A086618 -id:A086618 - cp's OEIS frontend

A349768 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * binomial(2*k,k) / (k+1).

1, 3, 19, 173, 1881, 22655, 291775, 3940725, 55149025, 793387235, 11668476579, 174735112997, 2656296912361, 40897718776647, 636588467802679, 10002872642155085, 158483629611962025, 2529389028336106475, 40631849127696017275, 656509442594976984405, 10663184061320964941761

Offset: 0

Ilya Gutkovskiy, Nov 29 2021

Cf. A000108, A086618, A086621, A243945.

A349768 := proc(n)
    hypergeom([1/2,-n,n+1],[1,2],-4) ;
    simplify(%) ;
end proc:
seq(A349768(n),n=0..20) ; # R. J. Mathar, Mar 02 2023

Table[Sum[Binomial[n, k] Binomial[n + k, k] Binomial[2 k, k]/(k + 1), {k, 0, n}], {n, 0, 20}]
Table[HypergeometricPFQ[{1/2, -n, n + 1}, {1, 2}, -4], {n, 0, 20}]

a(n) = sum(k=0, n, binomial(n,k)*binomial(n+k,k)*binomial(2*k,k)/(k+1)); \\ Michel Marcus, Nov 29 2021

From Vaclav Kotesovec, Nov 29 2021: (Start)
D-finite recurrence: n*(n+1)*(2*n - 3)*a(n) = (2*n - 1)*(19*n^2 - 37*n + 12)*a(n-1) - (2*n - 3)*(19*n^2 - 39*n + 14)*a(n-2) + (n-3)*(n-2)*(2*n - 1)*a(n-3).
a(n) ~ sqrt(5) * phi^(6*n + 3) / (8*Pi*n^2), where phi = A001622 is the golden ratio. (End)
D-finite with recurrence n*(n+1)*a(n) +(n+1)*(n-4)*a(n-1) +2*(-171*n^2 +512*n -388)*a(n-2) +2*(9*n^2 +296*n -796)*a(n-3) +(341*n^2 -2425*n +4320)*a(n-4) -19*(n-4)*(n-5)*a(n-5)=0. - R. J. Mathar, Mar 02 2023

A367178 Triangle read by rows. T(n, k) = binomial(n, k)^2 * CatalanNumber(k).

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 9, 18, 5, 1, 16, 72, 80, 14, 1, 25, 200, 500, 350, 42, 1, 36, 450, 2000, 3150, 1512, 132, 1, 49, 882, 6125, 17150, 18522, 6468, 429, 1, 64, 1568, 15680, 68600, 131712, 103488, 27456, 1430, 1, 81, 2592, 35280, 222264, 666792, 931392, 555984, 115830, 4862

Offset: 0

Peter Luschny, Nov 07 2023

			Triangle T(n, k) starts:
  [0] 1;
  [1] 1,  1;
  [2] 1,  4,    2;
  [3] 1,  9,   18,     5;
  [4] 1, 16,   72,    80,     14;
  [5] 1, 25,  200,   500,    350,     42;
  [6] 1, 36,  450,  2000,   3150,   1512,    132;
  [7] 1, 49,  882,  6125,  17150,  18522,   6468,    429;
  [8] 1, 64, 1568, 15680,  68600, 131712, 103488,  27456,   1430;
  [9] 1, 81, 2592, 35280, 222264, 666792, 931392, 555984, 115830, 4862;

Cf. A086618 (row sums), A186415 (central column), A000108 (main diagonal).

Cf. A098474, A367022, A367023, A387024, A387024, A367177.

T := (n, k) -> binomial(n, k)^2 * binomial(2*k, k) / (k + 1):
seq(seq(T(n, k), k = 0..n), n = 0..9);

T(n, k) = binomial(n, k)^2 * binomial(2*k, k) / (k + 1).

T(n, k) = [x^n] hypergeom([1/2, -n, -n], [1, 2], 4*x).

cp's OEIS Frontend

A349768 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * binomial(2*k,k) / (k+1).

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