A005810 -id:A005810 - cp's OEIS frontend

A359108 a(n) = A128899(2n, n) = 2binomial(4n - 1, 3n) for n >= 1 and a(0) = 1.

1, 2, 14, 110, 910, 7752, 67298, 592020, 5259150, 47071640, 423830264, 3834669566, 34834267234, 317506779800, 2902365981900, 26597044596360, 244263468539790, 2247575790712824, 20716044882791720, 191230475831922200, 1767658071106087160, 16359617358545329440

Offset: 0

Peter Luschny, Dec 27 2022

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A005810, A128899, A224274.

a := n -> ifelse(n = 0, 1, 2*binomial(4*n - 1, 3*n)):
# Alternative:
a := proc(n) option remember; ifelse(n < 2, n + 1, (8*(2*n - 1) * (4*n - 3) * (4*n - 1) * a(n - 1)) / (3 * n * (3*n - 2) * (3*n - 1))) end:
seq(a(n), n = 0..19);

A359108[n_] := If[n == 0, 1, 2*Binomial[4*n - 1, 3*n]];
Array[A359108, 25, 0] (* Paolo Xausa, Sep 18 2024 *)

a(n) = (8*(2*n - 1) * (4*n - 3) * (4*n - 1) * a(n - 1)) / (3*n * (3*n - 2) * (3*n - 1)) for n >= 2.
a(n) = (1/2)*A005810(n) = 2*A224274(n) for n >= 1. - Peter Bala, Feb 08 2023
a(n) = [x^n] C(x)^(2*n), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. - Peter Bala, Apr 27 2023

A364519 Square array read by ascending antidiagonals: T(n,k) = [x^(3*k)] ( (1 + x)^(n+3)/(1 - x)^(n-3) )^k for n, k >= 0.

Original entry on oeis.org

1, 1, 0, 1, -4, -20, 1, 0, 28, 0, 1, 20, -84, -220, 924, 1, 64, 924, 0, 1820, 0, 1, 140, 12012, 48620, 16796, -15504, -48620, 1, 256, 60060, 2621440, 2704156, 0, 134596, 0, 1, 420, 204204, 29745716, 608435100, 155117520, -3801900, -1184040, 2704156, 1, 640, 554268, 187432960, 15628090140, 146028888064, 9075135300, 0, 10518300, 0

Offset: 0

Peter Bala, Aug 07 2023

Compare with A364303 and A364518.
Given two sequences of integers c = (c_1, c_2, ..., c_K) and d = (d_1, d_2, ..., d_L), where c_1 + ... + c_K = d_1 + ... + d_L, we can define the factorial ratio sequence u_n(c, d) = (c_1*n)!*(c_2*n)!* ... *(c_K*n)!/ ( (d_1*n)!*(d_2*n)!* ... *(d_L*n)! ) and ask whether it is integral for all n >= 0. The integer L - K is called the height of the sequence. Bober completed the classification of integral factorial ratio sequences of height 1 (see A295431).
Each row of the present table is an integral factorial ratio sequence of height 1. It is usually assumed that the c's and d's are integers but here some of the c's and d's are half-integers. See A276098 and the cross references there for further examples of this type.
It is known that A005810, the unsigned version of row 1, satisfies the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and all positive integers n and r. We conjecture that each row sequence of the table satisfies the same supercongruences.

			Square array begins:
 n\k| 0    1       2          3             4                5
  - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  0 | 1    0     -20          0           924                0  ... see A066802
  1 | 1   -4      28       -220          1820           -15504  ... see A005810
  2 | 1    0     -84          0         16796                0
  3 | 1   20     924      48620       2704156        155117520  ... A066802
  4 | 1   64   12012    2621440     608435100     146028888064  ... A364520
  5 | 1  140   60060   29745716   15628090140    8480843582640  ... A211420

Peter Bala, Some integer ratios of factorials
J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc. (2) 79 2009, 422-444.
Wikipedia, Hypergeometric function

Cf. A066802 (row 3, also row 0 unsigned and without 0's), A005810 (row 1 unsigned), A364520 (row 4), A211420 (row 5).

Cf. A330843, A364303, A364518.

