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

Author

Seiichi Manyama, Feb 01 2021

Keywords

• nonn
• tabl

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

Columns k=0..3 give A000012, A026375, A084771, A340973.

Rows n=0..2 give A000012, A005408, A080859.

Main diagonal gives A340971.

Cf. A340968.

Programs

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));

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)