A364185 -id:A364185 - cp's OEIS frontend

A364173 a(n) = (9n)!(2n)!(3n/2)!/((9n/2)!(4n)!(3n)!*n!).

1, 128, 43758, 17039360, 7012604550, 2976412336128, 1288415796384780, 565399665327996928, 250622090889055155270, 111950839825145979207680, 50312973039218473430585508, 22723567527558510746926055424, 10304958075870392958137083227804

Offset: 0

Peter Bala, Jul 13 2023

A295440, defined by A295440(n) = (18*n)!*(4*n)!*(3*n)! / ((9*n)!*(8*n)!*(6*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 10). Here we are essentially considering the sequence {A295440(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, A295440, A347854, A347855, A347856, A347857, A347858, A364172 - A364185.

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

a(n) ~ c^n * 1/sqrt(4*Pi*n), where c = (3^7)/(2^3) * sqrt(3) = 473.4993895191418....

a(n) = 108*(9*n - 1)*(9*n - 5)*(9*n - 7)*(9*n - 11)*(9*n - 13)*(9*n - 17)/(n*(n - 1)*(4*n - 1)*(4*n - 3)*(4*n - 5)*(4*n - 7))*a(n-2) for n >= 2 with a(0) = 1 and a(1) = 128.

A363249 Leading digit of 9^n.

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Jul 15 2023

He, Xinwei; Hildebrand, A J; Li, Yuchen; Zhang, Yunyi, Complexity of Leading Digit Sequences, Discrete Mathematics and Theoretical Computer Science; 22 (2020), #14.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A008952, A060956, A111395, A362871, A363093, A364185.

Cf. A000030, A001019, A008570.

a[n_] := IntegerDigits[9^n][[1]]; Array[a, 100, 0] (* Amiram Eldar, Jul 15 2023 *)

PARI
```
a(n) = digits(9^n)[1];
```

a(n) = A000030(A001019(n)).

a(n) = A060956(2*n).

A362871 Leading digit of 6^n.

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Jul 15 2023

Cf. A008952, A060956, A111395, A363093, A363249, A364185.

Cf. A000030, A000400, A008567.

a[n_] := IntegerDigits[6^n][[1]]; Array[a, 100, 0] (* Amiram Eldar, Jul 15 2023 *)

PARI
```
a(n) = digits(6^n)[1];
```

a(n) = A000030(A000400(n)).

A363093 Leading digit of 7^n.

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Jul 15 2023

Cf. A008952, A060956, A111395, A362871, A363249, A364185.

Cf. A000030, A000420, A001903, A008568.

a[n_] := IntegerDigits[7^n][[1]]; Array[a, 100, 0] (* Amiram Eldar, Jul 15 2023 *)

PARI
```
a(n) = digits(7^n)[1];
```

a(n) = A000030(A000420(n)).

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

Original entry on oeis.org

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.

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.

A364175 a(n) = (6n)!(2n/3)!/((3n)!(2n)!(5n/3)!).

Original entry on oeis.org

1, 36, 3564, 408408, 49697388, 6249195036, 802241960520, 104466877291260, 13746018177013356, 1823169705017624880, 243331037661693468564, 32641262295291161362656, 4396944340992842923469640, 594371374049863341847620936, 80586283761263090599592845140

Offset: 0

Peter Bala, Jul 13 2023

A295445, defined by A295445(n) = (18*n)!*(2*n)! / ((9*n)!*(6*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 15). Here we are essentially considering the sequence {A295445(n/3) : n >= 0}. Fractional factorials are defined in terms of the gamma function; for example, (2*n/3)! := Gamma(1 + 2*n/3).
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, A295445, A347854, A347855, A347856, A347857, A347858, A364172 - A364185.

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

a(n) ~ c^n * 1/sqrt(5*Pi*n) where c = (1296/25)*20^(1/3) = 140.7154092442799....

a(n) = 93312*(2*n - 3)*(6*n - 1)*(6*n - 5)*(6*n - 7)*(6*n - 11)*(6*n - 13)*(6*n - 17)/(5*n*(n - 1)*(n - 2)*(5*n - 3)*(5*n - 6)*(5*n - 9)*(5*n - 12))*a(n-3) with a(0) = 1, a(1) = 36 and a(2) = 3564.

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.

cp's OEIS Frontend

A364173 a(n) = (9*n)!*(2*n)!*(3*n/2)!/((9*n/2)!*(4*n)!*(3*n)!*n!).

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A363249 Leading digit of 9^n.

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A362871 Leading digit of 6^n.

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A363093 Leading digit of 7^n.

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

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A364175 a(n) = (6*n)!*(2*n/3)!/((3*n)!*(2*n)!*(5*n/3)!).

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A364173 a(n) = (9n)!(2n)!(3n/2)!/((9n/2)!(4n)!(3n)!*n!).

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

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)!).

A364175 a(n) = (6n)!(2n/3)!/((3n)!(2n)!(5n/3)!).

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

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