A377661 - OEIS

%I #14 Nov 12 2024 20:20:59

%S 1,2,1,5,8,1,16,45,18,1,65,256,180,32,1,326,1625,1600,500,50,1,1957,

%T 11736,14625,6400,1125,72,1,13700,95893,143766,79625,19600,2205,98,1,

%U 109601,876800,1534288,1022336,318500,50176,3920,128,1

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

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

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

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

%e Triangle starts:

%e [0]      1;

%e [1]      2,      1;

%e [2]      5,      8,       1;

%e [3]     16,     45,      18,       1;

%e [4]     65,    256,     180,      32,      1;

%e [5]    326,   1625,    1600,     500,     50,     1;

%e [6]   1957,  11736,   14625,    6400,   1125,    72,    1;

%e [7]  13700,  95893,  143766,   79625,  19600,  2205,   98,   1;

%e [8] 109601, 876800, 1534288, 1022336, 318500, 50176, 3920, 128, 1;

%p T := (n, k) -> exp(1)*GAMMA(n - k + 1, 1)*binomial(n, k)^2:

%p seq(seq(simplify(T(n, k)), k = 0..n), n=0..8);

%p # Alternative:

%p A377661 := (n, k) -> n!*binomial(n,k)*add(1/(k!*(j-k)!), j = k..n):

%p for n from 0 to 8 do seq(A377661(n, k), k = 0..n) od;

%p # Or:

%p T := (n, k) -> binomial(n, k)^2 * KummerU(k - n, k - n, 1):

%p for n from 0 to 8 do seq(simplify(T(n, k)), k= 0..n) od;

%t T[n_, k_] := E Binomial[n, k]^2 Gamma[1 - k + n, 1];

%t Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten

%o (Python)

%o from math import comb, isqrt, factorial

%o def A377661(n):

%o     a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1))

%o     b = n-comb(a+1,2)

%o     fa, fb = factorial(a), factorial(b)

%o     return comb(a,b)*sum(fa//(fb*factorial(j-b)) for j in range(b,a+1)) # _Chai Wah Wu_, Nov 12 2024

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

%K nonn,tabl

%O 0,2

%A _Peter Luschny_, Nov 03 2024