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A304184 G.f. A(x) satisfies: 0 = [x^n] (1+x)^(n*(n-1)/2) / A(x) for n>0.

%I A304184 #14 May 10 2018 22:55:47
%S A304184 1,0,0,1,9,117,1851,34923,765933,19155084,538051164,16771165230,
%T A304184 574424285076,21443516818065,866521903003641,37683366660458208,
%U A304184 1754777541925339779,87115221430910051901,4592968693335470802627,256294382115032521083411,15090698035153332532531074
%N A304184 G.f. A(x) satisfies: 0 = [x^n] (1+x)^(n*(n-1)/2) / A(x) for n>0.
%F A304184 [x^n] (1+x)^(n*(n+1)/2) / A(x)  =  A101481(n+1)  =  A101479(n+1,0) for n>=0.
%F A304184 [x^n] (1+x)^((n+1)*(n+2)/2) / A(x)  =  Sum_{k=0..n} A101479(n+2,k+1) * A101479(k+1,0) for n>=0.
%e A304184 G.f.: A(x) = 1 + x^3 + 9*x^4 + 117*x^5 + 1851*x^6 + 34923*x^7 + 765933*x^8 + 19155084*x^9 + 538051164*x^10 + 16771165230*x^11 + 574424285076*x^12 + ...
%e A304184 ILLUSTRATION OF DEFINITION.
%e A304184 The table of coefficients of x^k in (1+x)^(n*(n-1)/2) / A(x) begins:
%e A304184 n=0: [1, 0, 0, -1, -9, -117, -1850, -34905, -765618, ...];
%e A304184 n=1: [1, 0, 0, -1, -9, -117, -1850, -34905, -765618, ...];
%e A304184 n=2: [1, 1, 0, -1, -10, -126, -1967, -36755, -800523, ...];
%e A304184 n=3: [1, 3, 3, 0, -12, -147, -2229, -40815, -876000, ...];
%e A304184 n=4: [1, 6, 15, 19, 0, -180, -2706, -47955, -1005279, ...];
%e A304184 n=5: [1, 10, 45, 119, 191, 0, -3335, -59840, -1214055, ...];
%e A304184 n=6: [1, 15, 105, 454, 1341, 2646, 0, -73965, -1545531, ...];
%e A304184 n=7: [1, 21, 210, 1329, 5955, 19833, 46737, 0, -1913457, ...];
%e A304184 n=8: [1, 28, 378, 3275, 20438, 97533, 364936, 1003150, 0, ...]; ...
%e A304184 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 A304184 RELATED SEQUENCES.
%e A304184 The secondary diagonal in the above table that begins
%e A304184 [1, 1, 3, 19, 191, 2646, 46737, 1003150, 25330125,  ...]
%e A304184 yields A101481, column 0 of triangle A101479.
%e A304184 Related triangular matrix T = A101479 begins:
%e A304184 1;
%e A304184 1, 1;
%e A304184 1, 1, 1;
%e A304184 3, 2, 1, 1;
%e A304184 19, 9, 3, 1, 1;
%e A304184 191, 70, 18, 4, 1, 1;
%e A304184 2646, 795, 170, 30, 5, 1, 1;
%e A304184 46737, 11961, 2220, 335, 45, 6, 1, 1;
%e A304184 1003150, 224504, 37149, 4984, 581, 63, 7, 1, 1; ...
%e A304184 in which row n equals row (n-1) of T^(n-1) followed by '1' for n>0.
%o A304184 (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-1)*(m-2)/2)/Ser(A) )[m] );A[n+1]}
%o A304184 for(n=0,30, print1(a(n),", "))
%Y A304184 Cf. A101481, A304185, A304186, A304187, A304189, A101479.
%K A304184 nonn
%O A304184 0,5
%A A304184 _Paul D. Hanna_, May 08 2018