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A304185 G.f. A(x) satisfies: 0 = [x^n] (1+x)^(n*(n+1)/2) / A(x) for n>0.

%I A304185 #13 May 10 2018 22:59:21
%S A304185 1,1,1,6,51,609,9284,171779,3729929,92828134,2602268335,81082146565,
%T A304185 2778537934965,103826098384590,4200631499008965,182912327481135887,
%U A304185 8528448938007615767,423908532350099458532,22375155669520993773517,1249871928128324730985042,73662719257076005954584046
%N A304185 G.f. A(x) satisfies: 0 = [x^n] (1+x)^(n*(n+1)/2) / A(x) for n>0.
%F A304185 [x^n] (1+x)^((n+1)*(n+2)/2) / A(x)  =  A101482(n+1)  =  A101479(n+2,1) for n>=0.
%F A304185 [x^n] (1+x)^((n+2)*(n+3)/2) / A(x)  =  Sum_{k=0..n} A101479(n+3,k+2) * A101479(k+2,1) for n>=0.
%e A304185 G.f.: A(x) = 1 + x + x^2 + 6*x^3 + 51*x^4 + 609*x^5 + 9284*x^6 + 171779*x^7 + 3729929*x^8 + 92828134*x^9 + 2602268335*x^10 + ...
%e A304185 ILLUSTRATION OF DEFINITION.
%e A304185 The table of coefficients of x^k in (1+x)^(n*(n+1)/2) / A(x) begins:
%e A304185 n=0: [1, -1, 0, -5, -40, -513, -8092, -153395, -3388500, ...];
%e A304185 n=1: [1, 0, -1, -5, -45, -553, -8605, -161487, -3541895, ...];
%e A304185 n=2: [1, 2, 0, -7, -56, -648, -9756, -179250, -3873474, ...];
%e A304185 n=3: [1, 5, 9, 0, -75, -837, -11875, -210518, -4441140, ...];
%e A304185 n=4: [1, 9, 35, 70, 0, -1096, -15664, -263340, -5357885, ...];
%e A304185 n=5: [1, 14, 90, 345, 795, 0, -20260, -352235, -6842115, ...];
%e A304185 n=6: [1, 20, 189, 1115, 4510, 11961, 0, -452166, -9245340, ...];
%e A304185 n=7: [1, 27, 350, 2893, 17019, 74282, 224504, 0, -11809259, ...];
%e A304185 n=8: [1, 35, 594, 6505, 51545, 312984, 1483340, 5051866, 0, ...]; ...
%e A304185 in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] (1+x)^(n*(n+1)/2) / A(x) for n>0.
%e A304185 RELATED SEQUENCES.
%e A304185 The secondary diagonal in the above table that begins
%e A304185 [1, 2, 9, 70, 795, 11961, 224504, 5051866, 132523155, ...]
%e A304185 yields A101482, column 1 of triangle A101479.
%e A304185 Related triangular matrix T = A101479 begins:
%e A304185 1;
%e A304185 1, 1;
%e A304185 1, 1, 1;
%e A304185 3, 2, 1, 1;
%e A304185 19, 9, 3, 1, 1;
%e A304185 191, 70, 18, 4, 1, 1;
%e A304185 2646, 795, 170, 30, 5, 1, 1;
%e A304185 46737, 11961, 2220, 335, 45, 6, 1, 1;
%e A304185 1003150, 224504, 37149, 4984, 581, 63, 7, 1, 1; ...
%e A304185 in which row n equals row (n-1) of T^(n-1) followed by '1' for n>0.
%o A304185 (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( (1+x +x*O(x^m))^(m*(m-1)/2)/Ser(A) )[m] );A[n+1]}
%o A304185 for(n=0,30, print1(a(n),", "))
%Y A304185 Cf. A101481, A304184, A304186, A304187, A101479.
%K A304185 nonn
%O A304185 0,4
%A A304185 _Paul D. Hanna_, May 08 2018