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

%I A141905 #38 Jan 31 2025 23:44:40
%S A141905 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,
%T A141905 1,49,336,525,280,84,14,1,1,64,616,1456,1120,448,112,16,1,1,81,1044,
%U A141905 3528,3906,2016,672,144,18,1,1,100,1665,7680,11970,8064,3360,960,180,20,1
%N A141905 Triangle read by rows, T(n, k) = binomial(n, k)*A052509(n, k) for 0 <= k <= n.
%C A141905 Original definition: A skew trinomial summed triangular sequence of coefficients: T(n, k) = Sum_{j=0..k} n!/((n - k - j)!*j!*k!).
%C A141905 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
%C A141905 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
%H A141905 G. C. Greubel, <a href="/A141905/b141905.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A141905 T(n, k) = Sum_{j=0..k} n!/((n - k - j)!*j!*k!).
%F A141905 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
%F A141905 T(n, k) = binomial(n, k)*hypergeom([-k, -n + k], [-k], -1). - _Peter Luschny_, Nov 28 2021
%e A141905 Triangle begins as:
%e A141905 [0]  1;
%e A141905 [1]  1,   1;
%e A141905 [2]  1,   4,    1;
%e A141905 [3]  1,   9,    6,    1;
%e A141905 [4]  1,  16,   24,    8,     1;
%e A141905 [5]  1,  25,   70,   40,    10,    1;
%e A141905 [6]  1,  36,  165,  160,    60,   12,    1;
%e A141905 [7]  1,  49,  336,  525,   280,   84,   14,   1;
%e A141905 [8]  1,  64,  616, 1456,  1120,  448,  112,  16,   1;
%e A141905 [9]  1,  81, 1044, 3528,  3906, 2016,  672, 144,  18,  1;
%p A141905 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
%t A141905 T[n_, k_]:= Sum[n!/((n-k-j)!*j!*k!), {j,0,k}];
%t A141905 Table[T[n, k], {n, 0, 10}, {k,0,n}] // Flatten
%o A141905 (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
%o A141905 (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
%Y A141905 Row sums are A027914.
%Y A141905 Cf. A038207, A052509.
%K A141905 nonn,tabl
%O A141905 0,5
%A A141905 _Roger L. Bagula_ and _Gary W. Adamson_, Sep 14 2008
%E A141905 Edited by _G. C. Greubel_, Mar 29 2021
%E A141905 New name by _Peter Luschny_, Nov 26 2021