A358501 Irregular triangle read by rows. Coefficients of the polynomials (-1)^n*binomial(-x - 1, -x - n - 1) * binomial(n + x, x) * (n!)^2 in ascending order of powers.

1, 1, 2, 1, 4, 12, 13, 6, 1, 36, 132, 193, 144, 58, 12, 1, 576, 2400, 4180, 3980, 2273, 800, 170, 20, 1, 14400, 65760, 129076, 143700, 100805, 46710, 14523, 3000, 395, 30, 1, 518400, 2540160, 5450256, 6787872, 5482456, 3034920, 1184153, 328986, 64743, 8820, 791, 42, 1

Offset

0,3

Links

Table of n, a(n) for n=0..48.

Formula

p(n, x) = (-1)^n*(-x - 1)!*(n + x)!/((-x - n - 1)!*x!).

p(n, x) = (-1)^n*Pochhammer(-x - n, n) * Pochhammer(1 + x, n).

Example

Polynomials p(n, x) begin:
[0]                                    1;
[1]                           -(-x - 1)*(x + 1);
[2]                   (-x - 2)*(-x - 1)*(x + 1)*(x + 2);
[3]         -(-x - 3)*(-x - 2)*(-x - 1)*(x + 1)*(x + 2)*(x + 3);
[4] (-x - 4)*(-x - 3)*(-x - 2)*(-x - 1)*(x + 1)*(x + 2)*(x + 3)*(x + 4);
.
Triangle T(n, k) begins:
[0]     1;
[1]     1,     2,      1;
[2]     4,    12,     13,      6,      1;
[3]    36,   132,    193,    144,     58,    12,     1;
[4]   576,  2400,   4180,   3980,   2273,   800,   170,   20,   1;
[5] 14400, 65760, 129076, 143700, 100805, 46710, 14523, 3000, 395, 30, 1;

Maple

p := (n, x) -> (-1)^n*binomial(-x - 1, -x - n - 1)*binomial(n + x, x)*(n!)^2:
for n from 0 to 7 do seq(coeff(expand(p(n, x)), x, k), k = 0..2*n) od;

Crossrefs

Cf. A001044 (row sums), A000007 (alternating row sums).

Sequence in context: A371062 A354116 A064297 * A323493 A052661 A225117

Adjacent sequences: A358498 A358499 A358500 * A358502 A358503 A358504

Keyword

nonn,tabf

Author

Peter Luschny, Nov 26 2022

Status

approved