A386389 -id:A386389 - cp's OEIS frontend

A386362 Expansion of (1/x) * Series_Reversion( x/(1+7x+9x^2) ).

1, 7, 58, 532, 5209, 53347, 564499, 6123481, 67732483, 761052565, 8662502212, 99671232514, 1157409133831, 13546774268125, 159649564550746, 1892849564159596, 22562032457415067, 270209749616920813, 3249905798884688038, 39237866746912398292, 475388228365424562019

Offset: 0

Seiichi Manyama, Aug 20 2025

nonn

Column k=3 of A386408.
Cf. A002212, A127846, A386389.
Cf. A000108, A337167, A385716.

my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+7*x+9*x^2))/x)

G.f.: 2/(1 - 7*x + sqrt((1-x) * (1-13*x))).
a(n) = (A337167(n+1) - A337167(n))/3.
(n+2)*a(n) = 7*(2*n+1)*a(n-1) - 13*(n-1)*a(n-2) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 9^k * 7^(n-2*k) * binomial(n,2*k) * Catalan(k).
a(n) = Sum_{k=0..n} 3^k * binomial(n,k) * Catalan(k+1).

A386387 a(n) = Sum_{k=0..n} 4^k * binomial(n,k) * Catalan(k).

Original entry on oeis.org

1, 5, 41, 429, 5073, 64469, 859385, 11853949, 167763361, 2422342053, 35543185353, 528450589005, 7943934373233, 120537517728117, 1843702988611737, 28397640862311453, 440070304667718465, 6856488470912854853, 107340528355762710377, 1687682549936270584045

Offset: 0

Seiichi Manyama, Aug 20 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..600

Column k=4 of A340968.

Cf. A386389, A007317.

[&+[4^k*Binomial(n,k) * Catalan(k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 27 2025

Table[Sum[4^k*Binomial[n,k]*CatalanNumber[k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 27 2025 *)
A386387[n_] := Hypergeometric2F1[1/2, -n, 2, -16]; Table[A386387[n], {n, 0, 19}]  (* Peter Luschny, Aug 27 2025 *)

a(n) = sum(k=0, n, 4^k*binomial(n, k)*(2*k)!/(k!*(k+1)!));

G.f.: 2/(1 - x + sqrt((1-x) * (1-17*x))).
G.f. A(x) satisfies A(x) = 1/(1 - x) + 4*x*A(x)^2.
a(n) = 1 + 4 * Sum_{k=0..n-1} a(k) * a(n-1-k).
(n+1)*a(n) = (18*n-8)*a(n-1) - 17*(n-1)*a(n-2) for n > 1.
a(n) ~ 17^(n + 3/2) / (64*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 20 2025
a(n) = hypergeom([1/2, -n], [2], -16). - Peter Luschny, Aug 27 2025

A386408 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) = Sum_{j=0..n} k^j * binomial(n,j) * Catalan(j+1).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 10, 1, 1, 7, 29, 36, 1, 1, 9, 58, 185, 137, 1, 1, 11, 97, 532, 1257, 543, 1, 1, 13, 146, 1161, 5209, 8925, 2219, 1, 1, 15, 205, 2156, 14849, 53347, 65445, 9285, 1, 1, 17, 274, 3601, 34041, 198729, 564499, 491825, 39587, 1

Offset: 0

Seiichi Manyama, Aug 20 2025

			Square array begins:
  1,    1,     1,      1,       1,       1,        1, ...
  1,    3,     5,      7,       9,      11,       13, ...
  1,   10,    29,     58,      97,     146,      205, ...
  1,   36,   185,    532,    1161,    2156,     3601, ...
  1,  137,  1257,   5209,   14849,   34041,    67657, ...
  1,  543,  8925,  53347,  198729,  562551,  1330693, ...
  1, 2219, 65445, 564499, 2748641, 9608811, 27053749, ...

Columns k=0..4 give A000012, A002212(n+1), A127846(n+1), A386362, A386389.
Main diagonal gives A386432.
Cf. A000108, A340968.

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

G.f. of column k: (1/x) * Series_Reversion( x/(1+(2*k+1)*x+(k*x)^2) ).
G.f. of column k: 2/(1 - (2*k+1)*x + sqrt((1-x) * (1-(4*k+1)*x))).
A(n,k) = (A340968(n+1,k) - A340968(n,k))/k for k > 0.
(n+2)*A(n,k) = (2*k+1)*(2*n+1)*A(n-1,k) - (4*k+1)*(n-1)*A(n-2,k) for n > 1.
A(n,k) = Sum_{j=0..floor(n/2)} k^(2*j) * (2*k+1)^(n-2*j) * binomial(n,2*j) * Catalan(j).

cp's OEIS Frontend

A386362 Expansion of (1/x) * Series_Reversion( x/(1+7x+9x^2) ).

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A386387 a(n) = Sum_{k=0..n} 4^k * binomial(n,k) * Catalan(k).

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A386408 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) = Sum_{j=0..n} k^j * binomial(n,j) * Catalan(j+1).

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cp's OEIS Frontend

A386362 Expansion of (1/x) * Series_Reversion( x/(1+7*x+9*x^2) ).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A386387 a(n) = Sum_{k=0..n} 4^k * binomial(n,k) * Catalan(k).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A386408 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) = Sum_{j=0..n} k^j * binomial(n,j) * Catalan(j+1).

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Examples

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PARI

Formula

A386362 Expansion of (1/x) * Series_Reversion( x/(1+7x+9x^2) ).