A340973 -id:A340973 - 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

A340970 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) * binomial(2*j,j).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 11, 1, 1, 7, 33, 45, 1, 1, 9, 67, 245, 195, 1, 1, 11, 113, 721, 1921, 873, 1, 1, 13, 171, 1593, 8179, 15525, 3989, 1, 1, 15, 241, 2981, 23649, 95557, 127905, 18483, 1, 1, 17, 323, 5005, 54691, 361449, 1137709, 1067925, 86515, 1

Offset: 0

Seiichi Manyama, Feb 01 2021

			Square array begins:
  1,   1,     1,     1,      1,       1, ...
  1,   3,     5,     7,      9,      11, ...
  1,  11,    33,    67,    113,     171, ...
  1,  45,   245,   721,   1593,    2981, ...
  1, 195,  1921,  8179,  23649,   54691, ...
  1, 873, 15525, 95557, 361449, 1032801, ...

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

Columns k=0..3 give A000012, A026375, A084771, A340973.
Rows n=0..2 give A000012, A005408, A080859.
Main diagonal gives A340971.
Cf. A340968.

T[n_, k_] := Sum[If[j == k == 0, 1, k^j] * Binomial[n, j] * Binomial[2*j, j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 01 2021 *)

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

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

G.f. of column k: 1/sqrt((1 - x) * (1 - (4*k+1)*x)).
T(n,k) = [x^n] (1+(2*k+1)*x+(k*x)^2)^n.
n * T(n,k) = (2*k+1) * (2*n-1) * 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). - Ilya Gutkovskiy, Feb 01 2021
From Seiichi Manyama, Aug 19 2025: (Start)
T(n,k) = (1/4)^n * Sum_{j=0..n} (4*k+1)^j * binomial(2*j,j) * binomial(2*(n-j),n-j).
T(n,k) = Sum_{j=0..n} (-k)^j * (4*k+1)^(n-j) * binomial(n,j) * binomial(2*j,j). (End)

A383951 Expansion of 1/sqrt((1-x)^3 * (1-13*x)).

Original entry on oeis.org

1, 8, 75, 796, 8975, 104532, 1242241, 14967680, 182172627, 2234388520, 27572562017, 341929238196, 4257601409425, 53196292831052, 666600870098895, 8374220026541296, 105432936550339523, 1329984626694890760, 16805828389675759921, 212684693606424187460, 2695304533342226489229

Offset: 0

Seiichi Manyama, Aug 19 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Partial sums of A340973.

R := PowerSeriesRing(Rationals(), 34); f := 1/Sqrt((1- x)^3 * (1-13*x)); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 27 2025

CoefficientList[Series[1/Sqrt[(1-x)^3*(1-13*x)],{x,0,33}],x] (* Vincenzo Librandi, Aug 27 2025 *)

my(N=30, x='x+O('x^N)); Vec(1/sqrt((1-x)^3*(1-13*x)))

n*a(n) = (14*n-6)*a(n-1) - 13*n*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 13^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} (-3)^k * 13^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+1,n-k).
a(n) = Sum_{k=0..n} 3^k * binomial(2*k,k) * binomial(n+1,n-k).

cp's OEIS Frontend

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

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A340970 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) * binomial(2*j,j).

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A383951 Expansion of 1/sqrt((1-x)^3 * (1-13*x)).

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