T(n,k) := add( binomial((n+3)*k, j)*binomial(n*k-j-1, 3*k-j), j = 0..3*k):
# display as a square array
seq(print(seq(T(n, k), k = 0..10)), n = 0..10);
# display as a sequence
seq(seq(T(n-k, k), k = 0..n), n = 0..10);

T(n,k) = sum(j = 0, 3*k, binomial((n+3)*k, j)*binomial(n*k-j-1, 3*k-j));
lista(nn) = for( n=0, nn, for (k=0, n, print1(T(n-k, k), ", "))); \\ Michel Marcus, Aug 13 2023

T(n,k) = Sum_{j = 0..3*k} binomial((n+3)*k, j)*binomial(n*k-j-1, 3*k-j).
For n >= 3, T(n,k) = binomial(n*k-1,3*k) * hypergeom([-(n+3)*k, -3*k], [1 - n*k], -1) = ((n+3)*k)!*((n-3)*k/2)!/(((n+3)*k/2)!*((n-3)*k)!*(3*k)!) by Kummer's Theorem.
The row generating functions are algebraic functions over the field of rational functions Q(x).

A374522 a(n) = n(binomial(4n, n) - 1).

Original entry on oeis.org

0, 3, 54, 657, 7276, 77515, 807570, 8288273, 84146392, 847289511, 8476605270, 84362730441, 836022413604, 8255176274787, 81266247493186, 797911337890785, 7816430993273264, 76417576884235999, 745777615780501902, 7266758081613043581, 70706322844243486380, 687103929058903836459

Offset: 0

Michel Marcus, Jul 10 2024

For n>1, a(n) is the number of vertices of the graph for the "Left, Center, Right" game with n players.

Benjamin Richeson and David Richeson, What's the Best Seat in the Game Left, Center, Right?, arXiv:2407.05069 [math.HO], 2024. See Table 1 p. 4.

Cf. A005810.

[n*(Binomial(4*n, n) - 1): n in [0..25]]; // Vincenzo Librandi, Nov 07 2024

CoefficientList[Series[4x*HypergeometricPFQ[{5/4,3/2,7/4},{4/3,5/3},256x/27]-x/(1-x)^2,{x,0,21}],x] (* Stefano Spezia, Jul 10 2024 *)

PARI
```
a(n) = n*(binomial(4*n, n) - 1);
```

from math import comb
def A374522(n): return n*(comb((n<<2),n)-1) # Chai Wah Wu, Jul 10 2024

G.f.: 4*x*hypergeom([5/4, 3/2, 7/4], [4/3, 5/3], 4^4*x/3^3) - x/(1 - x)^2. - Stefano Spezia, Jul 10 2024

D-finite with recurrence -3*(n-1)*(3*n-1)*(3*n-2)*a(n) +(283*n^3-519*n^2+398*n-144)*a(n-1) -8*(4*n-5)*(4*n-7)*(2*n-3)*a(n-2) +4*(229*n^3-783*n^2+932*n-384)=0. - R. J. Mathar, Jul 10 2024

A378803 a(n) = n^2 * binomial(4*n, n).

Original entry on oeis.org

0, 4, 112, 1980, 29120, 387600, 4845456, 58017960, 673171200, 7625605680, 84766052800, 927990034972, 10032268963392, 107317291572400, 1137727464904800, 11968670068362000, 125062895892372480, 1299098807032012272, 13423997084049034560, 138068403550647828400, 1414126456884869728000

Offset: 0

Amiram Eldar, Dec 07 2024

Amiram Eldar, Table of n, a(n) for n = 0..500
Necdet Batir and Anthony Sofo, On some series involving reciprocals of binomial coefficients, Appl. Math. Comp., Vol. 220 (2013), pp. 331-338.

Cf. A005810, A378802.

a[n_] := n^2 * Binomial[4*n, n]; Array[a, 20, 0]

PARI
```
a(n) = n^2 * binomial(4*n, n);
```

a(n) = n^2 * A005810(n).
a(n) = n * A378802(n).
a(n) == 0 (mod 4).
Sum_{n>=1} 1/a(n) = -(3/2)*log((c-1)/(c+1))^2 + (3/4) * arctan(2*sqrt(c^2+2*c)/(c^2+2*c-1))^2 + (3/4) * arctan(2*sqrt(c^2-2*c)/(c^2-2*c-1))^2 = 0.25947076781691783..., where c = sqrt(1 + (16/sqrt(3))*cos(arctan(sqrt(229/27))/3)) (Batir and Sofo, 2013, p. 336, Example 3).
Sum_{n>=1} (-1)^n/a(n) = -(3/2)*log((1-d)/(1+d))^2 + (3/4) * arctan(2*sqrt(d^2+2*d)/(d^2+2*d-1))^2 + (3/4) * arctan(2*sqrt(d^2-2*d)/(d^2-2*d-1))^2 = -0.24154452788843591937..., where d = sqrt(1 - (8/sqrt(3))*(((3*sqrt(3)+sqrt(283))/16)^(1/3) - (((3*sqrt(3)+sqrt(283))/16)^(-1/3)))) (Batir and Sofo, 2013, pp. 336-337, Example 4).

