A304189 G.f. A(x) satisfies: [x^n] (1+x)^(n*(n-1)) / A(x) = 0 for n>0.

1, 0, 1, 14, 262, 6512, 202194, 7540004, 328229124, 16332497152, 914162756076, 56834335366552, 3885119345623448, 289588265286519808, 23372826192097312232, 2030600572225893011568, 188934550189205698385072, 18743556336897311790277824, 1974977055586233987489048976, 220268077592251409442788164320, 25923441737544899398961718119392

Offset: 0

Author

Paul D. Hanna, May 08 2018

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x^2 + 14*x^3 + 262*x^4 + 6512*x^5 + 202194*x^6 + 7540004*x^7 + 328229124*x^8 + 16332497152*x^9 + 914162756076*x^10 + 56834335366552*x^11 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in (1+x)^(n*(n-1)) / A(x) begins:
  n=0: [1, 0, -1, -14, -261, -6484, -201475, -7519686, ...];
  n=1: [1, 0, -1, -14, -261, -6484, -201475, -7519686, ...];
  n=2: [1, 2, 0, -16, -290, -7020, -214704, -7929120, ...];
  n=3: [1, 6, 14, 0, -345, -8274, -244588, -8831232, ...];
  n=4: [1, 12, 65, 194, 0, -9968, -299160, -10429680, ...];
  n=5: [1, 20, 189, 1106, 4114, 0, -362790, -13084500, ...];
  n=6: [1, 30, 434, 4016, 26289, 118042, 0, -15934512, ...];
  n=7: [1, 42, 860, 11424, 110220, 809688, 4274612, 0, ...];
  n=8: [1, 56, 1539, 27650, 364705, 3749436, 30746547, 186932958, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that [x^n] (1+x)^(n*(n-1)) / A(x) = 0 for n>0.
RELATED SEQUENCES.
The secondary diagonal in the above table that begins
[1, 2, 14, 194, 4114, 118042, 4274612, 186932958, 9577713250, ...]
yields A132611, column 0 of triangle A132610.
Related triangular matrix T = A132610 begins:
  1;
  1, 1;
  2, 1, 1;
  14, 4, 1, 1;
  194, 39, 6, 1, 1;
  4114, 648, 76, 8, 1, 1;
  118042, 15465, 1510, 125, 10, 1, 1;
  4274612, 483240, 41121, 2908, 186, 12, 1, 1;
  186932958, 18685905, 1424178, 89670, 4970, 259, 14, 1, 1; ...
in which row n+1 of T = row n of matrix power T^(2*n) with appended '1' for n>=0.
		

Crossrefs

Cf. A132611, A304192, A304188, A304184, A132610.

Programs

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))/Ser(A) )[m] ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

A132611(n+1) = [x^n] (1+x)^(n*(n+1)) / A(x) for n>0.