A323233 Coefficients of polynomials p(n, x) generating the columns of A323224, triangle read by rows, T(n, k) for n >= 1 and k >= 0.

1, 2, 2, 6, 15, 3, 24, 140, 48, 4, 120, 1750, 775, 110, 5, 720, 28644, 14550, 2670, 210, 6, 5040, 588588, 323008, 68775, 7105, 357, 7, 40320, 14592864, 8388800, 1962632, 239120, 16016, 560, 8, 362880, 423227376, 250742700, 62531532, 8502921, 680904, 32130, 828, 9

Offset: 1

Author

Peter Luschny, Jan 27 2019

Keywords

• nonn
• tabl

Examples

			The triangle starts:
[ 1]       1;
[ 2]       2,         2;
[ 3]       6,        15,         3;
[ 4]      24,       140,        48,        4;
[ 5]     120,      1750,       775,      110,       5;
[ 6]     720,     28644,     14550,     2670,     210,      6;
[ 7]    5040,    588588,    323008,    68775,    7105,    357,     7;
[ 8]   40320,  14592864,   8388800,  1962632,  239120,  16016,   560,   8;
[ 9]  362880, 423227376, 250742700, 62531532, 8502921, 680904, 32130, 828, 9;
The first few polynomials are:
p[1](x) = 1;
p[2](x) = 2*x + 2!;
p[3](x) = 3*x*(x + 5) + 3!;
p[4](x) = 4*x*(x + 5)*(x + 7) + 4!;
p[5](x) = 5*x*(x + 5)*(x + 7)*(x + 10) + 5!;
p[6](x) = 6*x*(x + 7)*(x + 11)*(x^2 + 17*x + 62) + 6!;
p[7](x) = 7*x*(x + 6)*(x + 7)*(x + 11)*(x + 13)*(x + 14) + 7!;
		

Crossrefs

Cf. A323224, A034856, A323221, A323220, A014137, A014138.

Programs

Mathematica

ogf[n_] := (2/(1 + Sqrt[1 - 4 x] ))^n  x/(1 - x);
ser[n_, len_] := CoefficientList[Series[ogf[n], {x, 0, (n + 1) len + 1}], x];
tab[k_, len_] := Table[{n, ser[n, k + 1][[k + 1]]}, {n, 0, len - 1}];
pol[n_] := n! InterpolatingPolynomial[tab[n, n + 1], x] // Expand;
row[n_] := CoefficientList[pol[n], x]; Table[row[n], {n, 1, 9}]

Formula

A323224(n, k) = p(k, n)/k!.

T(n, k) = [x^k] p(n, x).

p(n, 1)/n! and p(n, -1)/n! are versions of the partial sums of the Catalan numbers.