A386362 -id:A386362 - cp's OEIS frontend

A385716 Expansion of 1/((1-x) * (1-13*x))^(3/2).

1, 21, 348, 5320, 78135, 1120287, 15805972, 220445316, 3047961735, 41857891075, 571725145992, 7774356136092, 105324231178621, 1422411298153125, 19157947746089520, 257427540725705056, 3451990965984505251, 46205867184493459023, 617482101788090727220, 8239952016851603641320

Offset: 0

Seiichi Manyama, Aug 19 2025

Paolo Xausa, Table of n, a(n) for n = 0..800

Cf. A331516, A385563.

Cf. A340973, A386362.

Module[{x}, CoefficientList[Series[1/((1-x)*(1-13*x))^(3/2), {x, 0, 25}], x]] (* Paolo Xausa, Aug 25 2025 *)

my(N=20, x='x+O('x^N)); Vec(1/((1-x)*(1-13*x))^(3/2))

n*a(n) = (14*n+7)*a(n-1) - 13*(n+1)*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 13^k * (2*k+1) * (2*(n-k)+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} 3^k * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = Sum_{k=0..n} (-3)^k * 13^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = binomial(n+2,2) * A386362(n).
a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 9^k * 7^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = Sum_{k=0..n} (7/2)^k * (-13/14)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k).
a(n) ~ sqrt(3*n) * 13^(n + 3/2) / (36*sqrt(Pi)). - Vaclav Kotesovec, Aug 21 2025

A386389 Expansion of (1/x) * Series_Reversion( x/(1+9x+16x^2) ).

Original entry on oeis.org

1, 9, 97, 1161, 14849, 198729, 2748641, 38977353, 563644673, 8280210825, 123226850913, 1853870946057, 28148395838721, 430791367720905, 6638484468424929, 102918165951351753, 1604104541561284097, 25121009971212463881, 395085505395126968417, 6237523016309454855561

Offset: 0

Seiichi Manyama, Aug 20 2025

nonn

Column k=4 of A386408.
Cf. A002212, A127846, A386362.
Cf. A000108, A269796, A386387.

my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+9*x+16*x^2))/x)

G.f.: 2/(1 - 9*x + sqrt((1-x) * (1-17*x))).
a(n) = (A386387(n+1) - A386387(n))/4.
(n+2)*a(n) = 9*(2*n+1)*a(n-1) - 17*(n-1)*a(n-2) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 16^k * 9^(n-2*k) * binomial(n,2*k) * Catalan(k).
a(n) = Sum_{k=0..n} 4^k * binomial(n,k) * Catalan(k+1).

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

A385716 Expansion of 1/((1-x) * (1-13*x))^(3/2).

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A386389 Expansion of (1/x) * Series_Reversion( x/(1+9x+16x^2) ).

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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

A385716 Expansion of 1/((1-x) * (1-13*x))^(3/2).

Views

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Mathematica

PARI

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A386389 Expansion of (1/x) * Series_Reversion( x/(1+9*x+16*x^2) ).

Views

Author

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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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A386389 Expansion of (1/x) * Series_Reversion( x/(1+9x+16x^2) ).