A375854 Triangle read by rows: T(n, k) = 2^k * hypergeom([-n, -k], [], 1/2).

1, 1, 3, 1, 4, 14, 1, 5, 22, 86, 1, 6, 32, 152, 648, 1, 7, 44, 248, 1256, 5752, 1, 8, 58, 380, 2248, 12032, 58576, 1, 9, 74, 554, 3768, 23272, 130768, 671568, 1, 10, 92, 776, 5984, 42112, 270400, 1586944, 8546432, 1, 11, 112, 1052, 9088, 72032, 523072, 3479744, 21241984, 119401856

Offset: 0

Author

Detlef Meya, Aug 31 2024

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[0] 1;
[1] 1, 3;
[2] 1, 4, 14;
[3] 1, 5, 22, 86;
[4] 1, 6, 32, 152, 648;
[5] 1, 7, 44, 248, 1256, 5752;
[6] 1, 8, 58, 380, 2248, 12032, 58576;
[7] 1, 9, 74, 554, 3768, 23272, 130768, 671568;
[8] 1, 10, 92, 776, 5984, 42112, 270400, 1586944, 8546432;
[9] 1, 11, 112, 1052, 9088, 72032, 523072, 3479744, 21241984, 119401856;
...
		

Crossrefs

Cf. A375855, A000012, A087912 (main diagonal).

Programs

Maple

T := (n, k) -> 2^k * hypergeom([-n, -k], [], 1/2):
for n from 0 to 9 do seq(simplify(T(n, k)), k=0..n) od; # Peter Luschny, Sep 02 2024

Mathematica

T[n_, k_] := Sum[2^(k - j)*Binomial[n, j]*Binomial[k, j]*j!, {j, 0, k}];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten

Python

from math import isqrt, comb, factorial
def A375854(n):
    a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1))
    b = n-comb(a+1,2)
    return sum(comb(a,j)*comb(b,j)*factorial(j)<Chai Wah Wu, Nov 13 2024

Formula

T(n, k) = Sum_{j=0..k} 2^(k - j)*binomial(n, j)*binomial(k, j)*j!.