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

%I A305143 #11 Mar 16 2025 23:54:22
%S A305143 1,1,13,316,10667,447576,22094626,1242995118,78081518451,
%T A305143 5400194995057,406998451178896,33165909456647704,2904055577822356346,
%U A305143 271843880829531635092,27087966494039897011884,2862718283883222686998584,319838550858171357010036323,37670084296166551957561304631
%N A305143 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^2 * Integral A(x)^4 dx ) / A(x), for n > 0.
%C A305143 It is remarkable that this sequence should consist entirely of integers.
%C A305143 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.
%H A305143 Paul D. Hanna, <a href="/A305143/b305143.txt">Table of n, a(n) for n = 0..250</a>
%F A305143 (a(n)/n!)^(1/n) tends to 25/4. - _Vaclav Kotesovec_, Oct 19 2020
%e A305143 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 + ...
%e A305143 ILLUSTRATION OF DEFINITION.
%e A305143 The table of coefficients of x^k in exp(n^2*Integral A(x)^4 dx)/A(x) begins:
%e A305143   n=0: [1, -1, -12, -291, -9904, -419430, -20878908, ...];
%e A305143   n=1: [1, 0, -21/2, -284, -78945/8, -420765, -336068285/16, ...];
%e A305143   n=2: [1, 3, 0, -235, -9540, -421722, -21319776, ...];
%e A305143   n=3: [1, 8, 75/2, 0, -61985/8, -408126, -345111453/16, ...];
%e A305143   n=4: [1, 15, 132, 861, 0, -328662, -20947980, ...];
%e A305143   n=5: [1, 24, 651/2, 3384, 226143/8, 0, -268775133/16, ...];
%e A305143   n=6: [1, 35, 672, 9621, 117020, 1187142, 0, ...];
%e A305143   n=7: [1, 48, 2475/2, 23180, 2891295/8, 5049117, 960763011/16, 0, ...]; ...
%e A305143 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.
%e A305143 RELATED SERIES.
%e A305143 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 + ...
%e A305143 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 + ...
%e A305143 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 + ...
%e A305143 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! + ...
%e A305143 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 + ...
%o A305143 (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]}
%o A305143 for(n=0,20,print1(a(n),", "))
%Y A305143 Cf. A305137, A305138, A305139, A305140, A305142.
%K A305143 nonn
%O A305143 0,3
%A A305143 _Paul D. Hanna_, May 31 2018