A338022 Triangle T(n,m) = Sum_{i=0..n-m} C(2*m-1,n-i-1)*C(n-i-1,i)*C(n-i-1,n-m-i)/(2*m-1).

1, 1, 1, 1, 3, 2, 0, 5, 10, 5, 0, 5, 26, 35, 14, 0, 3, 44, 125, 126, 42, 0, 1, 52, 295, 574, 462, 132, 0, 0, 44, 505, 1736, 2562, 1716, 429, 0, 0, 26, 655, 3864, 9450, 11220, 6435, 1430, 0, 0, 10, 655, 6664, 25830, 48840, 48477, 24310, 4862

Offset

1,5

Links

Table of n, a(n) for n=1..55.

Formula

G.f.: (1-sqrt(1-4*x*(x+1)*(x^2+x+1)*y))/(2*(x+1)).

Example

The triangle begins
  1,
  1, 1,
  1, 3,  2,
  0, 5, 10,   5,
  0, 5, 26,  35,    14,
  0, 3, 44, 125,   126,    42,
  0, 1, 52, 295,   574,   462,    132,
  0, 0, 44, 505,  1736,  2562,   1716,    429,
  0, 0, 26, 655,  3864,  9450,  11220,   6435,    1430,
  0, 0, 10, 655,  6664, 25830,  48840,  48477,   24310,   4862,
  0, 0,  2, 505,  9156, 55314, 158136, 243243,  207350,  92378,  16796,
  0, 0,  0, 295, 10164, 95844, 403656, 909909, 1178320, 880022, ...
From Peter Luschny, Oct 07 2020: (Start)
Let C(n) denote the Catalan numbers, then the columns start, written as rows,
C(0)*[1, 1,  1],
C(1)*[1, 3,  5,   5,   3,   1],
C(2)*[1, 5, 13,  22,  26,  22,  13,   5,   1],
C(3)*[1, 7, 25,  59, 101, 131, 131, 101,  59,  25,   7,   1],
C(4)*[1, 9, 41, 124, 276, 476, 654, 726, 654, 476, 276, 124, 41, 9, 1], ... . (End)

Mathematica

T[n_, m_] := Sum[Binomial[2*m - 1, n - i - 1] * Binomial[n - i - 1, i] * Binomial[n - i - 1, n - m - i]/(2*m - 1), {i, 0, n - m}]; Table[T[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Amiram Eldar, Oct 07 2020 *)

Prog

(Maxima)
T(n, m):= sum(binomial(2*m-1, n-i-1)*binomial(n-i-1, i)*binomial(n-i-1, n-m-i), i, 0, n-m)/(2*m-1);

Crossrefs

Cf. A000108 (main diagonal), A001700 (1st subdiagonal).

Sequence in context: A246773 A359843 A364361 * A253176 A079408 A114376

Adjacent sequences: A338019 A338020 A338021 * A338023 A338024 A338025

Keyword

nonn,tabl

Author

Vladimir Kruchinin, Oct 06 2020

Status

approved