A344563 T(n, k) = binomial(n - 1, k - 1) * binomial(n, k) * 2^k, T(0, 0) = 1. Triangle read by rows, T(n, k) for 0 <= k <= n.

1, 0, 2, 0, 4, 4, 0, 6, 24, 8, 0, 8, 72, 96, 16, 0, 10, 160, 480, 320, 32, 0, 12, 300, 1600, 2400, 960, 64, 0, 14, 504, 4200, 11200, 10080, 2688, 128, 0, 16, 784, 9408, 39200, 62720, 37632, 7168, 256, 0, 18, 1152, 18816, 112896, 282240, 301056, 129024, 18432, 512

Offset: 0

Author

Peter Luschny, May 30 2021

Keywords

• nonn
• tabl

Examples

			[0] 1;
[1] 0,  2;
[2] 0,  4,    4;
[3] 0,  6,   24,     8;
[4] 0,  8,   72,    96,     16;
[5] 0, 10,  160,   480,    320,     32;
[6] 0, 12,  300,  1600,   2400,    960,     64;
[7] 0, 14,  504,  4200,  11200,  10080,   2688,    128;
[8] 0, 16,  784,  9408,  39200,  62720,  37632,   7168,   256;
[9] 0, 18, 1152, 18816, 112896, 282240, 301056, 129024, 18432, 512.
		

Links

• T. Amdeberhan, Power of 2 dividing a specialized Mittag-Leffler polynomial, MathOverflow.

Crossrefs

Row sums are A002003 with a(0) = 1, cf. also A047781.

The coefficients of the associated polynomials are in A103371.

Programs

Maple

aRow := n -> seq(binomial(n-1, k-1)*binomial(n,k)*2^k, k=0..n):
seq(print(aRow(n)), n=0..9);

Mathematica

T[n_, k_] := Binomial[n-1, k-1] * Binomial[n, k] * 2^k;
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten

Python

from math import comb
def T(n, k):
    return comb(n-1, k-1)*comb(n, k)*2**k if k > 0 else k**n
print([T(n, k) for n in range(10) for k in range(n+1)]) # Michael S. Branicky, May 30 2021