A364173 -id:A364173 - cp's OEIS frontend

A364172 a(n) = (6n)!(n/3)!/((3n)!(2n)!(4*n/3)!).

1, 45, 6237, 1021020, 178719453, 32427545670, 6016814703900, 1133540594837892, 215925912619400925, 41477110789150966020, 8019784929635201045862, 1558875476359831844951100, 304331361887290342345862940, 59629409730107012112361325820

Offset: 0

Peter Bala, Jul 12 2023

Fractional factorials are defined in terms of the gamma function; for example, (n/3)! := Gamma(n/3 + 1).
Given two sequences of numbers 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. For a list of the 52 sporadic integral factorial ratio sequences see A295431.
It is usually assumed that the c's and d's are integers but here we allow for some of the c's and d's to be rational numbers.
A295437, defined by A295437(n) = (18*n)!*n! / ((9*n)!*(6*n)!*(4*n)!) is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin (see Bober, Table 2, Entry 7). Here we are essentially considering the sequence {A295437(n/3) : n >= 0}. This sequence is only conjecturally an integer sequence.
Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and all positive integers n and r.

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., 79, Issue 2, (2009), 422-444.

Cf. A276100, A276101, A276102, A295431, A295437, A347854, A347855, A347856, A347857, A347858, A364173 - A364185.

seq( simplify((6*n)!*(n/3)!/((3*n)!*(2*n)!*(4*n/3)!)), n = 0..15);

Table[Product[36*(6*k - 5)*(6*k - 1)/(k*(3*k + n)), {k, 1, n}], {n, 0, 20}] (*  Vaclav Kotesovec, Jul 13 2023 *)

a(n) ~ 2^(4*n/3 - 3/2) * 3^(4*n) / sqrt(Pi*n). - Vaclav Kotesovec, Jul 13 2023

a(n) = 5832*(6*n - 1)*(6*n - 5)*(6*n - 7)*(6*n - 11)*(6*n - 13)*(6*n - 17)/(n*(n - 1)*(n - 2)*(2*n - 3)*(4*n - 3)*(4*n - 9))*a(n-3) for n >= 3 with a(0) = 1, a(1) = 45 and a(2) = 6237.

A364183 a(n) = (12n)!(2n)!(n/2)!/((6n)!(4n)!(7n/2)!n!).

Original entry on oeis.org

1, 4224, 76488984, 1626105446400, 36856530424884600, 864687003650148532224, 20728451893251973782071160, 504292670666772382512278667264, 12401082728528113445556802226795640, 307453669544695584297743425538327838720, 7671567513095586883562392061857092727662984

Offset: 0

Peter Bala, Jul 13 2023

A295479, defined by A295479(n) = (24*n)!*(4*n)!*n! / ((12*n)!*(8*n)!*(7*n)!*(2*n)!), is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin (see Bober, Table 2, Entry 49). Here we are essentially considering the sequence {A295479(n/2) : n >= 0}. Fractional factorials are defined in terms of the gamma function; for example, (7*n/2)! := Gamma(1 + 7*n/2).
This sequence is only conjecturally an integer sequence.
Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and all positive integers n and r.

J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc., 79, Issue 2, (2009), 422-444.

Cf. A276100, A276101, A276102, A295431, A295479, A347854, A347855, A347856, A347857, A347858, A364173 - A364185.

seq( simplify((12*n)!*(2*n)!*(n/2)!/((6*n)!*(4*n)!*(7*n/2)!*n!)), n = 0..15);

a(n) ~ c^n * 1/sqrt(14*Pi*n), where c = (2^15)*(3^6)/(7^4) * sqrt(7).

a(n) = 1327104*(12*n - 1)*(12*n - 5)*(12*n - 7)*(12*n - 11)*(12*n - 13)*(12*n - 17)*(12*n - 19)*(12*n - 23)/(7*n*(n - 1)*(7*n - 2)*(7*n - 4)*(7*n - 6)*(7*n - 8)*(7*n - 10)*(7*n - 12))*a(n-2) with a(0) = 1 and a(1) = 4224.

A364176 a(n) = (15n)!(5n/2)!(2n)!/((15n/2)!(6n)!(5n)!*n!).

