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%I #7 Sep 03 2020 02:09:58
%S 1,1,14,947,157190,47437866,22437363324,15246207565643,
%T 14053536511674526,16868801353366004990,25566893078760354005252,
%U 47761059837097334197007118,107843046053558916525978556156,289613430019820775682179202404084
%N O.g.f. A(x) satisfies: [x^n] exp( n^2*(n+1) * x ) / A(x)^(n+1) = 0 for n > 0.
%C It is remarkable that this sequence consists entirely of integers.
%F Given o.g.f. A(x), define B(x) = A(x/B(x)), then B(x) is the o.g.f. of A337575 and satisfies [x^n] exp( n*(n-1)^2 * x/B(x) ) = 0 for n>0.
%e O.g.f.: A(x) = 1 + x + 14*x^2 + 947*x^3 + 157190*x^4 + 47437866*x^5 + 22437363324*x^6 + 15246207565643*x^7 + 14053536511674526*x^8 + ...
%e ILLUSTRATION OF DEFINITION.
%e The table of coefficients of x^k/k! in exp( n^2*(n+1)*x ) / A(x)^(n+1) begins:
%e n=0: [1, -1, -26, -5520, -3723384, -5652041280, -16083171669600, ...];
%e n=1: [1, 0, -54, -11200, -7486872, -11338403328, -32230618603040, ...];
%e n=2: [1, 9, 0, -18258, -11861352, -17522277048, -49272492906432, ...];
%e n=3: [1, 32, 916, 0, -17438424, -25288921344, -69043257103968, ...];
%e n=4: [1, 75, 5490, 363500, 0, -35101453320, -94993441197200, ...];
%e n=5: [1, 144, 20574, 2882400, 368064576, 0, -127110906431280, ...];
%e n=6: [1, 245, 59836, 14528010, 3470388768, 759773089152, 0, ...];
%e n=7: [1, 384, 147240, 56329472, 21453513648, 8058471570432, 2785824326725888, 0, ...]; ...
%e in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n^2*(n+1)*x ) / A(x)^(n+1) = 0 for n>0.
%e RELATED SERIES.
%e Define B(x) = A(x/B(x)), which begins
%e B(x) = 1 + x + 13*x^2 + 907*x^3 + 153145*x^4 + 46602295*x^5 + 22140651001*x^6 + 15084920403375*x^7 + ... + A337575(n)*x^n + ...
%e then the table of coefficients of x^k/k! in exp( n*(n-1)^2 * x/B(x) ) begins:
%e n=0: [1, 0, 0, 0, 0, 0, 0, 0, ...];
%e n=1: [1, 0, 0, 0, 0, 0, 0, 0, ...];
%e n=2: [1, 2, 0, -160, -43520, -36711168, -67072065536, ...];
%e n=3: [1, 12, 120, 0, -293760, -234067968, -415963247616, ...];
%e n=4: [1, 36, 1224, 36288, 0, -792405504, -1355831322624, ...];
%e n=5: [1, 80, 6240, 467840, 31356160, 0, -3403785728000, ...];
%e n=6: [1, 150, 22200, 3229200, 456364800, 58514400000, 0, ...];
%e n=7: [1, 252, 63000, 15603840, 3817860480, 913835768832, 200316485182464, 0, ...]; ...
%e in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n*(n-1)^2 * x/B(x) ) = 0 for n>0.
%e Also note that B(x) = x/Series_Reversion( x*A(x) ) and A(x) = B(x*A(x)).
%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(m*(m-1)^2*x +x*O(x^m)) / Ser(A)^m )[m]/m);A[n+1]}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A337575, A337458, A337578.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Sep 02 2020
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