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A317343 E.g.f. A(x) satisfies: [x^n] exp(n^2*x) / A(x)^n  = 0  for n >= 1. 4
1, 1, 3, 43, 1945, 178041, 26792971, 5940440563, 1812303908913, 725818277844145, 368664266359757971, 231291789356545214331, 175516846850044015048393, 158383499766971409675254953, 167568661592262324239839114395, 205391322843896611716680298647491, 288693442696565330249751284373878881, 461220054642000786943064376254671287393 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

It is remarkable that the logarithm of the e.g.f. A(x) should be an integer series.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..300

FORMULA

a(n) ~ sqrt(1-c) * 2^(2*n - 3/2) * n^(2*n-1) / (exp(2*n) * c^(n - 1/2) * (2-c)^(n-1)), where c = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599... - Vaclav Kotesovec, Aug 06 2018

EXAMPLE

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 43*x^3/3! + 1945*x^4/4! + 178041*x^5/5! + 26792971*x^6/6! + 5940440563*x^7/7! + 1812303908913*x^8/8! + ...

ILLUSTRATION OF DEFINITION.

The table of coefficients of x^k/k! in exp(n^2*x) / A(x)^n begins:

n=1: [1, 0, -2, -36, -1764, -167280, -25620600, -5737974480, ...];

n=2: [1, 2, 0, -88, -4160, -371328, -55329536, -12201990400, ...];

n=3: [1, 6, 30, 0, -7812, -698184, -97733304, -20791334880, ...];

n=4: [1, 12, 136, 1296, 0, -1171968, -168658176, -33909447168, ...];

n=5: [1, 20, 390, 7220, 113020, 0, -265712600, -55963975600, ...];

n=6: [1, 30, 888, 25704, 709056, 16600320, 0, -84622337280, ...];

n=7: [1, 42, 1750, 72072, 2909340, 112245672, 3684715944, 0, ...];

n=8: [1, 56, 3120, 172640, 9455488, 508540416, 26199517696, 1150524892160, 0, ...]; ...

in which the coefficient of x^n in row n forms a diagonal of zeros.

LOGARITHMIC PROPERTY.

The logarithm of the e.g.f. is the integer series:

log(A(x)) = x + x^2 + 6*x^3 + 74*x^4 + 1400*x^5 + 35676*x^6 + 1140328*x^7 + 43740848*x^8 + 1954336608*x^9 + 99561874080*x^10 + ... + A317344(n)*x^n + ...

PROG

(PARI) {a(n) = my(A=[1], m); for(i=1, n+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m^2*x +x*O(x^#A)) / Ser(A)^m )[m+1]/m ); n!*A[n+1]}

for(n=0, 20, print1(a(n), ", "))

CROSSREFS

Cf. A317344, A317345.

Sequence in context: A300988 A136648 A114337 * A307248 A009720 A300873

Adjacent sequences:  A317340 A317341 A317342 * A317344 A317345 A317346

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 26 2018

STATUS

approved

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Last modified April 10 15:47 EDT 2021. Contains 342845 sequences. (Running on oeis4.)