Original entry on oeis.org

1, 7168, 168043980, 4488240824320, 126694219977836700, 3688258943632086663168, 109504706026534324525391988, 3295939064766794222800490987520, 100204869963549181630558779565943580, 3070025447039504554088467623457608171520, 94632263448378916462441320194245442445186480

Offset: 0

Peter Bala, Jul 13 2023

A295456, defined by A295456(n) = (30*n)!*(5*n)!*(4*n)! / ((15*n)!*(12*n)!*(10*n)!*(2*n)!), is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin (see Bober, Table 2, Entry 26). Here we are essentially considering the sequence {A295456(n/2) : n >= 0}. Fractional factorials are defined in terms of the gamma function; for example, (5*n/2)! := Gamma(1 + 5*n/2).
This sequence is only conjecturally an integer sequence.
Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and all positive integers n and r.

J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc., 79, Issue 2, (2009), 422-444.

Cf. A276100, A276101, A276102, A295431, A295456, A347854, A347855, A347856, A347857, A347858, A364173 - A364185.

seq( simplify((15*n)!*(5*n/2)!*(2*n)!/((15*n/2)!*(6*n)!*(5*n)!*n!)), n = 0..15)

a(n) ~ c^n * 1/sqrt(6*Pi*n), where c = 18750*sqrt(3).

a(n) = 4800*(15*n - 1)*(15*n - 7)*(15*n - 11)*(15*n - 13)*(15*n - 17)*(15*n - 19)*(15*n - 23)*(15*n - 29)/(n*(n - 1)*(3*n - 2)*(3*n - 4)*(6*n - 1)*(6*n - 5)*(6*n - 7)*(6*n - 11))*a(n-2) with a(0) = 1 and a(1) = 7168.

A364177 a(n) = (15n)!(5n/2)!(2n)!/((15n/2)!(5n)!(4n)!(3n)!).

Original entry on oeis.org

1, 35840, 5545451340, 991901222174720, 188242272043069768860, 36901030731039027064995840, 7383354803839076831124554790900, 1498315221854950975184507333477662720, 307213802011837003346320048243705086348060

Offset: 0

Peter Bala, Jul 13 2023

A295458, defined by A295458(n) = (30*n)!*(5*n)!*(4*n)! / ((15*n)!*(10*n)!*(8*n)!*(6*n)!), is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin (see Bober, Table 2, Entry 28). Here we are essentially considering the sequence {A295458(n/2) : n >= 0}. Fractional factorials are defined in terms of the gamma function; for example, (5*n/2)! := Gamma(1 + 5*n/2).
This sequence is only conjecturally an integer sequence.
Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and all positive integers n and r.

J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc., 79, Issue 2, (2009), 422-444.

Cf. A276100, A276101, A276102, A295431, A295458, A347854, A347855, A347856, A347857, A347858, A364173 - A364185.

seq( simplify((15*n)!*(5*n/2)!*(2*n)!/((15*n/2)!*(5*n)!*(4*n)!*(3*n)!)), n = 0..15);

a(n) ~ c^n * 1/sqrt(12*Pi*n), where c = (3^4)*(5^5) * sqrt(3)/2.

a(n) = 43200*(15*n - 1)*(15*n - 7)*(15*n - 11)*(15*n - 13)*(15*n - 17)*(15*n - 19)*(15*n - 23)*(15*n - 29)/(n*(n - 1)*(3*n - 2)*(3*n - 4)*(4*n - 1)*(4*n - 3)*(4*n - 5)*(4*n - 7))*a(n-2) with a(0) = 1 and a(1) = 35840.

A364178 a(n) = (10n)!(3n)!(n/2)!/((6n)!(5n)!(3n/2)!n!).

Original entry on oeis.org

1, 168, 83980, 48664320, 29966636700, 19075222663168, 12398706131799988, 8175717823943147520, 5447952226877283703580, 3659442300478634742251520, 2473617870747229982625186480, 1680586987551894402985233481728, 1146602219745194113307246953503300

Offset: 0

Peter Bala, Jul 13 2023

A295470, defined by A295470(n) = (20*n)!*(6*n)!*n! / ((12*n)!*(10*n)!*(3*n)!*(2*n)!), is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin (see Bober, Table 2, Entry 40). Here we are essentially considering the sequence {A295470(n/2) : n >= 0}. Fractional factorials are defined in terms of the gamma function; for example, (3*n/2)! := Gamma(1 + 3*n/2).
This sequence is only conjecturally an integer sequence.
Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and all positive integers n and r.

