A305143 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^2 * Integral A(x)^4 dx ) / A(x), for n > 0.

1, 1, 13, 316, 10667, 447576, 22094626, 1242995118, 78081518451, 5400194995057, 406998451178896, 33165909456647704, 2904055577822356346, 271843880829531635092, 27087966494039897011884, 2862718283883222686998584, 319838550858171357010036323, 37670084296166551957561304631

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)^4 dx ) / F(x) holds for n > 0 when F(x) = 1 + x*F(x)^5 is a g.f. of A002294.

Examples

			O.g.f.: A(x) = 1 + x + 13*x^2 + 316*x^3 + 10667*x^4 + 447576*x^5 + 22094626*x^6 + 1242995118*x^7 + 78081518451*x^8 + 5400194995057*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in exp(n^2*Integral A(x)^4 dx)/A(x) begins:
  n=0: [1, -1, -12, -291, -9904, -419430, -20878908, ...];
  n=1: [1, 0, -21/2, -284, -78945/8, -420765, -336068285/16, ...];
  n=2: [1, 3, 0, -235, -9540, -421722, -21319776, ...];
  n=3: [1, 8, 75/2, 0, -61985/8, -408126, -345111453/16, ...];
  n=4: [1, 15, 132, 861, 0, -328662, -20947980, ...];
  n=5: [1, 24, 651/2, 3384, 226143/8, 0, -268775133/16, ...];
  n=6: [1, 35, 672, 9621, 117020, 1187142, 0, ...];
  n=7: [1, 48, 2475/2, 23180, 2891295/8, 5049117, 960763011/16, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp(n^2*Integral A(x)^4 dx)/A(x), for n > 0.
RELATED SERIES.
A(x)^2 = 1 + 2*x + 27*x^2 + 658*x^3 + 22135*x^4 + 924702*x^5 + 45461602*x^6 + 2548558008*x^7 + 159620140335*x^8 + ...
A(x)^3 = 1 + 3*x + 42*x^2 + 1027*x^3 + 34443*x^4 + 1432833*x^5 + 70159774*x^6 + 3919323204*x^7 + 244746587643*x^8 + ...
A(x)^4 = 1 + 4*x + 58*x^2 + 1424*x^3 + 47631*x^4 + 1973476*x^5 + 96250266*x^6 + 5358025992*x^7 + 333596305267*x^8 + ...
exp( Integral A(x)^4 dx) = 1 + x + 9*x^2/2! + 373*x^3/3! + 35809*x^4/4! + 5918961*x^5/5! + 1461206521*x^6/6! + 496585571749*x^7/7! + 220438988917953*x^8/8! + ...
A'(x)/A(x) = 1 + 25*x + 910*x^2 + 41117*x^3 + 2166366*x^4 + 128865058*x^5 + 8487954042*x^6 + 611163126189*x^7 + 47668752953875*x^8 + ...
		

Links

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

Crossrefs

Cf. A305137, A305138, A305139, A305140, A305142.

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^2*intformal(Ser(A)^4)) / Ser(A) )[m+1] );A[n+1]}
for(n=0,20,print1(a(n),", "))

Formula

(a(n)/n!)^(1/n) tends to 25/4. - Vaclav Kotesovec, Oct 19 2020