A304186 - OEIS

%I #22 May 10 2018 23:05:59

%S 1,3,6,24,189,2199,33495,625743,13778469,348258723,9916111584,

%T 313642702743,10901017499472,412730651160567,16902257604014685,

%U 744247858158013245,35058132248539742325,1758994171367292095805,93646661943861501833100,5272500660870261306912750

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

%F [x^n] (1+x)^((n+2)*(n+3)/2) / A(x)  =  A101483(n+1)  =  A101479(n+3,2) for n>=0.

%F [x^n] (1+x)^((n+3)*(n+4)/2) / A(x)  =  Sum_{k=0..n} A101479(n+4,k+3) * A101479(k+3,2) for n>=0.

%e G.f.: A(x) = 1 + 3*x + 6*x^2 + 24*x^3 + 189*x^4 + 2199*x^5 + 33495*x^6 + 625743*x^7 + 13778469*x^8 + 348258723*x^9 + 9916111584*x^10 + ...

%e ILLUSTRATION OF DEFINITION.

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

%e n=0: [1, -2, 0, -12, -105, -1434, -23877, -473730, -10881882, ...];

%e n=1: [1, 0, -3, -14, -129, -1656, -26850, -522918, -11853219, ...];

%e n=2: [1, 3, 0, -22, -180, -2088, -32219, -608565, -13504179, ...];

%e n=3: [1, 7, 18, 0, -255, -2937, -41739, -750711, -16140285, ...];

%e n=4: [1, 12, 63, 170, 0, -3996, -58877, -991308, -20341875, ...];

%e n=5: [1, 18, 150, 748, 2220, 0, -78435, -1401570, -27251715, ...];

%e n=6: [1, 25, 297, 2211, 11271, 37149, 0, -1843458, -38615364, ...];

%e n=7: [1, 33, 525, 5343, 38745, 207663, 758814, 0, -50361381, ...];

%e n=8: [1, 42, 858, 11340, 108630, 797100, 4541805, 18301950, 0, ...]; ...

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

%e RELATED SEQUENCES.

%e The secondary diagonal in the above table that begins

%e [1, 3, 18, 170, 2220, 37149, 758814, 18301950, 508907970, ...]

%e yields A101483, column 2 of triangle A101479.

%e Related triangular matrix T = A101479 begins:

%e 1;

%e 1, 1;

%e 1, 1, 1;

%e 3, 2, 1, 1;

%e 19, 9, 3, 1, 1;

%e 191, 70, 18, 4, 1, 1;

%e 2646, 795, 170, 30, 5, 1, 1;

%e 46737, 11961, 2220, 335, 45, 6, 1, 1;

%e 1003150, 224504, 37149, 4984, 581, 63, 7, 1, 1; ...

%e in which row n equals row (n-1) of T^(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*(m+1)/2)/Ser(A) )[m] );A[n+1]}

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

%Y Cf. A101481, A304184, A304185, A304187, A101479.

%K nonn

%O 0,2

%A _Paul D. Hanna_, May 08 2018