A295431 -id:A295431 - cp's OEIS frontend

A295466 a(n) = (15n)!(6n)!n!/((12n)!(5n)!(3n)!(2*n)!).

1, 1365, 6531525, 36155455908, 212422672667205, 1289347174337943240, 7988578284829726217700, 50202452144162092092126900, 318773474802860214828983563845, 2040213003374489097738224937634800, 13139471029644309699515037254016803400, 85049362365373576290876509860293890825640

Offset: 0

Gheorghe Coserea, Nov 28 2017

nonn

Cf. A295431.

G.f.: hypergeom([1/15, 2/15, 4/15, 7/15, 8/15, 11/15, 13/15, 14/15], [1/12, 1/4, 5/12, 1/2, 7/12, 3/4, 11/12], 7119140625/1048576*x).

A295467 a(n) = (15n)!n!/((8n)!(5n)!(3*n)!).

Original entry on oeis.org

1, 45045, 9704539845, 2437781614596900, 651257537955459119685, 179888173423256639605246920, 50740927694468549286238961648100, 14520123213959375344187178875204676660, 4199008318481460680983513774150641241683525, 1224055446916973458193847787407419946733555964400

Offset: 0

Gheorghe Coserea, Nov 28 2017

nonn

Cf. A295431.

G.f.: hypergeom([1/15, 2/15, 4/15, 7/15, 8/15, 11/15, 13/15, 14/15], [1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8], 5189853515625/16777216*x).

A295468 a(n) = (30n)!(5n)!(3n)!(2n)!/((15n)!(10n)!(8n)!(6n)!*n!).

Original entry on oeis.org

1, 2772725670, 40337629723514950470, 671214073076279711920414071900, 11815915461993730897935386470911734090310, 214599520915474479942307452790907076147031402434420, 3975780607168011583140781349809187812166883920578195778723100

Offset: 0

Gheorghe Coserea, Nov 28 2017

nonn

Cf. A295431.

G.f.: hypergeom([1/30, 7/30, 11/30, 13/30, 17/30, 19/30, 23/30, 29/30], [1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8], 5189853515625/256*x).

A295472 a(n) = (20n)!(3n)!(2n)!/((10n)!(8n)!(6n)!*n!).

Original entry on oeis.org

1, 277134, 289119667590, 343884311368413900, 432365948166678100986950, 560667054442505107362561718884, 741516037730263944066134968787982492, 994213662322534065297017671627388489353560, 1346503768567844163328344622811399413074559930950

Offset: 0

Gheorghe Coserea, Nov 28 2017

nonn

Cf. A295431.

G.f.: hypergeom([1/20, 3/20, 7/20, 9/20, 11/20, 13/20, 17/20, 19/20], [1/8, 1/6, 3/8, 1/2, 5/8, 5/6, 7/8], 39062500/27*x).

A295474 a(n) = (20n)!(7n)!(2n)!/((14n)!(10n)!(4n)!*n!).

Original entry on oeis.org

1, 3230, 28539654, 285775341740, 3020260664350790, 32901338385286966980, 365436597301046276738460, 4114133256443603065995388440, 46780615678270404496871840084550, 536005188259178667341474315575006676, 6178692020351504220300189963052978836404, 71573683420303750522221857739718191020815080

Offset: 0

Gheorghe Coserea, Nov 28 2017

nonn

Cf. A295431.

G.f.: hypergeom([1/20, 3/20, 7/20, 9/20, 11/20, 13/20, 17/20, 19/20], [1/14, 3/14, 5/14, 1/2, 9/14, 11/14, 13/14], 10000000000/823543*x).

A295476 a(n) = (20n)!(9n)!(6n)!/((18n)!(10n)!(4n)!(3n)!).

Original entry on oeis.org

1, 190, 95238, 54503020, 32989143110, 20598742103940, 13119794010495900, 8472058672655664024, 5526367916474032140870, 3632882290435237504630420, 2402807931482709593805719988, 1597125851104281978332538662760, 1065954690102347966626416591930140

Offset: 0

Gheorghe Coserea, Nov 28 2017

nonn

Cf. A295431.

Table[Times@@(#!&/@(n{20,9,6}))/Times@@(#!&/@(n{18,10,4,3})),{n,0,20}] (* Harvey P. Dale, May 12 2018 *)

G.f.: hypergeom([1/20, 3/20, 7/20, 9/20, 11/20, 13/20, 17/20, 19/20], [1/18, 5/18, 7/18, 1/2, 11/18, 13/18, 17/18], 10000000000/14348907*x).

A295480 a(n) = (24n)!(7n)!(4n)!/((14n)!(12n)!(8n)!*n!).

Original entry on oeis.org

1, 44574, 5512410886, 770205244644780, 113493385509758105670, 17232366122881987992336324, 2667342121225036254551827535260, 418446897672874336912068988103878680, 66297659655162356848194576725889346695750, 10584121412183743053352396456036397314784501460

Offset: 0

Gheorghe Coserea, Nov 28 2017

nonn

Cf. A295431.

