login
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.
0
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
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
KEYWORD
nonn,tabf
AUTHOR
Peter Luschny, Nov 26 2022
STATUS
approved