A304188 - OEIS

%I #7 May 09 2018 10:18:38

%S 1,6,30,264,4179,97758,3000084,113020056,5018695542,255724146876,

%T 14671199172480,934467807541824,65366076594301044,4978197982191048600,

%U 409875168025688997456,36268233577292228677728,3431775207222740657912472,345742547371677388835049744,36948141363745699171977916032,4174429749114285739841190548928

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

%F A132613(n+1) = [x^n] (1+x)^((n+2)*(n+3)) / A(x) for n>0.

%e G.f.: A(x) = 1 + 6*x + 30*x^2 + 264*x^3 + 4179*x^4 + 97758*x^5 + 3000084*x^6 + 113020056*x^7 + 5018695542*x^8 + 255724146876*x^9 + 14671199172480*x^10 + ...

%e ILLUSTRATION OF DEFINITION.

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

%e n=0: [1, -4, -5, -114, -2289, -62568, -2113983, -84889290, ...];

%e n=1: [1, 0, -15, -154, -2790, -72432, -2378450, -93729900, ...];

%e n=2: [1, 6, 0, -224, -3924, -91776, -2858196, -109145280, ...];

%e n=3: [1, 14, 76, 0, -5310, -128964, -3714456, -134815824, ...];

%e n=4: [1, 24, 261, 1510, 0, -169752, -5223348, -178378752, ...];

%e n=5: [1, 36, 615, 6446, 41121, 0, -6779045, -251285430, ...];

%e n=6: [1, 50, 1210, 18696, 201435, 1424178, 0, -323428800, ...];

%e n=7: [1, 66, 2130, 44616, 675591, 7663626, 59857416, 0, ...]; ...

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

%e RELATED SEQUENCES.

%e The secondary diagonal in the above table that begins

%e [1, 6, 76, 1510, 41121, 1424178, 59857416, 2957282370, ...]

%e yields A132613, column 2 of triangle A132610.

%e Related triangular matrix T = A132610 begins:

%e 1;

%e 1, 1;

%e 2, 1, 1;

%e 14, 4, 1, 1;

%e 194, 39, 6, 1, 1;

%e 4114, 648, 76, 8, 1, 1;

%e 118042, 15465, 1510, 125, 10, 1, 1;

%e 4274612, 483240, 41121, 2908, 186, 12, 1, 1;

%e 186932958, 18685905, 1424178, 89670, 4970, 259, 14, 1, 1; ...

%e in which  row n+1 of T = row n of matrix power T^(2*n) with appended '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))/Ser(A) )[m] ); A[n+1]}

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

%Y Cf. A132613, A304189, A304192, A304193, A132610.

%K nonn

%O 0,2

%A _Paul D. Hanna_, May 09 2018