J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc., 79, Issue 2, (2009), 422-444.

Cf. A276100, A276101, A276102, A295431, A295470, A347854, A347855, A347856, A347857, A347858, A364173 - A364185.

seq( simplify((10*n)!*(3*n)!*(n/2)!/((6*n)!*(5*n)!*(3*n/2)!*n!)), n = 0..15);

a(n) ~ c^n * 1/sqrt(6*Pi*n), where c = (10/3)^5 * sqrt(3).

a(n) = 1600*(10*n - 1)*(10*n - 3)*(10*n - 7)*(10*n - 9)*(10*n - 11)*(10*n - 13)*(10*n - 17)*(10*n - 19)/(27*n*(n - 1)*(3*n - 2)*(3*n - 4)*(6*n - 1)*(6*n - 5)*(6*n - 7)*(6*n - 11))*a(n-2) with a(0) = 1 and a(1) = 168.

A364179 a(n) = (10n)!(n/2)!/((5n)!(4n)!(3*n/2)!).

Original entry on oeis.org

1, 840, 2771340, 10754814720, 44524428808860, 190847602744995840, 835982760936614190900, 3716634993696885851422720, 16702642470437308383606668060, 75679458912906782280286032887808, 345116202503279265243707597937393840, 1581997780375359530321517073184807976960

Offset: 0

Peter Bala, Jul 13 2023

A295471, defined by A295471(n) = (20*n)!*n! / ((10*n)!*(8*n)!*(3*n)!), is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin (see Bober, Table 2, Entry 41). Here we are essentially considering the sequence {A295471(n/2) : n >= 0}. Fractional factorials are defined in terms of the gamma function; for example, (3*n/2)! := Gamma(1 + 3*n/2).
This sequence is only conjecturally an integer sequence.
Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and all positive integers n and r.

J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc., 79, Issue 2, (2009), 422-444.

Cf. A276100, A276101, A276102, A295431, A295471, A347854, A347855, A347856, A347857, A347858, A364173 - A364185.

seq( simplify((10*n)!*(n/2)!/((5*n)!*(4*n)!*(3*n/2)!)), n = 0..15);

a(n) ~ c^n * 1 /sqrt(12*Pi*n), where c = (2^3)*(5^5)/(3^2) * sqrt(3).

a(n) = 1600*(10*n - 1)*(10*n - 3)*(10*n - 7)*(10*n - 9)*(10*n - 11)*(10*n - 13)*(10*n - 17)*(10*n - 19)/(n*(3*n - 1)*(3*n - 2)*(3*n - 4)*(4*n - 1)*(4*n - 3)*(4*n - 5)*(4*n - 7))*a(n-2) with a(0) = 1 and a(1) = 840.

A364180 a(n) = (10n)!(n/2)!/((5n)!(7n/2)!(2*n)!).

Original entry on oeis.org

1, 1152, 5542680, 31473008640, 190818980609400, 1198265754978353152, 7691041400616850556280, 50107639155283424528302080, 330014847932376708502470210680, 2191489080600524699617120065945600, 14647137653300940580784413641872332680

Offset: 0

Peter Bala, Jul 13 2023

A061164, defined by A061164(n) = (20*n)!*n! / ((10*n)!*(7*n)!*(4*n)!), is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin (see Bober, Table 2, Entry 43). Here we are essentially considering the sequence {A061164(n/2) : n >= 0}. Fractional factorials are defined in terms of the gamma function; for example, (7*n/2)! := Gamma(1 + 7*n/2).
This sequence is only conjecturally an integer sequence.
Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and all positive integers n and r.

J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc., 79, Issue 2, (2009), 422-444.

