A377661 Triangle read by rows: T(n, k) = e*Gamma(n - k + 1, 1)*binomial(n, k)^2.

1, 2, 1, 5, 8, 1, 16, 45, 18, 1, 65, 256, 180, 32, 1, 326, 1625, 1600, 500, 50, 1, 1957, 11736, 14625, 6400, 1125, 72, 1, 13700, 95893, 143766, 79625, 19600, 2205, 98, 1, 109601, 876800, 1534288, 1022336, 318500, 50176, 3920, 128, 1

Offset: 0

Author

Peter Luschny, Nov 03 2024

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[0]      1;
[1]      2,      1;
[2]      5,      8,       1;
[3]     16,     45,      18,       1;
[4]     65,    256,     180,      32,      1;
[5]    326,   1625,    1600,     500,     50,     1;
[6]   1957,  11736,   14625,    6400,   1125,    72,    1;
[7]  13700,  95893,  143766,   79625,  19600,  2205,   98,   1;
[8] 109601, 876800, 1534288, 1022336, 318500, 50176, 3920, 128, 1;
		

Crossrefs

Cf. A000522 (column 0), A001105 (subdiagonal), A377662 (row sums), A073107.

Programs

Maple

T := (n, k) -> exp(1)*GAMMA(n - k + 1, 1)*binomial(n, k)^2:
seq(seq(simplify(T(n, k)), k = 0..n), n=0..8);
# Alternative:
A377661 := (n, k) -> n!*binomial(n,k)*add(1/(k!*(j-k)!), j = k..n):
for n from 0 to 8 do seq(A377661(n, k), k = 0..n) od;
# Or:
T := (n, k) -> binomial(n, k)^2 * KummerU(k - n, k - n, 1):
for n from 0 to 8 do seq(simplify(T(n, k)), k= 0..n) od;

Mathematica

T[n_, k_] := E Binomial[n, k]^2 Gamma[1 - k + n, 1];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten

Python

from math import comb, isqrt, factorial
def A377661(n):
    a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1))
    b = n-comb(a+1,2)
    fa, fb = factorial(a), factorial(b)
    return comb(a,b)*sum(fa//(fb*factorial(j-b)) for j in range(b,a+1)) # Chai Wah Wu, Nov 12 2024

Formula

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

T(n, k) = binomial(n, k)^2 * KummerU(k - n, k - n, 1).

T(n, k) = binomial(n, k) * A073107(n, k).