A141905 Triangle read by rows, T(n, k) = binomial(n, k)*A052509(n, k) for 0 <= k <= n.

1, 1, 1, 1, 4, 1, 1, 9, 6, 1, 1, 16, 24, 8, 1, 1, 25, 70, 40, 10, 1, 1, 36, 165, 160, 60, 12, 1, 1, 49, 336, 525, 280, 84, 14, 1, 1, 64, 616, 1456, 1120, 448, 112, 16, 1, 1, 81, 1044, 3528, 3906, 2016, 672, 144, 18, 1, 1, 100, 1665, 7680, 11970, 8064, 3360, 960, 180, 20, 1

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Sep 14 2008

Keywords

• nonn
• tabl

Comments

Original definition: A skew trinomial summed triangular sequence of coefficients: T(n, k) = Sum_{j=0..k} n!/((n - k - j)!*j!*k!).

It is obscure how the above formula is used for the region where the sum reaches k > n-m, which needs a definition of the factorials at negative integer argument. If we trust the author's Mma implementation, Mma throws in some magic renormalization to cover these arguments. If we define, properly, t(n, k) = Sum_{j=0..n-k} n!/((n-k-j)!*j!*k!), then we recover just A038207. - R. J. Mathar, Feb 07 2014

Let p(n, k, j) = n!/((n-k-j)!*j!*k!), for j<=n-k and 0<= k <=n and p(n, k, j) = 0, for j > n-k and 0<= k <=n. It seems that T(n, k) coincides with Sum_{j=0..k} p(n, k, j). - Luis Manuel Rivera Martínez, Mar 04 2014

Examples

			Triangle begins as:
[0]  1;
[1]  1,   1;
[2]  1,   4,    1;
[3]  1,   9,    6,    1;
[4]  1,  16,   24,    8,     1;
[5]  1,  25,   70,   40,    10,    1;
[6]  1,  36,  165,  160,    60,   12,    1;
[7]  1,  49,  336,  525,   280,   84,   14,   1;
[8]  1,  64,  616, 1456,  1120,  448,  112,  16,   1;
[9]  1,  81, 1044, 3528,  3906, 2016,  672, 144,  18,  1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Row sums are A027914.

Cf. A038207, A052509.

Programs

Magma

[Binomial(n,k)*(&+[Binomial(n-k,j): j in [0..k]]): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 29 2021

Maple

A052509 := proc(n, k) option remember: if k = 0 or k = n then 1 else A052509(n-1, k) + A052509(n-2, k-1) fi end: T := (n, k) -> binomial(n, k)*A052509(n, k): seq(seq(T(n, k), k=0..n), n=0..10); # Peter Luschny, Nov 26 2021

Mathematica

T[n_, k_]:= Sum[n!/((n-k-j)!*j!*k!), {j,0,k}];
Table[T[n, k], {n, 0, 10}, {k,0,n}] // Flatten

Sage

flatten([[binomial(n,k)*sum(binomial(n-k,j) for j in (0..k)) for k in [0..n]] for n in [0..12]]) # G. C. Greubel, Mar 29 2021

Formula

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

G.f.: (2*x)/((3*x - 1)*sqrt(-4*x^2*y + x^2 - 2*x + 1) - 4*x^2*y + x^2 - 2*x +1). - Vladimir Kruchinin, Oct 05 2020

T(n, k) = binomial(n, k)*hypergeom([-k, -n + k], [-k], -1). - Peter Luschny, Nov 28 2021

Extensions

Edited by G. C. Greubel, Mar 29 2021

New name by Peter Luschny, Nov 26 2021