Cf. A061164, A276100, A276101, A276102, A295431, A347854, A347855, A347856, A347857, A347858, A364173 - A364185.

seq( simplify((10*n)!*(n/2)!/((5*n)!*(7*n/2)!*(2*n)!)), n = 0..15);

a(n) ~ c^n * 1/sqrt(14*Pi*n), where c = (2^11)*(5^5)/(7^4) * sqrt(7).

a(n) = 409600*(10*n - 1)*(10*n - 3)*(10*n - 7)*(10*n - 9)*(10*n - 11)*(10*n - 13)*(10*n - 17)*(10*n - 19)/(7*n*(n - 1)*(7*n - 2)*(7*n - 4)*(7*n - 6)*(7*n - 8)*(7*n - 10)*(7*n - 12))*a(n-2) with a(0) = 1 and a(1) = 1152.

A364181 a(n) = (10n)!(3n/2)!/((5n)!(9n/2)!(2n)!).

Original entry on oeis.org

1, 384, 461890, 638582784, 935387159850, 1414457284624384, 2182519096151533552, 3414991108739243704320, 5398397695681095146608490, 8600772808890306913527398400, 13787702861800799166026014363140, 22213518902232966637201617101783040, 35936545440404705429404600374145350960

Offset: 0

Peter Bala, Jul 13 2023

A295475, defined by A295475(n) = (20*n)!*(3*n)! / ((10*n)!*(9*n)!*(4*n)!), is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin (see Bober, Table 2, Entry 45). Here we are essentially considering the sequence {A295475(n/2) : n >= 0}. Fractional factorials are defined in terms of the gamma function; for example, (3*n/2)! := Gamma(1 + 3*n/2).
This sequence is only conjecturally an integer sequence.
Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and all positive integers n and r.

J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc., 79, Issue 2, (2009), 422-444.

Cf. A276100, A276101, A276102, A295431, A295475, A347854, A347855, A347856, A347857, A347858, A364173 - A364185.

seq( simplify((10*n)!*(3*n/2)!/((5*n)!*(9*n/2)!*(2*n)!)), n = 0..15);

a(n) ~ c^n * 1/sqrt(6*Pi*n), where c = (2^11)*(5^5)/(3^8)*sqrt(3).

a(n) = 409600*(10*n - 1)*(10*n - 3)*(10*n - 7)*(10*n - 9)*(10*n - 11)*(10*n - 13)*(10*n - 17)*(10*n - 19)/(27*n*(n - 1)*(9*n - 2)*(9*n - 4)*(9*n - 8)*(9*n - 10)*(9*n - 14)*(9*n - 16))*a(n-2) with a(0) = 1 and a(1) = 384

A364182 a(n) = (12n)!(n/2)!/((6n)!(4n)!(5*n/2)!).

Original entry on oeis.org

1, 7392, 267711444, 11489451294720, 527048385075849780, 25051434899696246587392, 1217325447549161369383451760, 60050961586064738516089033457664, 2994861478939539397101967737771147060, 150602318360773064327512837557840362078208

Offset: 0

Peter Bala, Jul 13 2023

A295477, defined by A295477(n) = (24*n)!*n! / ((12*n)!*(8*n)!*(5*n)!), is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin (see Bober, Table 2, Entry 47). Here we are essentially considering the sequence {A295477(n/2) : n >= 0}. Fractional factorials are defined in terms of the gamma function; for example, (5*n/2)! := Gamma(1 + 5*n/2).
This sequence is only conjecturally an integer sequence.
Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and all positive integers n and r.

J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc., 79, Issue 2, (2009), 422-444.

Cf. A276100, A276101, A276102, A295431, A295477, A347854, A347855, A347856, A347857, A347858, A364173 - A364185.

seq( simplify((12*n)!*(n/2)!/((6*n)!*(4*n)!*(5*n/2)!)), n = 0..15);

a(n) ~ c^n * 1/sqrt(20*Pi*n), where c = (2^12)*(3^6)/(5^3) * sqrt(5).

a(n) = 82944*(12*n - 1)*(12*n - 5)(12*n - 7)*(12*n - 11)*(12*n - 13)*(12*n - 17)*(12*n - 19)*(12*n - 23)/(5*n*(n - 1)*(2*n - 1)*(2*n - 3)*(5*n - 2)*(5*n - 4)*(5*n - 6)*(5*n - 8))*a(n-2) with a(0) = 1 and a(1) = 7392

A364184 a(n) = (12n)!(2n)!(3n/2)!/((6n)!(9n/2)!(4n)!*n!).

