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A141905
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Triangle read by rows, T(n, k) = binomial(n, k)*A052509(n, k) for 0 <= k <= n.
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1
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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
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OFFSET
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0,5
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COMMENTS
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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
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LINKS
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FORMULA
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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
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EXAMPLE
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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;
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MAPLE
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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
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MATHEMATICA
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T[n_, k_]:= Sum[n!/((n-k-j)!*j!*k!), {j, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten
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PROG
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(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
(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
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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