A378806 Decimal expansion of Sum_{k>=1} 1/binomial(4*k, k).

Original entry on oeis.org

2, 9, 0, 8, 8, 2, 0, 7, 1, 5, 2, 1, 2, 8, 7, 2, 1, 2, 7, 6, 2, 5, 9, 7, 2, 5, 6, 6, 8, 6, 8, 1, 0, 3, 5, 7, 7, 3, 3, 6, 8, 1, 7, 6, 1, 6, 7, 6, 0, 9, 7, 9, 2, 7, 5, 8, 2, 3, 7, 9, 3, 5, 9, 2, 6, 2, 2, 8, 4, 8, 1, 2, 4, 6, 8, 0, 2, 5, 4, 2, 5, 5, 0, 5, 5, 9, 3, 3, 9, 1, 8, 9, 7, 1, 6, 4, 9, 5, 6, 0, 3, 0, 3, 3, 4

Offset: 0

Amiram Eldar, Dec 07 2024

			0.29088207152128721276259725668681035773368176167609...

Necdet Batir and Anthony Sofo, On some series involving reciprocals of binomial coefficients, Appl. Math. Comp., Vol. 220 (2013), pp. 331-338.

Cf. A005810, A073016, A229705, A378807.

RealDigits[HypergeometricPFQ[{1, 4/3, 5/3, 2}, {5/4, 3/2, 7/4}, 27/256] / 4, 10, 120][[1]]

Equals 4F3(1, 4/3, 5/3, 2; 5/4, 3/2, 7/4; 27/256) / 4, where 4F3 is a generalized hypergeometric function.

Equals 27*c^2/((c^2-4)*(2*c^2+1)^2) + (3*c*(c^2-1)*(2*c^2-1)/(2*(2*c^2+1)^3)) * log((c-1)/(c+1)) + (3*(c^2-1)*(2*c^4-2*c^3-7*c^2-3*c+1)/(4*c*(2*c^2+1)^3)) * (c/(c+2))^(3/2) * arctan(2*sqrt(c^2+2*c)/(c^2+2*c-1)) + (3*(c^2-1)*(2*c^4+2*c^3-7*c^2+3*c+1)/(4*c*(2*c^2+1)^3)) * (c/(c-2))^(3/2) * arctan(2*sqrt(c^2-2*c)/(c^2-2*c-1)), where c = sqrt(1 + (16/sqrt(3))*cos(arctan(sqrt(229/27))/3)) (Batir and Sofo, 2013, p. 337, Example 9).

A378807 Decimal expansion of Sum_{k>=1} (-1)^k/binomial(4*k, k) (negated).

Original entry on oeis.org

2, 1, 8, 3, 3, 9, 5, 4, 7, 1, 7, 7, 9, 3, 4, 4, 3, 6, 8, 7, 0, 9, 9, 8, 3, 2, 1, 0, 2, 7, 8, 8, 5, 3, 9, 1, 9, 8, 3, 0, 4, 8, 6, 4, 0, 2, 9, 2, 2, 6, 2, 2, 7, 0, 0, 1, 3, 2, 5, 6, 8, 5, 4, 9, 8, 0, 6, 6, 7, 9, 6, 6, 1, 3, 5, 9, 0, 4, 2, 7, 6, 1, 3, 1, 7, 0, 9, 3, 7, 4, 0, 2, 9, 0, 7, 9, 6, 3, 9, 3, 9, 6, 3, 3, 2

Offset: 0

Amiram Eldar, Dec 07 2024

			-0.21833954717793443687099832102788539198304864029226...

Necdet Batir and Anthony Sofo, On some series involving reciprocals of binomial coefficients, Appl. Math. Comp., Vol. 220 (2013), pp. 331-338.

Cf. A005810, A086465, A229703, A378806.

RealDigits[HypergeometricPFQ[{1, 4/3, 5/3, 2}, {5/4, 3/2, 7/4}, -27/256]/ 4, 10, 120][[1]]

Equals 4F3(1, 4/3, 5/3, 2; 5/4, 3/2, 7/4; -27/256) / 4, where 4F3 is a generalized hypergeometric function.

