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%I #10 Oct 06 2020 03:12:23
%S 1,2,7,40,341,3958,58231,1033124,21365965,503022874,13253293847,
%T 385800906976,12285311512117,424560956348894,15819575740546279,
%U 632092618823958364,26957367530649232781,1222184752998639598978,58698481264946403235495,2977081912253813732898008,159007077603468166439812885,8920982827223292597938207302,524554918971298203370379111063
%N G.f. A(x) satisfies: [x^n] 1/(1 - (n+1)*x) / A(x) = 0 for n >= 1.
%H Paul D. Hanna, <a href="/A319945/b319945.txt">Table of n, a(n) for n = 0..300</a>
%F a(n) ~ (exp(1) - 1) * n^n. - _Vaclav Kotesovec_, Oct 06 2020
%e G.f.: A(x) = 1 + 2*x + 7*x^2 + 40*x^3 + 341*x^4 + 3958*x^5 + 58231*x^6 + 1033124*x^7 + 21365965*x^8 + 503022874*x^9 + 13253293847*x^10 + ...
%e ILLUSTRATION OF DEFINITION.
%e The table of coefficients of x^k in 1/(1 - (n+1)*x) / A(x) begins:
%e n=1: [1, -1, -4, -24, -224, -2840, -44700, -832636, -17836768, ...];
%e n=2: [1, 0, -3, -26, -252, -3120, -48100, -884136, -18772404, ...];
%e n=3: [1, 1, 0, -20, -260, -3396, -52048, -944080, -19836372, ...];
%e n=4: [1, 2, 5, 0, -200, -3416, -55524, -1010032, -21044260, ...];
%e n=5: [1, 3, 12, 40, 0, -2616, -54940, -1062636, -22317312, ...];
%e n=6: [1, 4, 21, 106, 436, 0, -41860, -1039096, -23238708, ...];
%e n=7: [1, 5, 32, 204, 1228, 5980, 0, -787936, -22519684, ...];
%e n=8: [1, 6, 45, 340, 2520, 17544, 98492, 0, -17004132, ...];
%e n=9: [1, 7, 60, 520, 4480, 37704, 297476, 1889348, 0, ...];
%e n=10:[1, 8, 77, 750, 7300, 70384, 661980, 5831864, 41314508, 0, ...]; ...
%e in which the coefficient of x^(n-1) in row n forms a diagonal of zeros after an initial '1'.
%e RELATED SERIES.
%e The square-root of the g.f. is an integer series that begins
%e A(x)^(1/2) = 1 + x + 3*x^2 + 17*x^3 + 149*x^4 + 1779*x^5 + 26745*x^6 + 481947*x^7 + 10079457*x^8 + 239266403*x^9 + ... + A319946(n)*x^n + ...
%e The series 1/A(x) begins
%e 1/A(x) = 1 - 2*x - 3*x^2 - 20*x^3 - 200*x^4 - 2616*x^5 - 41860*x^6 - 787936*x^7 - 17004132*x^8 - 413145080*x^9 - 11149376612*x^10 + ...
%e The integral of 1/A(x) is an integer series that begins
%e Integral 1/A(x) dx = x - x^2 - x^3 - 5*x^4 - 40*x^5 - 436*x^6 - 5980*x^7 - 98492*x^8 - 1889348*x^9 - 41314508*x^10 - 1013579692*x^11 - 27557806012*x^12 + ...
%e the coefficients of which form a diagonal in the above table.
%o (PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); m=#A; A[m] = Vec( 1/Ser(A)/(1-m*x +x^2*O(x^m)))[m] ); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A319946.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Oct 11 2018
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