A340970 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^j * binomial(n,j) * binomial(2*j,j).
1, 1, 1, 1, 3, 1, 1, 5, 11, 1, 1, 7, 33, 45, 1, 1, 9, 67, 245, 195, 1, 1, 11, 113, 721, 1921, 873, 1, 1, 13, 171, 1593, 8179, 15525, 3989, 1, 1, 15, 241, 2981, 23649, 95557, 127905, 18483, 1, 1, 17, 323, 5005, 54691, 361449, 1137709, 1067925, 86515, 1
Offset: 0
Examples
Square array begins: 1, 1, 1, 1, 1, 1, ... 1, 3, 5, 7, 9, 11, ... 1, 11, 33, 67, 113, 171, ... 1, 45, 245, 721, 1593, 2981, ... 1, 195, 1921, 8179, 23649, 54691, ... 1, 873, 15525, 95557, 361449, 1032801, ...
Links
- Seiichi Manyama, Antidiagonals n = 0..139, flattened
Crossrefs
Programs
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Mathematica
T[n_, k_] := Sum[If[j == k == 0, 1, k^j] * Binomial[n, j] * Binomial[2*j, j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 01 2021 *) -
PARI
T(n, k) = sum(j=0, n, k^j*binomial(n, j)*binomial(2*j, j));
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PARI
T(n, k) = polcoef((1+(2*k+1)*x+(k*x)^2)^n, n);
Formula
G.f. of column k: 1/sqrt((1 - x) * (1 - (4*k+1)*x)).
T(n,k) = [x^n] (1+(2*k+1)*x+(k*x)^2)^n.
n * T(n,k) = (2*k+1) * (2*n-1) * T(n-1,k) - (4*k+1) * (n-1) * T(n-2,k) for n > 1.
E.g.f. of column k: exp((2*k+1)*x) * BesselI(0,2*k*x). - Ilya Gutkovskiy, Feb 01 2021
From Seiichi Manyama, Aug 19 2025: (Start)
T(n,k) = (1/4)^n * Sum_{j=0..n} (4*k+1)^j * binomial(2*j,j) * binomial(2*(n-j),n-j).
T(n,k) = Sum_{j=0..n} (-k)^j * (4*k+1)^(n-j) * binomial(n,j) * binomial(2*j,j). (End)