A338979 -id:A338979 - cp's OEIS frontend

A337167 a(n) = 1 + 3 * Sum_{k=0..n-1} a(k) * a(n-k-1).

1, 4, 25, 199, 1795, 17422, 177463, 1870960, 20241403, 223438852, 2506596547, 28494103183, 327507800725, 3799735202218, 44440058006593, 523388751658831, 6201937444137619, 73888034816382820, 884517283667145259, 10634234680321209373, 128347834921058404249

Offset: 0

Ilya Gutkovskiy, Jan 28 2021

nonn

Binomial transform of A005159.

Seiichi Manyama, Table of n, a(n) for n = 0..901
N. J. A. Sloane, Transforms

Column k=3 of A340968.

Cf. A000108, A005159, A007317, A162326, A337169, A338979.

a[n_] := a[n] = 1 + 3 Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 20}]
Table[Sum[Binomial[n, k] 3^k CatalanNumber[k], {k, 0, n}], {n, 0, 20}]
Table[Hypergeometric2F1[1/2, -n, 2, -12], {n, 0, 20}]

{a(n) = sum(k=0, n, 3^k*binomial(n, k)*(2*k)!/(k!*(k+1)!))} \\ Seiichi Manyama, Jan 31 2021

my(N=20, x='x+O('x^N)); Vec(2/(1-x+sqrt((1-x)*(1-13*x)))) \\ Seiichi Manyama, Feb 01 2021

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + 3*x*A(x)^2.
G.f.: (1 - sqrt(1 - 12*x / (1 - x))) / (6*x).
E.g.f.: exp(7*x) * (BesselI(0,6*x) - BesselI(1,6*x)).
a(n) = Sum_{k=0..n} binomial(n,k) * 3^k * Catalan(k).
a(n) = 2F1([1/2, -n], [2], -12), where 2F1 is the hypergeometric function.
D-finite with recurrence (n+1) * a(n) = 2 * (7*n-3) * a(n-1) - 13 * (n-1) * a(n-2) for n > 1. - Seiichi Manyama, Jan 31 2021
a(n) ~ 13^(n + 3/2) / (8 * 3^(3/2) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 14 2021

A340968 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^jbinomial(n,j)Catalan(j).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 13, 15, 1, 1, 5, 25, 71, 51, 1, 1, 6, 41, 199, 441, 188, 1, 1, 7, 61, 429, 1795, 2955, 731, 1, 1, 8, 85, 791, 5073, 17422, 20805, 2950, 1, 1, 9, 113, 1315, 11571, 64469, 177463, 151695, 12235, 1

Offset: 0

Seiichi Manyama, Jan 31 2021

			Square array begins:
  1,   1,    1,     1,     1,      1, ...
  1,   2,    3,     4,     5,      6, ...
  1,   5,   13,    25,    41,     61, ...
  1,  15,   71,   199,   429,    791, ...
  1,  51,  441,  1795,  5073,  11571, ...
  1, 188, 2955, 17422, 64469, 181776, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..4 give A000012, A007317(n+1), A162326(n+1), A337167, A386387.
Main diagonal gives A338979.
Cf. A000108, A290605, A340970.

T_row := n -> k -> hypergeom([1/2, -n], [2], -4*k): for n from 0 to 6 do seq(simplify(T_row(n)(k)), k = 0..6) od; # Peter Luschny, Aug 27 2025

T[n_, k_] := Sum[If[j == k == 0, 1, k^j] * Binomial[n, j] * CatalanNumber[j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 01 2021 *)
A340968[n_, k_] := Hypergeometric2F1[1/2, -n, 2, -4*k]; Table[A340968[n, k], {n, 0, 6}, {k, 0, 7}] (* row-wise *) (* Peter Luschny, Aug 27 2025 *)

T(n, k) = sum(j=0, n, k^j*binomial(n, j)*(2*j)!/(j!*(j+1)!));

T(n, k) = 1+k*sum(j=0, n-1, T(j, k)*T(n-1-j, k));

G.f. A_k(x) of column k satisfies A_k(x) = 1/(1 - x) + k*x*A_k(x)^2.
A_k(x) = 2/( 1 - x + sqrt((1 - x) * (1 - (4*k+1)*x)) ).
T(n,k) = 1 + k * Sum_{j=0..n-1} T(j,k) * T(n-1-j,k).
(n+1) * T(n,k) = 2 * ((2*k+1) * n - k) * 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) - BesselI(1,2*k*x)). - Ilya Gutkovskiy, Feb 01 2021
T_row(n) = k -> hypergeom([1/2, -n], [2], -4*k). - Peter Luschny, Aug 27 2025

A340971 a(n) = Sum_{k=0..n} n^k * binomial(n,k) * binomial(2*k,k).

