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A304400 O.g.f. A(x) satisfies: [x^n] exp( n^2 * x*A(x) ) * (2 - A(x)) = 0 for n > 0. 5
1, 1, 8, 153, 4736, 205125, 11606832, 826208992, 72258829312, 7635270104361, 961709587281200, 142709474491679777, 24684776053129473408, 4928830965337886481836, 1126011129156595573835552, 291967631033958376653342600, 85304359600279978669204291584, 27900684466477404020849587348577 (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.
A304857(n) = a(n) / n^2 for n >= 1.
LINKS
FORMULA
a(n) ~ c * n!^2 * n^2, where c = 0.777184293541721432034108670879422244... - Vaclav Kotesovec, Oct 06 2020
EXAMPLE
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 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^2 * x*A(x) ) * (2 - A(x)) begins:
n=0: [1, -1, -16, -918, -113664, -24615000, -8356919040, ...];
n=1: [1, 0, -15, -920, -113955, -24650904, -8363901035, ...];
n=2: [1, 3, 0, -830, -113088, -24636696, -8363675648, ...];
n=3: [1, 8, 65, 0, -97923, -23962896, -8273887803, ...];
n=4: [1, 15, 240, 3850, 0, -19894104, -7851595520, ...];
n=5: [1, 24, 609, 16432, 444861, 0, -6241325915, ...];
n=6: [1, 35, 1280, 49410, 2034240, 84952296, 0, ...];
n=7: [1, 48, 2385, 123640, 6775197, 399396504, 24384667957, 0, ...]; ...
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.
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, ...].
RELATED SERIES.
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! + ...
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)^2 * x * Ser(A) ) * (2 - Ser(A)) )[m] ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
Sequence in context: A171202 A247538 A360575 * A174845 A352253 A302259
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 25 2018
STATUS
approved

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Last modified July 21 05:28 EDT 2024. Contains 374463 sequences. (Running on oeis4.)