A163869 -id:A163869 - cp's OEIS frontend

A163842 Triangle interpolating the swinging factorial (A056040) restricted to odd indices with its binomial transform. Same as interpolating the beta numbers 1/beta(n,n) (A002457) with (A163869). Triangle read by rows, for n >= 0, k >= 0.

1, 7, 6, 43, 36, 30, 249, 206, 170, 140, 1395, 1146, 940, 770, 630, 7653, 6258, 5112, 4172, 3402, 2772, 41381, 33728, 27470, 22358, 18186, 14784, 12012, 221399, 180018, 146290, 118820, 96462, 78276, 63492, 51480

Offset: 0

Peter Luschny, Aug 06 2009

			Triangle begins:
      1;
      7,     6;
     43,    36,    30;
    249,   206,   170,   140;
   1395,  1146,   940,   770,   630;
   7653,  6258,  5112,  4172,  3402,  2772;
  41381, 33728, 27470, 22358, 18186, 14784, 12012;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.

Row sums are A163845. Cf. A056040, A163650, A163841, A163842, A163840, A002426, A000984.

# Computes n rows of the triangle. For the functions 'SumTria' and 'swing' see A163840.
a := n -> SumTria(k->swing(2*k+1),n,true);

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[Binomial[n-k, n-i]*sf[2*i+1], {i, k, n}]; Table[t[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)

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

A387228 Expansion of sqrt((1-x) / (1-5*x)^5).

Original entry on oeis.org

1, 12, 103, 764, 5215, 33728, 210021, 1271504, 7532547, 43859460, 251809701, 1428911652, 8028877233, 44734340784, 247433518875, 1359902816880, 7432212863235, 40416897046740, 218812616979845, 1179889937796900, 6339243523221245, 33947223885549040, 181245459484155935

Offset: 0

Seiichi Manyama, Aug 23 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A387229, A387230.
Cf. A085362, A163869.
Cf. A377199.

R := PowerSeriesRing(Rationals(), 34); f := Sqrt((1- x) / (1-5*x)^5); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 24 2025

CoefficientList[Series[Sqrt[(1-x)/(1-5*x)^5],{x,0,33}],x] (* Vincenzo Librandi, Aug 24 2025 *)

my(N=30, x='x+O('x^N)); Vec(sqrt((1-x)/(1-5*x)^5))

n*a(n) = (6*n+6)*a(n-1) - 5*n*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 5^k * ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} ((2*k+1) * (2*k+3)/3) * binomial(2*k,k) * binomial(n+1,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 5^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n+1,n-k).
a(n) ~ 8 * 5^(n - 1/2) * n^(3/2) / (3*sqrt(Pi)). - Vaclav Kotesovec, Aug 23 2025

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

A387211 Expansion of sqrt((1-2x) / (1-6x)^3).

Original entry on oeis.org

1, 8, 58, 400, 2678, 17584, 113892, 730272, 4646310, 29380912, 184867148, 1158418144, 7233806524, 45038743520, 279704675464, 1733203476288, 10718950211334, 66176597723184, 407931346057020, 2511127341708384, 15438601388617044, 94810212917983392, 581639541983344632

Offset: 0

Seiichi Manyama, Aug 22 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A163869, A387212.

Cf. A360317.

R := PowerSeriesRing(Rationals(), 34); f := Sqrt((1- 2*x) / (1-6*x)^3); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 23 2025

CoefficientList[Series[Sqrt[(1-2*x)/(1-6*x)^3],{x,0,33}],x] (* Vincenzo Librandi, Aug 23 2025 *)

my(N=30, x='x+O('x^N)); Vec(sqrt((1-2*x)/(1-6*x)^3))

n*a(n) = 8*n*a(n-1) - 12*(n-1)*a(n-2) for n > 1.
a(n) = (1/2)^n * Sum_{k=0..n} 3^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^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 6^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n,n-k).

A387212 Expansion of sqrt((1-3x) / (1-7x)^3).

Original entry on oeis.org

1, 9, 75, 599, 4659, 35595, 268485, 2005785, 14873715, 109643195, 804354417, 5877232773, 42798735805, 310767250773, 2250899498763, 16267896905895, 117347641620435, 845043416086635, 6076092412278465, 43629213402099045, 312892629725930121, 2241442380182752209

Offset: 0

Seiichi Manyama, Aug 22 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A163869, A387211.

Cf. A360318, A385813.

R := PowerSeriesRing(Rationals(), 34); f := Sqrt((1- 3*x) / (1-7*x)^3); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 23 2025

CoefficientList[Series[Sqrt[(1-3*x)/(1-7*x)^3],{x,0,33}],x] (* Vincenzo Librandi, Aug 23 2025 *)

my(N=30, x='x+O('x^N)); Vec(sqrt((1-3*x)/(1-7*x)^3))

n*a(n) = (10*n-1)*a(n-1) - 21*(n-1)*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 7^k * 3^(n-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^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 7^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n,n-k).

cp's OEIS Frontend

A163842 Triangle interpolating the swinging factorial (A056040) restricted to odd indices with its binomial transform. Same as interpolating the beta numbers 1/beta(n,n) (A002457) with (A163869). Triangle read by rows, for n >= 0, k >= 0.

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A387228 Expansion of sqrt((1-x) / (1-5*x)^5).

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

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Magma

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

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Magma

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A387211 Expansion of sqrt((1-2*x) / (1-6*x)^3).

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Magma

Mathematica

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A387212 Expansion of sqrt((1-3*x) / (1-7*x)^3).

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Magma

Mathematica

PARI

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A387211 Expansion of sqrt((1-2x) / (1-6x)^3).

A387212 Expansion of sqrt((1-3x) / (1-7x)^3).