A304190 - OEIS

%I #5 May 08 2018 23:21:49

%S 1,0,0,4,90,2448,79716,3058740,135637242,6835557984,386119895256,

%T 24170805494868,1661105052140226,124342746871407984,

%U 10070793262851698412,877493877654988612836,81848857574562663295026,8137513480199793111630528,859067817713438540813133744,95973644392465888508242272804,11312379843382901418721437545706

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

%F A132616(n+1) = [x^n] (1+x)^(n^2) / A(x) for n>=0.

%e G.f.: A(x) = 1 + 4*x^3 + 90*x^4 + 2448*x^5 + 79716*x^6 + 3058740*x^7 + 135637242*x^8 + 6835557984*x^9 + 386119895256*x^10 + 24170805494868*x^11 + ...

%e ILLUSTRATION OF DEFINITION.

%e The table of coefficients of x^k in (1+x)^((n-1)^2) / A(x) begins:

%e n=0: [1, 1, 0, -4, -94, -2538, -82148, -3137720, ...];

%e n=1: [1, 0, 0, -4, -90, -2448, -79700, -3058020, ...];

%e n=2: [1, 1, 0, -4, -94, -2538, -82148, -3137720, ...];

%e n=3: [1, 4, 6, 0, -105, -2832, -90048, -3391872, ...];

%e n=4: [1, 9, 36, 80, 0, -3276, -105224, -3871476, ...];

%e n=5: [1, 16, 120, 556, 1666, 0, -123900, -4673220, ...];

%e n=6: [1, 25, 300, 2296, 12460, 47232, 0, -5561820, ...];

%e n=7: [1, 36, 630, 7136, 58671, 368784, 1694704, 0,  ...];

%e n=8: [1, 49, 1176, 18420, 211590, 1895322, 13604628, 73552752, 0, ...]; ...

%e in which the main diagonal is all zeros after the initial term, illustrating that [x^n] (1+x)^((n-1)^2) / A(x) = 0 for n>0.

%e RELATED SEQUENCES.

%e The secondary diagonal in the above table that begins

%e [1, 1, 6, 80, 1666, 47232, 1694704, 73552752, ...]

%e yields A132616, column 0 of triangle A132615.

%e Related triangular matrix T = A132615 begins:

%e 1;

%e 1, 1;

%e 1, 1, 1;

%e 6, 3, 1, 1;

%e 80, 25, 5, 1, 1;

%e 1666, 378, 56, 7, 1, 1;

%e 47232, 8460, 1020, 99, 9, 1, 1;

%e 1694704, 252087, 26015, 2134, 154, 11, 1, 1;

%e 73552752, 9392890, 855478, 61919, 3848, 221, 13, 1, 1; ...

%e in which row n equals row (n-1) of T^(2*n-1) followed by '1' for n > 0.

%o (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-2)^2)/Ser(A) )[m] ); A[n+1]}

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

%Y Cf. A132616, A304191, A304192, A304193, A132615.

%K nonn

%O 0,4

%A _Paul D. Hanna_, May 08 2018