A305142 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^3 * Integral A(x)^3 dx ) / A(x), for n > 0.

1, 1, 37, 4096, 878619, 306873869, 158938884952, 114993958088544, 111352808890827351, 139608635486408132803, 220605354590414591998297, 429593550416513276960527556, 1011544195064396609819653321932, 2833764097327349890282080026444076, 9314700709596523207841989131758528948, 35498787449426898120781594428097022541008

Offset: 0

Author

Paul D. Hanna, May 31 2018

Keywords

• nonn

Comments

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

Note: 0 = [x^n] exp( n * Integral F(x)^3 dx ) / F(x) holds for n > 0 when F(x) = 1 + x*F(x)^4 is a g.f. of A002293.

Examples

			O.g.f.: A(x) = 1 + x + 37*x^2 + 4096*x^3 + 878619*x^4 + 306873869*x^5 + 158938884952*x^6 + 114993958088544*x^7 + 111352808890827351*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in exp(n^3*Integral A(x)^3 dx)/A(x) begins:
  n=0: [1, -1, -36, -4023, -869168, -304829775, ...];
  n=1: [1, 0, -35, -12064/3, -870135, -915526348/3, ...];
  n=2: [1, 7, 0, -11609/3, -2626022/3, -307526817, ...];
  n=3: [1, 26, 342, 0, -847892, -312911550, ...];
  n=4: [1, 63, 2044, 131387/3, 0, -919948381/3, ...];
  n=5: [1, 124, 7839, 1011556/3, 31877746/3, 0, ...];
  n=6: [1, 215, 23400, 1722357, 96411130, 4177156347, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp(n^3*Integral A(x)^3 dx)/A(x), for n > 0.
RELATED SERIES.
A(x)^2 = 1 + 2*x + 75*x^2 + 8266*x^3 + 1766799*x^4 + 615808080*x^5 + 318573312664*x^6 + 230335700260146*x^7 + 222950653057400247*x^8 + ...
A(x)^3 = 1 + 3*x + 114*x^2 + 12511*x^3 + 2664651*x^4 + 926819028*x^5 + 478906878958*x^6 + 346026409343751*x^7 + 334794104506072215*x^8 + ...
exp( Integral A(x)^3 dx) = 1 + x + 7*x^2/2! + 703*x^3/3! + 303145*x^4/4! + 321307921*x^5/5! + 669264720031*x^6/6! + 2418416266536607*x^7/7! + 13971240948079459633*x^8/8! + ...
A'(x)/A(x) = 1 + 73*x + 12178*x^2 + 3495501*x^3 + 1529245631*x^4 + 951553836400*x^5 + 803743212623394*x^6 + 889843851811684197*x^7 + ...
		

Links

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

Crossrefs

Cf. A305137, A305138, A305139, A305140, A305141, A305143.

Programs

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

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 = 0.0710327332647009858916047504... - Vaclav Kotesovec, Oct 20 2020