A305138 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^3 * Integral A(x) dx ) / A(x), for n > 0.

1, 1, 29, 2829, 574365, 198036917, 103683001241, 76732157303677, 76332092063105089, 98308205372662506789, 159202381329276508297845, 316697346580111339298407645, 759241108161224318180212593541, 2158996472269125350886176802590109, 7185071088438157708730937024732618961, 27665076438562174334456341670646413685101

Offset

0,3

Comments

It is remarkable that this sequence should consist entirely of integers.

Note: 0 = [x^n] exp( n * Integral C(x) dx ) / C(x) holds for n > 0 when C(x) = 1 + x*C(x)^2 is a g.f. of the Catalan numbers (A000108).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..250

Formula

a(n) ~ c * d^n * n!^2 / n^(3/2), where d = 27 / (-LambertW(-3*exp(-3)) * (3 + LambertW(-3*exp(-3)))^2) and c = 0.0614343742940117549375894089... - Vaclav Kotesovec, Oct 20 2020

Example

O.g.f.: A(x) = 1 + x + 29*x^2 + 2829*x^3 + 574365*x^4 + 198036917*x^5 + 103683001241*x^6 + 76732157303677*x^7 + 76332092063105089*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in exp(n^3*Integral A(x) dx)/A(x) begins:
n=0: [1, -1, -28, -2772, -567952, -196735000, -103247834508, ...];
n=1: [1, 0, -28, -8372/3, -570045, -2957867108/15, -930709619938/9, ...];
n=2: [1, 7, 0, -8512/3, -1754242/3, -1002113658/5, -4705750542584/45, ...];
n=3: [1, 26, 323, 0, -603988, -1046990252/5, -539032922442/5, ...];
n=4: [1, 63, 1988, 119140/3, 0, -3286250072/15, -5146650533948/45, ...];
n=5: [1, 124, 7722, 961184/3, 28298144/3, 0, -1087966178572/9, ...];
n=6: [1, 215, 23192, 1672776, 90300350, 18408322114/5, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp(n^3*Integral A(x) dx)/A(x), for n > 0.
RELATED SERIES.
exp( Integral A(x) dx) = 1 + x + 2*x^2/2! + 62*x^3/3! + 17216*x^4/4! + 13870816*x^5/5! + 23847686176*x^6/6! + 74818727953312*x^7/7! + 387328675940041472*x^8/8! + ...
A'(x)/A(x) = 1 + 57*x + 8401*x^2 + 2284577*x^3 + 986920761*x^4 + 620790291801*x^5 + 536349430717661*x^6 + 610013974179245217*x^7 + ...

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^3*intformal(Ser(A))) / Ser(A) )[m+1] ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A305137, A305139, A305140, A305141, A305142, A305143.

Sequence in context: A316333 A281440 A171720 * A091751 A144233 A125074

Adjacent sequences: A305135 A305136 A305137 * A305139 A305140 A305141

Keyword

nonn

Author

Paul D. Hanna, May 31 2018

Status

approved