A347855 -id:A347855 - cp's OEIS frontend

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

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

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

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

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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!).