A358501 - OEIS

%I #7 Nov 26 2022 09:59:35

%S 1,1,2,1,4,12,13,6,1,36,132,193,144,58,12,1,576,2400,4180,3980,2273,

%T 800,170,20,1,14400,65760,129076,143700,100805,46710,14523,3000,395,

%U 30,1,518400,2540160,5450256,6787872,5482456,3034920,1184153,328986,64743,8820,791,42,1

%N 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.

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

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

%e Polynomials p(n, x) begin:

%e [0]                                    1;

%e [1]                           -(-x - 1)*(x + 1);

%e [2]                   (-x - 2)*(-x - 1)*(x + 1)*(x + 2);

%e [3]         -(-x - 3)*(-x - 2)*(-x - 1)*(x + 1)*(x + 2)*(x + 3);

%e [4] (-x - 4)*(-x - 3)*(-x - 2)*(-x - 1)*(x + 1)*(x + 2)*(x + 3)*(x + 4);

%e .

%e Triangle T(n, k) begins:

%e [0]     1;

%e [1]     1,     2,      1;

%e [2]     4,    12,     13,      6,      1;

%e [3]    36,   132,    193,    144,     58,    12,     1;

%e [4]   576,  2400,   4180,   3980,   2273,   800,   170,   20,   1;

%e [5] 14400, 65760, 129076, 143700, 100805, 46710, 14523, 3000, 395, 30, 1;

%p p := (n, x) -> (-1)^n*binomial(-x - 1, -x - n - 1)*binomial(n + x, x)*(n!)^2:

%p for n from 0 to 7 do seq(coeff(expand(p(n, x)), x, k), k = 0..2*n) od;

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

%K nonn,tabf

%O 0,3

%A _Peter Luschny_, Nov 26 2022