Equals 27*d^2/((d^2-4)*(2*d^2+1)^2) + (3*d*(d^2-1)*(2*d^2-1)/(2*(2*d^2+1)^3)) * log(abs((d-1)/(d+1))) + (3*(d^2-1)*(2*d^4-2*d^3-7*d^2-3*d+1)/(4*d*(2*d^2+1)^3)) * (d/(d+2))^(3/2) * arctan(2*sqrt(d^2+2*d)/(d^2+2*d-1)) - (3*(d^2-1)*(2*d^4+2*d^3-7*d^2+3*d+1)/(4*d*(2*d^2+1)^3)) * (d/(d-2))^(3/2) * arctan(2*sqrt(d^2-2*d)/(d^2-2*d-1)), where d = sqrt(1 - (8/sqrt(3))*(((3*sqrt(3)+sqrt(283))/16)^(1/3) - (((3*sqrt(3)+sqrt(283))/16)^(-1/3)))) (Batir and Sofo, 2013, pp. 336-337, Example 4).

A383119 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(4*k,k).

Original entry on oeis.org

1, 3, 21, 147, 1093, 8343, 64869, 510891, 4062277, 32539647, 262181601, 2122581123, 17252278789, 140695104943, 1150670390541, 9433965332127, 77512716483461, 638080242074447, 5261486780929209, 43450477494413751, 359308411992366513, 2974886601163646379, 24657831769475675253

Offset: 0

Ilya Gutkovskiy, Apr 17 2025

nonn

Inverse binomial transform of A005810.

Cf. A002426, A005810, A346664, A359643, A383118.

Cf. A317133.

Table[Sum[(-1)^(n - k) Binomial[n, k] Binomial[4 k, k], {k, 0, n}], {n, 0, 22}]
Table[(-1)^n HypergeometricPFQ[{1/4, 1/2, 3/4, -n}, {1/3, 2/3, 1}, 256/27], {n, 0, 22}]
nmax = 22; CoefficientList[Series[(1/x) Sum[Binomial[4 k, k] (x/(1 + x))^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*binomial(4*k, k)); \\ Seiichi Manyama, Apr 17 2025

G.f.: (1/x) * Sum_{k>=0} binomial(4*k,k) * (x/(1 + x))^(k+1).
a(n) = [x^n] (1 + 3*x + 6*x^2 + 4*x^3 + x^4)^n.
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x/((1+x)^4 - x) ). See A317133. - Seiichi Manyama, Apr 17 2025
a(n) ~ 229^(n + 1/2) / (2^(7/2) * sqrt(Pi*n) * 3^(3*n + 1/2)). - Vaclav Kotesovec, Apr 17 2025

A378805 a(n) = n^2 * 2^n * binomial(4*n, n).

Original entry on oeis.org

0, 8, 448, 15840, 465920, 12403200, 310109184, 7426298880, 172331827200, 3904310108160, 86800438067200, 1900523591622656, 41092173674053632, 879143252561100800, 18640526785000243200, 392189380800086016000, 8196121945202522849280, 170275478835299912515584, 3519020291600950115696640

Offset: 0

Amiram Eldar, Dec 07 2024

Amiram Eldar, Table of n, a(n) for n = 0..500
Necdet Batir and Anthony Sofo, On some series involving reciprocals of binomial coefficients, Appl. Math. Comp., Vol. 220 (2013), pp. 331-338.

Cf. A005810, A007758, A378803, A378804.

a[n_] := n^2 * 2^n * Binomial[4*n, n]; Array[a, 20, 0]

PARI
```
a(n) = n^2 * 2^n * binomial(4*n, n);
```

a(n) = A007758(n) * A005810(n).
a(n) = 2^n * A378803(n).
a(n) = n * A378804(n).
a(n) == 0 (mod 8).
Sum_{n>=1} 1/a(n) = -(3/2)*log((c-1)/(c+1))^2 + (3/4) * arctan(2*sqrt(c^2+2*c)/(c^2+2*c-1))^2 + (3/4) * arctan(2*sqrt(c^2-2*c)/(c^2-2*c-1))^2 = 0.12729750445123620540..., where c = sqrt(1 + (16*sqrt(2/3))*cos(arctan(sqrt(485/27))/3)) (Batir and Sofo, 2013, p. 336, Example 1).

A380113 Triangle read by rows: The inverse matrix of the central factorials A370707, row n normalized by (-1)^(n - k)*A370707(n, n).

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 10, 15, 6, 1, 35, 56, 28, 8, 1, 126, 210, 120, 45, 10, 1, 462, 792, 495, 220, 66, 12, 1, 1716, 3003, 2002, 1001, 364, 91, 14, 1, 6435, 11440, 8008, 4368, 1820, 560, 120, 16, 1, 24310, 43758, 31824, 18564, 8568, 3060, 816, 153, 18, 1

Offset: 0

Peter Luschny, Jan 12 2025

The inverse matrix of A370707 is a rational matrix and the normalization serves to make it a matrix over the integers. Note that the normalization factor A370707(n, n) = FallingFactorial(n, n) * RisingFactorial(n, n) extends A002674 to n = 0.