G.f.: hypergeom([1/24, 5/24, 7/24, 11/24, 13/24, 17/24, 19/24, 23/24], [1/14, 3/14, 5/14, 1/2, 9/14, 11/14, 13/14], 139314069504/823543*x).

A295482 a(n) = (24n)!(9n)!(6n)!(4n)!/((18n)!(12n)!(8n)!(3n)!(2n)!).

Original entry on oeis.org

1, 2622, 18395142, 146893400940, 1239644505129030, 10788772828908810372, 95762108788362083967900, 861689801350732909790930328, 7831988825786019814338666836550, 71735998233558658268051328890615700, 661068900973188409771193653810093715892

Offset: 0

Gheorghe Coserea, Nov 28 2017

nonn

Cf. A295431.

G.f.: hypergeom([1/24, 5/24, 7/24, 11/24, 13/24, 17/24, 19/24, 23/24], [1/18, 5/18, 7/18, 1/2, 11/18, 13/18, 17/18], 262144/27*x).

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.

A364174 a(n) = (9n)!(5n/2)!(3n/2)!/((5n)!(9n/2)!(3n)!*(n/2)!).

Original entry on oeis.org

1, 48, 4862, 549120, 65132550, 7945986048, 987291797996, 124259864002560, 15789207515217990, 2021092963752345600, 260227401685879140612, 33665720694993527504896, 4372592850984736084611996, 569819472537519480058675200, 74468439316740019538310543000

Offset: 0

Peter Bala, Jul 13 2023

A295442, defined by A295442(n) = (18*n)!*(5*n)!*(3*n)!/((10*n)!*(9*n)!*(6*n)!*n!), is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin (see Bober, Table 2, Entry 12). Here we are essentially considering the sequence {A295442(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, A295442, A347854, A347855, A347856, A347857, A347858, A364172 - A364185.

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

a(n) ~ c^n * 1/sqrt(2*Pi*n), where c = 2*(3^7)/(5^3) * sqrt(15) = 135.5234332504899....

a(n) = 108*(9*n - 1)*(9*n - 5)*(9*n - 7)*(9*n - 11)*(9*n - 13)*(9*n - 17)/(5*n*(n - 1)*(5*n - 1)*(5*n - 3)*(5*n - 7)*(5*n - 9))*a(n-2) for n >= 2 with a(0) = 1 and a(1) = 48.

cp's OEIS Frontend

A295466 a(n) = (15*n)!*(6*n)!*n!/((12*n)!*(5*n)!*(3*n)!*(2*n)!).

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A295467 a(n) = (15*n)!*n!/((8*n)!*(5*n)!*(3*n)!).

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A295468 a(n) = (30*n)!*(5*n)!*(3*n)!*(2*n)!/((15*n)!*(10*n)!*(8*n)!*(6*n)!*n!).

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A295472 a(n) = (20*n)!*(3*n)!*(2*n)!/((10*n)!*(8*n)!*(6*n)!*n!).

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A295474 a(n) = (20*n)!*(7*n)!*(2*n)!/((14*n)!*(10*n)!*(4*n)!*n!).

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A295476 a(n) = (20*n)!*(9*n)!*(6*n)!/((18*n)!*(10*n)!*(4*n)!*(3*n)!).

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Mathematica

Formula

A295480 a(n) = (24*n)!*(7*n)!*(4*n)!/((14*n)!*(12*n)!*(8*n)!*n!).

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A295482 a(n) = (24*n)!*(9*n)!*(6*n)!*(4*n)!/((18*n)!*(12*n)!*(8*n)!*(3*n)!*(2*n)!).

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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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Maple

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A364174 a(n) = (9*n)!*(5*n/2)!*(3*n/2)!/((5*n)!*(9*n/2)!*(3*n)!*(n/2)!).

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Maple

Formula

A295466 a(n) = (15n)!(6n)!n!/((12n)!(5n)!(3n)!(2*n)!).

A295467 a(n) = (15n)!n!/((8n)!(5n)!(3*n)!).

A295468 a(n) = (30n)!(5n)!(3n)!(2n)!/((15n)!(10n)!(8n)!(6n)!*n!).

A295472 a(n) = (20n)!(3n)!(2n)!/((10n)!(8n)!(6n)!*n!).

A295474 a(n) = (20n)!(7n)!(2n)!/((14n)!(10n)!(4n)!*n!).

A295476 a(n) = (20n)!(9n)!(6n)!/((18n)!(10n)!(4n)!(3n)!).

A295480 a(n) = (24n)!(7n)!(4n)!/((14n)!(12n)!(8n)!*n!).

A295482 a(n) = (24n)!(9n)!(6n)!(4n)!/((18n)!(12n)!(8n)!(3n)!(2n)!).

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

A364174 a(n) = (9n)!(5n/2)!(3n/2)!/((5n)!(9n/2)!(3n)!*(n/2)!).