A066802 -id:A066802 - cp's OEIS frontend

A277520 Denominator of 3F2([3n, -n, n+1],[2n+1, n+1/2], 1).

1, 3, 25, 147, 1089, 20449, 48841, 312987, 55190041, 14322675, 100100025, 32065374675, 4546130625, 29873533563, 1859904071089, 4089135109921, 9399479144449, 22568149425822049, 1293753708921104809, 2835106739783283, 3289668853728536041

Offset: 0

Seiichi Manyama, Oct 19 2016

Neil Calkin found the closed forms of 3F2([3*n, -n, n+1],[2*n+1, n+1/2], 1) in 2007.

Jonathan Borwein, David Bailey, Mathematics by Experiment, 2nd Edition: Plausible Reasoning in the 21st Century.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A005810, A052203, A066802, A187364, A277170 (numerators).

a[n_] := HypergeometricPFQ[{3n, -n, n+1}, {2n+1, n+1/2}, 1] // Denominator;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Oct 22 2016 *)

(s(n) =) 3F2([3*n, -n, n+1],[2*n+1, n+1/2], 1) = A277170(n) / a(n).
s(2k) = (A005810(k) / A066802(k))^2 = (((4k)! * (3k)!) / ((6k)! * k!))^2.
s(2k+1) = -1/3 * (A052203(k) / A187364(k))^2 = -1/3 * (((4k+1)! * (3k)!) / ((6k+1)! * k!))^2.

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).

A387249 a(n) = 10/(n + 1) * Catalan(3*n).

Original entry on oeis.org

10, 25, 440, 12155, 416024, 16158075, 682341000, 30582833775, 1433226830360, 69533550916004, 3468169547356640, 176946775343535925, 9199844912200348840, 486018122664268428850, 26029619941269629306160, 1410698658798280045783575, 77251704848334920869407000, 4269325372507953547350453420

Offset: 0

Peter Bala, Aug 24 2025

Compare with Catalan(n) = 1/(n + 1) * binomial(2*n, n).

For r >= 2, there is a constant K_r such that K_r/(n + 1) * Catalan(r*n) is integral for all n.

Paolo Xausa, Table of n, a(n) for n = 0..500

Cf. A000108, A000984, A007272, A066802, A387248, A387250.

seq( 10/((n+1)*(3*n+1)) * binomial(6*n, 3*n), n = 0..20);

A387249[n_] := 10*CatalanNumber[3*n]/(n + 1); Array[A387249, 20, 0] (* Paolo Xausa, Sep 02 2025 *)

a(n) = 10/((n + 1)*(3*n + 1)) * binomial(6*n, 3*n).
a(n) = (3*n + 2)/2 * (16*Catalan(3*n) - 8*Catalan(3*n+1) + Catalan(3n+2)) (shows a(n) to be an integer since Catalan(n) is odd iff n = 2^k - 1 for some k, so Catalan(3*n+2) is always even).
a(n) = (3*n + 2)/2 * A007272(3*n).
a(n) = 8*(2*n - 1)*(6*n - 1)*(6*n - 5)/((n + 1)*(3*n + 1)*(3*n - 1)) * a(n-1) with a(0) = 10.
a(n) ~ 10/(sqrt(27*Pi)) * 64^n/n^(5/2).
E.g.f.: 10*hypergeom([1/6, 1/2, 5/6], [2/3, 4/3, 2], 64*x). - Stefano Spezia, Aug 27 2025

A361949 Triangle read by rows. T(n, k) = binomial(3n - 1, 3k - 1).

Original entry on oeis.org

1, 10, 1, 28, 56, 1, 55, 462, 165, 1, 91, 2002, 3003, 364, 1, 136, 6188, 24310, 12376, 680, 1, 190, 15504, 125970, 167960, 38760, 1140, 1, 253, 33649, 490314, 1352078, 817190, 100947, 1771, 1, 325, 65780, 1562275, 7726160, 9657700, 3124550, 230230, 2600, 1

Offset: 1

Peter Luschny, Mar 31 2023

			Table T(n, k) starts:
  [1]   1;
  [2]  10,     1;
  [3]  28,    56,       1;
  [4]  55,   462,     165,       1;
  [5]  91,  2002,    3003,     364,       1;
  [6] 136,  6188,   24310,   12376,     680,       1;
  [7] 190, 15504,  125970,  167960,   38760,    1140,      1;
  [8] 253, 33649,  490314, 1352078,  817190,  100947,   1771,    1;
  [9] 325, 65780, 1562275, 7726160, 9657700, 3124550, 230230, 2600, 1.

Cf. A082365 (row sums), A228888 (subdiagonal), A060544 (column 1), A066802 (central column).

T := (n, k) -> binomial(3*n - 1, 3*k - 1):
seq(print(seq(T(n, k), k = 1..n)), n = 1..8);

cp's OEIS Frontend

A277520 Denominator of 3F2([3*n, -n, n+1],[2*n+1, n+1/2], 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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A387249 a(n) = 10/(n + 1) * Catalan(3*n).

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A361949 Triangle read by rows. T(n, k) = binomial(3*n - 1, 3*k - 1).

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A277520 Denominator of 3F2([3n, -n, n+1],[2n+1, n+1/2], 1).

A361949 Triangle read by rows. T(n, k) = binomial(3n - 1, 3k - 1).