			Triangle starts:
  [0] [    1]
  [1] [    1,     1]
  [2] [    3,     4,     1]
  [3] [   10,    15,     6,     1]
  [4] [   35,    56,    28,     8,    1]
  [5] [  126,   210,   120,    45,   10,    1]
  [6] [  462,   792,   495,   220,   66,   12,   1]
  [7] [ 1716,  3003,  2002,  1001,  364,   91,  14,   1]
  [8] [ 6435, 11440,  8008,  4368, 1820,  560, 120,  16,  1]
  [9] [24310, 43758, 31824, 18564, 8568, 3060, 816, 153, 18, 1]
.
Row 3 of the matrix inverse of the central factorials is [-1/36, 1/24, -1/60, 1/360]. Normalized with (-1)^(n-k)*360 gives row 3 of T.

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).

Variant: A094527.

Cf. A370707, A002674, A008311, A088218 and A110556 (column 0), A081294 (row sums), A000007 (alternating row sums), A005810 (central terms).

T := (n, k) -> if n = k then 1 elif k = 0 then binomial(2*n, n - k)/2 else binomial(2*n, n - k) fi: seq(seq(T(n, k), k = 0..n), n = 0..9);

A380113[n_, k_] := Binomial[2*n, n - k]/(Boole[k == 0 && n > 0] + 1);
Table[A380113[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 13 2025 *)

def Trow(n):
    def cf(n, k): return falling_factorial(n, k)*rising_factorial(n, k)
    def w(n): return factorial(n)*rising_factorial(n, n)
    m = matrix(QQ, n + 1, lambda x, y: cf(x, y)).inverse()
    return [(-1)^(n-k)*w(n)*m[n, k] for k in range(n+1)]
for n in range(10): print(Trow(n))

T(n, k) = (-1)^(n - k) * ff(n, n) * rf(n, n) * M^(-1)(ff(n, k) * rf(n, k)) where ff denotes the falling factorial, rf the rising factorial and M^(-1)(t(n, k)) the matrix inverse to the matrix with entries t(n, k).
T(n, k) = binomial(2*n, n - k) for 0 < k < n. T(n, n) = 1; T(n, 0) = (-1)^n*binomial(-n, n).
Sum_{k=0..n} T(n, k)*cos(k*x) = 2^(n-1)*(cos(x)+1)^n. (After Philippe Deléham in A008311).

A024496 a(n) = (3/(8n-4))*C(4n,n).

Original entry on oeis.org

3, 7, 33, 195, 1292, 9177, 68310, 525915, 4153380, 33460284, 273904969, 2271800037, 19050406788, 161242554550, 1375709203260, 11819200090635, 102162535941492, 887830494976788, 7752586858050900, 67986848888695660, 598522586288243760, 5287559893638230385

Offset: 1

Clark Kimberling

nonn

Cf. A005810.

Table[3/(8*n - 4)*Binomial[4*n, n], {n, 1, 20}] (* Vaclav Kotesovec, Mar 12 2019 *)

a(n) = 3*binomial(4*n, n)/(8*n-4); \\ Michel Marcus, Mar 12 2019

G.f.: (g+2)*(g-1)/g^2 where g = 1+x*g^4 is the g.f. of A002293. - Mark van Hoeij, Nov 11 2011

a(n) ~ 2^(8*n - 5/2) / (sqrt(Pi) * n^(3/2) * 3^(3*n - 1/2)). - Vaclav Kotesovec, Mar 12 2019

More terms from Vaclav Kotesovec, Mar 12 2019

cp's OEIS Frontend

A359108 a(n) = A128899(2*n, n) = 2*binomial(4*n - 1, 3*n) for n >= 1 and a(0) = 1.

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A364519 Square array read by ascending antidiagonals: T(n,k) = [x^(3*k)] ( (1 + x)^(n+3)/(1 - x)^(n-3) )^k for n, k >= 0.

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A374522 a(n) = n*(binomial(4*n, n) - 1).

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A378803 a(n) = n^2 * binomial(4*n, n).

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A378806 Decimal expansion of Sum_{k>=1} 1/binomial(4*k, k).

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A378807 Decimal expansion of Sum_{k>=1} (-1)^k/binomial(4*k, k) (negated).

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A383119 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(4*k,k).

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A378805 a(n) = n^2 * 2^n * binomial(4*n, n).

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A359108 a(n) = A128899(2n, n) = 2binomial(4n - 1, 3n) for n >= 1 and a(0) = 1.

A374522 a(n) = n(binomial(4n, n) - 1).