,
%U A304400 4928830965337886481836,1126011129156595573835552,291967631033958376653342600,85304359600279978669204291584,27900684466477404020849587348577
%N A304400 O.g.f. A(x) satisfies: [x^n] exp( n^2 * x*A(x) ) * (2 - A(x)) = 0 for n > 0.
%C A304400 Note: the factorial series, F(x) = Sum_{n>=0} n! * x^n, satisfies:
%C A304400 (1) [x^n] exp( n * x*F(x) ) * (2 - F(x)) = 0 for n > 0,
%C A304400 (2) [x^n] exp( x*F(x) ) * (n + 1 - F(x)) = 0 for n > 0.
%C A304400 It is remarkable that this sequence should consist entirely of integers.
%C A304400 A304857(n) = a(n) / n^2 for n >= 1.
%H A304400 Paul D. Hanna, <a href="/A304400/b304400.txt">Table of n, a(n) for n = 0..300</a>
%F A304400 a(n) ~ c * n!^2 * n^2, where c = 0.777184293541721432034108670879422244... - _Vaclav Kotesovec_, Oct 06 2020
%e A304400 O.g.f.: A(x) = 1 + x + 8*x^2 + 153*x^3 + 4736*x^4 + 205125*x^5 + 11606832*x^6 + 826208992*x^7 + 72258829312*x^8 + 7635270104361*x^9 + ...
%e A304400 ILLUSTRATION OF DEFINITION.
%e A304400 The table of coefficients of x^k/k! in exp( n^2 * x*A(x) ) * (2 - A(x)) begins:
%e A304400 n=0: [1, -1, -16, -918, -113664, -24615000, -8356919040, ...];
%e A304400 n=1: [1, 0, -15, -920, -113955, -24650904, -8363901035, ...];
%e A304400 n=2: [1, 3, 0, -830, -113088, -24636696, -8363675648, ...];
%e A304400 n=3: [1, 8, 65, 0, -97923, -23962896, -8273887803, ...];
%e A304400 n=4: [1, 15, 240, 3850, 0, -19894104, -7851595520, ...];
%e A304400 n=5: [1, 24, 609, 16432, 444861, 0, -6241325915, ...];
%e A304400 n=6: [1, 35, 1280, 49410, 2034240, 84952296, 0, ...];
%e A304400 n=7: [1, 48, 2385, 123640, 6775197, 399396504, 24384667957, 0, ...]; ...
%e A304400 in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n^2 * x*A(x) ) * (2 - A(x)) = 0 for n > 0.
%e A304400 Terms along the secondary diagonal in the above table are divisible by the odd numbers: [1, 3/3, 65/5, 3850/7, 444861/9, 84952296/11, 24384667957/13, ...] = [1, 1, 13, 550, 49429, 7722936, 1875743689, ...].
%e A304400 RELATED SERIES.
%e A304400 exp( x*A(x) ) = 1 + x + 3*x^2/2! + 55*x^3/3! + 3889*x^4/4! + 588201*x^5/5! + 151295251*x^6/6! + 59575340623*x^7/7! + 33795420271425*x^8/8! + ...
%e A304400 Note that the factorial series
%e A304400 F(x) = 1 + x + 2!*x^2 + 3!*x^3 + 4!*x^4 + 5!*x^5 + ... + n!*x^n + ...
%e A304400 satisfies [x^n] exp( n*x*F(x) ) * (2 - F(x)) = 0 for n > 0.
%o A304400 (PARI) {a(n) = my(A=[1],m); for(i=1,n, A=concat(A,0); m=#A; A[m] = Vec( exp( (m-1)^2 * x * Ser(A) ) * (2 - Ser(A)) )[m] );A[n+1]}
%o A304400 for(n=0,20, print1(a(n),", "))
%Y A304400 Cf. A304401, A304402, A304857, A305116.
%K A304400 nonn
%O A304400 0,3
%A A304400 _Paul D. Hanna_, May 25 2018