A342313 T(n, k) = (n + k - 1)*(n + k)*binomial(2*n + 1, n - k + 1) with T(0, 0) = T(1, 0) = 1. Triangle read by rows, T(n, k) for 0 <= k <= n.

1, 1, 6, 20, 60, 60, 210, 420, 420, 210, 1512, 2520, 2520, 1512, 504, 9240, 13860, 13860, 9240, 3960, 990, 51480, 72072, 72072, 51480, 25740, 8580, 1716, 270270, 360360, 360360, 270270, 150150, 60060, 16380, 2730, 1361360, 1750320, 1750320, 1361360, 816816, 371280, 123760, 28560, 4080

Offset: 0

Author

Peter Luschny, Mar 08 2021

Keywords

• nonn
• tabl
• frac

Comments

The triangle can be seen as representing the denominators of a sequence of rational polynomials. Let p_{n}(x) = Sum_{k=0..n} (A342312(n, k)/T(n, k))*x^k. Then p_{n}(1) = B_{n}(1), where B_{n}(x) are the Bernoulli polynomials. See A342312 for a formula for the polynomials.

Examples

			Triangle starts:
[0] 1
[1] 1,       6
[2] 20,      60,      60
[3] 210,     420,     420,     210
[4] 1512,    2520,    2520,    1512,    504
[5] 9240,    13860,   13860,   9240,    3960,   990
[6] 51480,   72072,   72072,   51480,   25740,  8580,   1716
[7] 270270,  360360,  360360,  270270,  150150, 60060,  16380,  2730
[8] 1361360, 1750320, 1750320, 1361360, 816816, 371280, 123760, 28560, 4080
		

Crossrefs

Cf. A069072 (main diagonal), A342312 (numerators).

Programs

Maple

T := (n, k) -> `if`(n=0, 1,`if`(n=1 and k=0, 1,
(n + k - 1)*(n + k)*binomial(2*n + 1, n - k + 1))):
seq(print(seq(T(n, k), k = 0..n)), n = 0..8);

Mathematica

T[0, 0] := 1; T[1, 0] := 1;
T[n_, k_] := (n - 1 + k) (n + k) Binomial[2n + 1, n - k + 1];
Table[T[n, k], {n, 0, 8}, {k, 0, n}]