A248978 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation of 1 + x + x^2 + ... + x^n to the polynomial A_k*(x-3k)^k for 0 <= k <= n.

1, 4, 1, 4, 13, 1, 4, 94, 28, 1, 4, 526, 460, 49, 1, 4, 2551, 5860, 1399, 76, 1, 4, 11299, 64180, 30559, 3316, 109, 1, 4, 47020, 635716, 566374, 109156, 6724, 148, 1, 4, 186988, 5861188, 9384358, 3012196, 309124, 12244, 193, 1, 4, 718429, 51210820, 143307490, 73556068, 11790874, 747076, 20605, 244, 1, 4, 2686729, 429124420, 2056495090, 1641197668, 394515874, 37488676, 1608205, 32644, 301, 1

Offset

0,2

Comments

Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x-0)^0 + A_1*(x-3)^1 + A_2*(x-6)^2 + ... + A_n*(x-3n)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0.

Links

Table of n, a(n) for n=0..65.

Formula

T(n,n-1) = 1 + 3n^2 for n > 0.

T(n,1) = (3^n*(n^2-n+1)-1)/2 for n > 0.

Example

1;
4,      1;
4,     13,        1;
4,     94,       28,         1;
4,    526,      460,        49,        1;
4,   2551,     5860,      1399,       76,        1;
4,  11299,    64180,     30559,     3316,      109,      1;
4,  47020,   635716,    566374,   109156,     6724,    148,     1;
4, 186988,  5861188,   9384358,  3012196,   309124,  12244,   193,   1;
4, 718429, 51210820, 143307490, 73556068, 11790874, 747076, 20605, 244, 1;

Prog

(PARI) for(n=0, 10, for(k=0, n, if(!k, if(n, print1(4, ", ")); if(!n, print1(1, ", "))); if(k, print1(sum(i=1, n, ((3*k)^(i-k)*i*binomial(i, k)))/k, ", "))))

Crossrefs

Cf. A248977, A248830, A242598, A193843, A153703, A056107.

Sequence in context: A208918 A228782 A205125 * A143461 A066808 A363935

Adjacent sequences: A248975 A248976 A248977 * A248979 A248980 A248981

Keyword

nonn,tabl

Author

Derek Orr, Oct 18 2014

Status

approved