Original entry on oeis.org

1, 3, 33, 721, 23649, 1032801, 56317969, 3682424775, 280767441537, 24456613613401, 2395993939827201, 260764460901476643, 31213273328323059169, 4075382667781540713807, 576394007453263029232497

Offset: 0

Seiichi Manyama, Feb 01 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..322

Main diagonal of A340970.

Cf. A338979, A340972.

a[0] = 1; a[n_] := Sum[n^k * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]; Array[a, 15, 0] (* Amiram Eldar, Feb 01 2021 *)

a(n) = sum(k=0, n, n^k*binomial(n, k)*binomial(2*k, k));

a(n) = polcoef(1/sqrt((1-x)*(1-(4*n+1)*x)+x*O(x^n)), n);

a(n) = polcoef((1+(2*n+1)*x+(n*x)^2)^n, n);

a(n) = [x^n] 1/sqrt((1-x)*(1-(4*n+1)*x)).
a(n) = [x^n] (1+(2*n+1)*x+(n*x)^2)^n.
a(n) = n! * [x^n] exp((2*n+1)*x) * BesselI(0,2*n*x). - Ilya Gutkovskiy, Feb 01 2021
a(n) ~ exp(1/4) * 4^n * n^(n - 1/2) / sqrt(Pi). - Vaclav Kotesovec, Nov 13 2021
From Seiichi Manyama, Aug 19 2025: (Start)
a(n) = (1/4)^n * Sum_{k=0..n} (4*n+1)^k * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} (-n)^k * (4*n+1)^(n-k) * binomial(n,k) * binomial(2*k,k). (End)

A339001 a(n) = (-1)^n * Sum_{k=0..n} (-n)^k * binomial(n,k) * Catalan(k).

Original entry on oeis.org

1, 0, 5, 89, 2481, 93274, 4450645, 258297570, 17689681345, 1397903887808, 125286890408901, 12562851683433765, 1393925069404093105, 169595051215441936902, 22454465186157134883285

Offset: 0

Seiichi Manyama, Jan 31 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..322

Cf. A000108, A005043, A337168, A337169, A338979.

a[0] = 1; a[n_] := (-1)^n * Sum[(-n)^k * Binomial[n, k] * CatalanNumber[k], {k, 0, n}]; Array[a, 15, 0] (* Amiram Eldar, Feb 01 2021 *)

{a(n) = (-1)^n*sum(k=0, n, (-n)^k*binomial(n, k)*(2*k)!/(k!*(k+1)!))}

a(n) = n! * [x^n] exp((2*n-1)*x) * (BesselI(0,2*n*x) - BesselI(1,2*n*x)). - Ilya Gutkovskiy, Feb 01 2021

a(n) ~ exp(-1/4) * 4^n * n^(n - 3/2) / sqrt(Pi). - Vaclav Kotesovec, Feb 14 2021

A386432 a(n) = Sum_{k=0..n} n^k * binomial(n,k) * Catalan(k+1).

Original entry on oeis.org

1, 3, 29, 532, 14849, 562551, 27053749, 1581258225, 108965790593, 8657148898585, 779508506302701, 78480330282178738, 8738801236865140417, 1066555304017996550265, 141604665239501105707269, 20321162053065050407161076, 3134730687100285268294654465, 517309567362171441488395248225

Offset: 0

Seiichi Manyama, Aug 20 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Main diagonal of A386408.

Cf. A000108, A338979.

[&+[n^k*Binomial(n, k) * Catalan (k+1): k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Aug 22 2025

Table[Sum[(n^k/. 0^0->1)*Binomial[n,k]*CatalanNumber[k+1],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 22 2025 *)

a(n) = sum(k=0, n, n^k*binomial(n, k)*(2*(k+1))!/((k+1)!*(k+2)!));

a(n) = Sum_{k=0..floor(n/2)} n^(2*k) * (2*n+1)^(n-2*k) * binomial(n,2*k) * Catalan(k).

cp's OEIS Frontend

A337167 a(n) = 1 + 3 * Sum_{k=0..n-1} a(k) * a(n-k-1).

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A340968 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)*Catalan(j).

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A340971 a(n) = Sum_{k=0..n} n^k * binomial(n,k) * binomial(2*k,k).

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A339001 a(n) = (-1)^n * Sum_{k=0..n} (-n)^k * binomial(n,k) * Catalan(k).

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A386432 a(n) = Sum_{k=0..n} n^k * binomial(n,k) * Catalan(k+1).

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A340968 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^jbinomial(n,j)Catalan(j).