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A304187 G.f. A(x) satisfies: 0 = [x^n] (1+x)^((n+2)*(n+3)/2) / A(x) for n>0.

%I A304187 #11 May 08 2018 23:46:07
%S A304187 1,6,21,86,606,6756,102316,1931046,43250376,1114876536,32394654066,
%T A304187 1045240099026,37027935179016,1427410628324856,59449956448178106,
%U A304187 2659215814433980056,127108810653344820456,6464722863550156435146,348541208165221134718986,19854709880058367829287716,1191556960000156185148449636
%N A304187 G.f. A(x) satisfies: 0 = [x^n] (1+x)^((n+2)*(n+3)/2) / A(x) for n>0.
%F A304187 A101479(n+4,3) = [x^n] (1+x)^((n+3)*(n+4)/2) / A(x) for n>=0.
%e A304187 G.f.: A(x) = 1 + 6*x + 21*x^2 + 86*x^3 + 606*x^4 + 6756*x^5 + 102316*x^6 + 1931046*x^7 + 43250376*x^8 + 1114876536*x^9 + 32394654066*x^10 + ...
%e A304187 ILLUSTRATION OF DEFINITION.
%e A304187 The table of coefficients of x^k in (1+x)^((n+1)*(n+2)/2) / A(x) begins:
%e A304187 n=0: [1, -3, 0, -22, -216, -3180, -56540, -1186170, -28599870, ...];
%e A304187 n=1: [1, 0, -6, -30, -285, -3894, -66750, -1365546, -32331180, ...];
%e A304187 n=2: [1, 4, 0, -50, -440, -5238, -84162, -1657080, -38209725, ...];
%e A304187 n=3: [1, 9, 30, 0, -645, -7917, -115248, -2134920, -47391375, ...];
%e A304187 n=4: [1, 15, 99, 335, 0, -11046, -171920, -2957874, -62097600, ...];
%e A304187 n=5: [1, 22, 225, 1378, 4984, 0, -233730, -4379370, -86791905, ...];
%e A304187 n=6: [1, 30, 429, 3850, 23610, 92652, 0, -5860422, -127938780, ...];
%e A304187 n=7: [1, 39, 735, 8875, 76350, 483684, 2065146, 0, -169402725, ...];
%e A304187 n=8: [1, 49, 1170, 18100, 203065, 1743201, 11567124, 53636520, 0, ...]; ...
%e A304187 in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] (1+x)^((n+2)*(n+3)/2) / A(x) for n>0.
%e A304187 RELATED SEQUENCES.
%e A304187 The secondary diagonal in the above table that begins
%e A304187 [1, 4, 30, 335, 4984, 92652, 2065146, 53636520, 1589752230, ...]
%e A304187 yields column 3 of triangle A101479.
%e A304187 Related triangular matrix T = A101479 begins:
%e A304187 1;
%e A304187 1, 1;
%e A304187 1, 1, 1;
%e A304187 3, 2, 1, 1;
%e A304187 19, 9, 3, 1, 1;
%e A304187 191, 70, 18, 4, 1, 1;
%e A304187 2646, 795, 170, 30, 5, 1, 1;
%e A304187 46737, 11961, 2220, 335, 45, 6, 1, 1;
%e A304187 1003150, 224504, 37149, 4984, 581, 63, 7, 1, 1; ...
%e A304187 in which row n equals row (n-1) of T^(n-1) followed by '1' for n>0.
%o A304187 (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 A304187 for(n=0,30, print1(a(n),", "))
%Y A304187 Cf. A101481, A304184, A304185, A304186, A101479.
%K A304187 nonn
%O A304187 0,2
%A A304187 _Paul D. Hanna_, May 08 2018