Original entry on oeis.org

1, 1408, 6374082, 32993443840, 180669266788650, 1020694137466257408, 5882199787281395215344, 34369110490167819009785856, 202857467914154836183288657770, 1206640354461153104738279049134080, 7221430962039777689508936047385667332

Offset: 0

Peter Bala, Jul 13 2023

A295481, defined by A295481(n) = (24*n)!*(4*n)!*(3*n)! / ((12*n)!*(9*n)!*(8*n)!*(2*n)!), is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin (see Bober, Table 2, Entry 51). Here we are essentially considering the sequence {A295481(n/2) : n >= 0}. Fractional factorials are defined in terms of the gamma function; for example, (9*n/2)! := Gamma(1 + 9*n/2).
This sequence is only conjecturally an integer sequence.
Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and all positive integers n and r.

J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc., 79, Issue 2, (2009), 422-444.

Cf. A276100, A276101, A276102, A295431, A295481, A347854, A347855, A347856, A347857, A347858, A364173 - A364183.

seq( simplify((12*n)!*(2*n)!*(3*n/2)!/((6*n)!*(9*n/2)!*(4*n)!*n!)), n = 0..15);

a(n) ~ c^n * 1/sqrt(6*Pi*n), where c = (2^15)/(3^2) * sqrt(3).

a(n) = 49152*(12*n - 1)*(12*n - 5)*(12*n - 7)*(12*n - 11)*(12*n - 13)*(12*n - 17)*(12*n - 19)*(12*n - 23)/(n*(n - 1)*(9*n - 2)*(9*n - 4)*(9*n - 8)*(9*n - 10)*(9*n - 14)*(9*n - 16))*a(n-2) with a(0) = 1 and a(1) = 1408.

cp's OEIS Frontend

A364172 a(n) = (6*n)!*(n/3)!/((3*n)!*(2*n)!*(4*n/3)!).

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A364183 a(n) = (12*n)!*(2*n)!*(n/2)!/((6*n)!*(4*n)!*(7*n/2)!*n!).

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A364176 a(n) = (15*n)!*(5*n/2)!*(2*n)!/((15*n/2)!*(6*n)!*(5*n)!*n!).

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A364177 a(n) = (15*n)!*(5*n/2)!*(2*n)!/((15*n/2)!*(5*n)!*(4*n)!*(3*n)!).

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A364178 a(n) = (10*n)!*(3*n)!*(n/2)!/((6*n)!*(5*n)!*(3*n/2)!*n!).

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A364179 a(n) = (10*n)!*(n/2)!/((5*n)!*(4*n)!*(3*n/2)!).

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Maple

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A364180 a(n) = (10*n)!*(n/2)!/((5*n)!*(7*n/2)!*(2*n)!).

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A364181 a(n) = (10*n)!*(3*n/2)!/((5*n)!*(9*n/2)!*(2*n)!).

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A364172 a(n) = (6n)!(n/3)!/((3n)!(2n)!(4*n/3)!).

A364183 a(n) = (12n)!(2n)!(n/2)!/((6n)!(4n)!(7n/2)!n!).

A364176 a(n) = (15n)!(5n/2)!(2n)!/((15n/2)!(6n)!(5n)!*n!).

A364177 a(n) = (15n)!(5n/2)!(2n)!/((15n/2)!(5n)!(4n)!(3n)!).

A364178 a(n) = (10n)!(3n)!(n/2)!/((6n)!(5n)!(3n/2)!n!).

A364179 a(n) = (10n)!(n/2)!/((5n)!(4n)!(3*n/2)!).

A364180 a(n) = (10n)!(n/2)!/((5n)!(7n/2)!(2*n)!).

A364181 a(n) = (10n)!(3n/2)!/((5n)!(9n/2)!(2n)!).

A364182 a(n) = (12n)!(n/2)!/((6n)!(4n)!(5*n/2)!).

A364184 a(n) = (12n)!(2n)!(3n/2)!/((6n)!(9n/2)!(4n)!*n!).