A163872 -id:A163872 - cp's OEIS frontend

A163869 Binomial transform of the beta numbers 1/beta(n+1,n+1) (A002457).

1, 7, 43, 249, 1395, 7653, 41381, 221399, 1175027, 6196725, 32512401, 169863147, 884318973, 4589954619, 23761814955, 122735222505, 632698778835, 3255832730565, 16728131746145, 85826852897675, 439793834236745, 2251006269442815, 11509340056410735, 58790764269668805

Offset: 0

Peter Luschny, Aug 06 2009

Also a(n) = Sum_{i=0..n} binomial(n,n-i) (2*i+1)$ where i$ denotes the swinging factorial of i (A056040).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.

Cf. A163842, A163872, A387208.

a := proc(n) local i; add(binomial(n,i)/Beta(i+1,i+1), i=0..n) end:

CoefficientList[Series[-Sqrt[x-1]/(5*x-1)^(3/2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 21 2012 *)
sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[n_] := Sum[ Binomial[n, n-i]*sf[2*i+1], {i, 0, n}]; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Jul 26 2013 *)
Table[Hypergeometric2F1[3/2, -n, 1, -4], {n, 0, 20}] (* Vladimir Reshetnikov, Apr 25 2016 *)

From Vaclav Kotesovec, Oct 21 2012: (Start)
G.f.: -sqrt(x-1)/(5*x-1)^(3/2).
Recurrence: n*a(n) = (6*n+1)*a(n-1) - 5*(n-1)*a(n-2).
a(n) ~ 4*5^(n-1/2)*sqrt(n)/sqrt(Pi).
(End)
a(n) = hypergeom([3/2, -n], [1], -4) = hypergeom([3/2, n+1], [1], 4/5)/(5*sqrt(5)). - Vladimir Reshetnikov, Apr 25 2016
E.g.f.: exp(3*x) * ((1 + 4*x) * BesselI(0,2*x) + 4 * x * BesselI(1,2*x)). - Ilya Gutkovskiy, Nov 19 2021
From Seiichi Manyama, Aug 22 2025: (Start)
a(n) = (1/4)^n * Sum_{k=0..n} 5^k * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(n,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 5^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n,n-k). (End)

A387208 Expansion of sqrt((1-x) / (1-9*x)^3).

Original entry on oeis.org

1, 13, 145, 1517, 15329, 151565, 1476465, 14228205, 135990465, 1291409165, 12199991633, 114761111789, 1075651464865, 10051341904141, 93677905064497, 871083359663085, 8083754402585985, 74885500462111245, 692624008942816785, 6397104350057979885, 59008673876627412321

Offset: 0

Seiichi Manyama, Aug 22 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A163869, A163872, A387210.

Cf. A085363, A383946.

R := PowerSeriesRing(Rationals(), 34); f := Sqrt((1-x) / (1-9*x)^3); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 23 2025

CoefficientList[Series[Sqrt[(1-x)/(1-9*x)^3],{x,0,33}],x] (* Vincenzo Librandi, Aug 23 2025 *)

my(N=30, x='x+O('x^N)); Vec(sqrt((1-x)/(1-9*x)^3))

n*a(n) = (10*n+3)*a(n-1) - 9*(n-1)*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 9^k * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} 2^k * (2*k+1) * binomial(2*k,k) * binomial(n,n-k).
a(n) = Sum_{k=0..n} (-2)^k * 9^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n,n-k).
a(n) ~ 2^(5/2) * sqrt(n) * 3^(2*n-1) / sqrt(Pi). - Vaclav Kotesovec, Aug 23 2025

A387210 Expansion of sqrt((1-x) / (1-13*x)^3).

Original entry on oeis.org

1, 19, 307, 4645, 67843, 969337, 13643533, 189953659, 2622877075, 35982412921, 491057325577, 6672763735183, 90347244052429, 1219537191931975, 16418449380961891, 220534056531679141, 2956293832279184659, 39559312793250153577, 528522358385088314425, 7051193680459915645903

Offset: 0

Seiichi Manyama, Aug 22 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A163869, A163872, A387208.

Cf. A385716.

R := PowerSeriesRing(Rationals(), 34); f := Sqrt(((1-x) / (1-13*x)^3)); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 23 2025

CoefficientList[Series[Sqrt[(1-x)/(1-13*x)^3],{x,0,33}],x] (* Vincenzo Librandi, Aug 23 2025 *)

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

n*a(n) = (14*n+5)*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) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} 3^k * (2*k+1) * binomial(2*k,k) * binomial(n,n-k).
a(n) = Sum_{k=0..n} (-3)^k * 13^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n,n-k).
a(n) ~ 4 * sqrt(3*n) * 13^(n - 1/2) / sqrt(Pi). - Vaclav Kotesovec, Aug 23 2025

A163945 Triangle interpolating between (-1)^n (A033999) and the swinging factorial function (A056040) restricted to odd indices (2n+1)$ (A002457), read by rows.

Original entry on oeis.org

1, -1, 6, 1, -12, 30, -1, 18, -90, 140, 1, -24, 180, -560, 630, -1, 30, -300, 1400, -3150, 2772, 1, -36, 450, -2800, 9450, -16632, 12012, -1, 42, -630, 4900, -22050, 58212, -84084, 51480, 1, -48, 840, -7840, 44100, -155232, 336336, -411840, 218790

Offset: 0

Peter Luschny, Aug 07 2009

			Triangle begins:
   1;
  -1,   6;
   1, -12,   30;
  -1,  18,  -90,   140;
   1, -24,  180,  -560,   630;
  -1,  30, -300,  1400, -3150,   2772;
   1, -36,  450, -2800,  9450, -16632, 12012;

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.

Row sums are the inverse binomial transform of the beta numbers (A163872).

Cf. A163649, A098473, A056040.

swing := proc(n) option remember; if n = 0 then 1 elif
irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
a := proc(n, k) (-1)^(n-k)*binomial(n,k)*swing(2*k+1) end:
seq(print(seq(a(n,k),k=0..n)),n=0..8);

T[n_,k_] := ((-1)^(Mod[k,2]+n)*((2*k+1)!/(k!)^2)*Binomial[n,n-k]);
Flatten[Table[T[n,k],{n,0,8},{k,0,n}]] (* Detlef Meya, Oct 07 2023 *)

For n >= 0, k >= 0, T(n, k) = (-1)^(n-k) binomial(n,k) (2*k+1)$ where i$ denotes the swinging factorial of i (A056040).
Conjectural g.f.: sqrt(1 + t)/(1 + (1 - 4*x)*t)^(3/2) = 1 + (-1 + 6*x)*t + (1 - 12*x + 30*x^2)*t^2 + .... - Peter Bala, Nov 10 2013
T(n, k) = ((-1)^(k mod 2) + n)*((2*k + 1)!/(k!)^2)*binomial(n, n - k). - Detlef Meya, Oct 07 2023

cp's OEIS Frontend

A163869 Binomial transform of the beta numbers 1/beta(n+1,n+1) (A002457).

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A387208 Expansion of sqrt((1-x) / (1-9*x)^3).

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A387210 Expansion of sqrt((1-x) / (1-13*x)^3).

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A163945 Triangle interpolating between (-1)^n (A033999) and the swinging factorial function (A056040) restricted to odd indices (2n+1)$ (A002457), read by rows.

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