%I #8 Oct 06 2020 07:22:22

%S 1,1,1,7,87,1667,42971,1387941,53739797,2421203261,124265293581,

%T 7150727869627,455701200668539,31846907669892495,2421141672213472919,

%U 198897819736367366729,17556316040185549675881,1656973308228250148662329,166509657562826568857464281,17749793745710561363581851663,2000554650909636157531234301439

%N G.f. A(x) satisfies: [x^(n-1)] (1+x)^(n*(n-1)) / A(x)^n = 0 for n>1.

%H Paul D. Hanna, <a href="/A303061/b303061.txt">Table of n, a(n) for n = 0..300</a>

%F a(n) + a(n-1) = A303060(n) for n>=0.

%F a(n) ~ c * d^n * n! / n^2, where d = -4 / (LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) = 6.17655460948348035823168... and c = 0.06049920104... - _Vaclav Kotesovec_, Oct 06 2020

%e G.f.: A(x) = 1 + x + x^2 + 7*x^3 + 87*x^4 + 1667*x^5 + 42971*x^6 + 1387941*x^7 + 53739797*x^8 + 2421203261*x^9 + 124265293581*x^10 + ...

%e ILLUSTRATION OF DEFINITION.

%e The table of coefficients in (1+x)^(n*(n-1)) / A(x)^n begins:

%e n=1: [1, -1, 0, -6, -74, -1500, -39688, -1302742, ...];

%e n=2: [1, 0, -2, -12, -159, -3136, -82180, -2680752, ...];

%e n=3: [1, 3, 0, -26, -300, -5454, -137764, -4398210, ...];

%e n=4: [1, 8, 24, 0, -548, -9576, -223760, -6847536, ...];

%e n=5: [1, 15, 100, 350, 0, -16022, -376660, -10771830, ...];

%e n=6: [1, 24, 270, 1844, 7641, 0, -596908, -17643792, ...];

%e n=7: [1, 35, 588, 6258, 46186, 224196, 0, -26940146, ...];

%e n=8: [1, 48, 1120, 16864, 182640, 1478160, 8281968, 0, ...]; ...

%e in which the main diagonal equals all zeros after the initial term, illustrating that [x^(n-1)] (1+x)^(n*(n-1)) / A(x)^n = 0 for n>1.

%o (PARI) {a(n) = my(A=[1]); for(m=1,n+1, A=concat(A,0); A[m] = Vec( (1+x +x*O(x^n))^(m*(m-1))/Ser(A)^m )[m]/m ); A[n+1]}

%o for(n=0,30, print1(a(n),", "))

%Y Cf. A303060.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Apr 17 2018