cp's OEIS Frontend

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

%I A377661 #14 Nov 12 2024 20:20:59
%S A377661 1,2,1,5,8,1,16,45,18,1,65,256,180,32,1,326,1625,1600,500,50,1,1957,
%T A377661 11736,14625,6400,1125,72,1,13700,95893,143766,79625,19600,2205,98,1,
%U A377661 109601,876800,1534288,1022336,318500,50176,3920,128,1
%N A377661 Triangle read by rows: T(n, k) = e*Gamma(n - k + 1, 1)*binomial(n, k)^2.
%F A377661 T(n, k) = binomial(n, k)*Sum_{j=k..n} n!/(k!*(j-k)!).
%F A377661 T(n, k) = binomial(n, k)^2 * KummerU(k - n, k - n, 1).
%F A377661 T(n, k) = binomial(n, k) * A073107(n, k).
%e A377661 Triangle starts:
%e A377661 [0]      1;
%e A377661 [1]      2,      1;
%e A377661 [2]      5,      8,       1;
%e A377661 [3]     16,     45,      18,       1;
%e A377661 [4]     65,    256,     180,      32,      1;
%e A377661 [5]    326,   1625,    1600,     500,     50,     1;
%e A377661 [6]   1957,  11736,   14625,    6400,   1125,    72,    1;
%e A377661 [7]  13700,  95893,  143766,   79625,  19600,  2205,   98,   1;
%e A377661 [8] 109601, 876800, 1534288, 1022336, 318500, 50176, 3920, 128, 1;
%p A377661 T := (n, k) -> exp(1)*GAMMA(n - k + 1, 1)*binomial(n, k)^2:
%p A377661 seq(seq(simplify(T(n, k)), k = 0..n), n=0..8);
%p A377661 # Alternative:
%p A377661 A377661 := (n, k) -> n!*binomial(n,k)*add(1/(k!*(j-k)!), j = k..n):
%p A377661 for n from 0 to 8 do seq(A377661(n, k), k = 0..n) od;
%p A377661 # Or:
%p A377661 T := (n, k) -> binomial(n, k)^2 * KummerU(k - n, k - n, 1):
%p A377661 for n from 0 to 8 do seq(simplify(T(n, k)), k= 0..n) od;
%t A377661 T[n_, k_] := E Binomial[n, k]^2 Gamma[1 - k + n, 1];
%t A377661 Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
%o A377661 (Python)
%o A377661 from math import comb, isqrt, factorial
%o A377661 def A377661(n):
%o A377661     a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1))
%o A377661     b = n-comb(a+1,2)
%o A377661     fa, fb = factorial(a), factorial(b)
%o A377661     return comb(a,b)*sum(fa//(fb*factorial(j-b)) for j in range(b,a+1)) # _Chai Wah Wu_, Nov 12 2024
%Y A377661 Cf. A000522 (column 0), A001105 (subdiagonal), A377662 (row sums), A073107.
%K A377661 nonn,tabl
%O A377661 0,2
%A A377661 _Peter Luschny_, Nov 03 2024