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A305137 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^2 * Integral A(x) dx ) / A(x), for n > 0. 13
1, 1, 7, 97, 1987, 53281, 1754245, 68228209, 3055471369, 154724090845, 8740256396563, 545005932104377, 37196779826275411, 2759229671824346893, 221140447146112986889, 19051164839221523341825, 1756309610450933072328241, 172576908229287147075691417, 18010455349270266144268806799, 1989930676607696867000687913025 (list; graph; refs; listen; history; text; internal format)
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..300

FORMULA

a(n) ~ c * d^n * (n-1)!, where d = 4 / (-LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) and c = 0.2658290856... - Vaclav Kotesovec, Oct 19 2020

EXAMPLE

O.g.f.: A(x) = 1 + x + 7*x^2 + 97*x^3 + 1987*x^4 + 53281*x^5 + 1754245*x^6 + 68228209*x^7 + 3055471369*x^8 + 154724090845*x^9 + 8740256396563*x^10 + ...

ILLUSTRATION OF DEFINITION.

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

n=0: [1, -1, -6, -84, -1764, -48360, -1620186, -63857556, ...];

n=1: [1, 0, -6, -88, -1830, -249144/5, -1661842, -2284994352/35, ...];

n=2: [1, 3, 0, -90, -2025, -272391/5, -8967968/5, -488439972/7, ...];

n=3: [1, 8, 30, 0, -2100, -311856/5, -10175418/5, -545952984/7, ...];

n=4: [1, 15, 114, 532, 0, -327564/5, -2389194, -3186733572/35, ...];

n=5: [1, 24, 294, 2416, 13536, 0, -2539746, -763395912/7, ...];

n=6: [1, 35, 624, 7542, 68415, 2234001/5, 0, -4102900932/35, ...];

n=7: [1, 48, 1170, 19320, 242550, 12134424/5, 90334582/5, 0, ...]; ...

in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp(n^2*Integral A(x) dx)/A(x), for n > 0.

RELATED SERIES.

exp(Integral A(x) dx) = 1 + x + 2*x^2/2! + 18*x^3/3! + 648*x^4/4! + 50904*x^5/5! + 6700464*x^6/6! + 1310200848*x^7/7! + 354395417472*x^8/8! + 126396068810112*x^9/9! + ...

A'(x)/A(x) = 1 + 13*x + 271*x^2 + 7489*x^3 + 253771*x^4 + 10113877*x^5 + 461995381*x^6 + 23766009457*x^7 + 1359214691545*x^8 + 85572483605593*x^9 + ...

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*intformal(Ser(A))) / Ser(A) )[m+1] ); A[n+1]}

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

CROSSREFS

Cf. A305138, A305139, A305140, A305141, A305142, A305143.

Sequence in context: A022007 A174516 A058805 * A132061 A013521 A003710

Adjacent sequences: A305134 A305135 A305136 * A305138 A305139 A305140

KEYWORD

nonn

AUTHOR

Paul D. Hanna, May 31 2018

STATUS

approved

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Last modified December 6 12:11 EST 2022. Contains 358634 sequences. (Running on oeis4.)