A008978 -id:A008978 - cp's OEIS frontend

A071552 a(n) = (10n)!/n!^10.

1, 3628800, 2375880867360000, 4386797336285844480000000, 12868639981414579848070084500000000, 49120458506088132224064306071170476903628800

Offset: 0

Benoit Cloitre, May 30 2002

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Sequences (k*n)!/n!^k: A000984 (k = 2), A006480 (k = 3), A008977 (k = 4), A008978 (k = 5), A008979 (k = 6), A071549 (k = 7), A071550 (k = 8), A071551 (k = 9).

[Factorial(10*n)/Factorial(n)^10: n in [0..20]]; // Vincenzo Librandi, Aug 13 2014

Table[(10n)!/(n)!^10, {n, 0, 20}] (* Vincenzo Librandi, Aug 13 2014 *)

A089659 a(n) = S1(n,2), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).

Original entry on oeis.org

0, 2, 19, 104, 440, 1600, 5264, 16128, 46848, 130560, 352000, 923648, 2369536, 5963776, 14766080, 36044800, 86900736, 207224832, 489357312, 1145569280, 2660761600, 6136266752, 14060355584, 32027705344, 72561459200, 163577856000, 367068708864, 820204535808

Offset: 0

N. J. A. Sloane, Jan 04 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jun Wang and Zhizheng Zhang, On extensions of Calkin's binomial identities, Discrete Math., 274 (2004), 331-342.
Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Columns : A000012, A000012, A000984, A006480, A008977, A008978.

Sequences of S1(n, t): A001792 (t=0), A089658 (t=1), this sequence (t=2), A089660 (t=3), A089661 (t=4), A089662 (t=5), A089663 (t=6).

I:=[0,2,19,104]; [n le 4 select I[n] else 8*Self(n-1)-24*Self(n-2)+32*Self(n-3)-16*Self(n-4): n in [1..41]]; // Vincenzo Librandi, Jun 22 2016

LinearRecurrence[{8,-24,32,-16}, {0,2,19,104}, 40] (* Vincenzo Librandi, Jun 22 2016 *)

[2^(n-3)*n*(7*n^2 + 12*n + 5)/3 for n in (0..40)] # G. C. Greubel, May 24 2022

a(n) = 2^(n-3)*n*(7*n^2 + 12*n + 5)/3. (see Wang and Zhang p. 333)
From Chai Wah Wu, Jun 21 2016: (Start)
a(n) = 8*a(n-1) - 24*a(n-2) + 32*a(n-3) - 16*a(n-4) for n > 3.
G.f.: x*(2 + 3*x)/(1 - 2*x)^4. (End)
E.g.f.: x*(12 + 33*x + 14*x^2)*exp(2*x)/6. - Ilya Gutkovskiy, Jun 21 2016

A322252 a(0) = 1 and a(n) = (5n)!/(5!n!^5) for n > 0.

Original entry on oeis.org

1, 1, 945, 1401400, 2546168625, 5194672859376, 11423951396577720, 26478825654361766400, 63805953776276649848625, 158421985022100255941485000, 402789797982510165934296910320, 1044048983553856888083223814102400, 2749848597736878877579660426025283000

Offset: 0

Seiichi Manyama, Nov 30 2018

nonn

Row 5 of A060540.

Cf. A000142, A100734, A008978, A060542, A082368.

[1] cat [Factorial(5*n)/(120*Factorial(n)^5):n in [1..12]]; // Marius A. Burtea, Feb 18 2020

a[n_]:=(5*n)!/(5!*n!^5); Array[a, 20] (* or *) CoefficientList[Series[HypergeometricPFQ[{1/5, 2/5, 3/5, 4/5}, {1, 1, 1}, 3125 x]/(120 x) , {x, 0, 20}], x] (* Stefano Spezia, Dec 01 2018 *)

O.g.f.: F({1/5, 2/5, 3/5, 4/5}, {1, 1, 1}, 3125*x)/(120*x), where F is the generalized hypergeometric function. - Stefano Spezia, Dec 01 2018

a(n) = (1/5!)*A008978(n) for n >= 1. - Peter Bala, Feb 18 2020

A352651 a(n) = ( binomial(5n,2n)binomial(5n/2,2n)binomial(2n,n)^2 ) / binomial(5n/2,n)^2.

Original entry on oeis.org

1, 12, 378, 14700, 629850, 28540512, 1341310320, 64676424384, 3178603964250, 158529793422000, 7999466594747628, 407514796591710600, 20924507330066816112, 1081581197431986720000, 56225684939117297889600, 2937292879652230377427200, 154108110471294720105987930

Offset: 0

Peter Bala, Mar 25 2022

We write x! as shorthand for Gamma(x+1) and binomial(x,y) as shorthand for x!/(y!*(x-y)!) = Gamma(x+1)/(Gamma(y+1)*Gamma(x-y+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. Soundararajan gives many examples of two-parameter families of integral factorial ratio sequences of height 2.
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. See A276098 and the cross references for further examples of this type.
Conjecture: the supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) hold for all primes p >= 5 and positive integers n and k. The case n = k = 1 is easily proved.
More generally, for an integer N not equal to 0 or 1, the height 2 factorial ratio sequence whose n-th term is given by ( binomial(N*n,2*n)* binomial(N*n/2,2*n)* binomial(2*n,n)^2 )/binomial(N*n/2,n)^2 is conjectured to be integral and satisfy the same supercongruences. This is the case N = 5. See A352652 (N = 7)

			Examples of supercongruences:
a(2*7) - a(2) = 56225684939117297889600 - 378 = 2*(3^3)*(7^4)*6553*411473* 160830097 == 0 (mod 7^4).
a(13) - a(1) = 1081581197431986720000 - 12 = (2^2)*3*(13^3)* 41024927834622467 == 0 (mod 13^3)

Peter Bala, Some integer ratios of factorials
J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, 2007, arXiv:0709.1977v1 [math.NT], 2007; J. London Math. Soc., 79, Issue 2, (2009), 422-444.
K. Soundararajan, Integral factorial ratios: irreducible examples with height larger than 1, Phil. Trans. R. Soc.A378: 2018044, 2019.

Cf. A008978, A262732, A275652, A276098, A276100, A276101, A276102, A352652.

a := n -> if n = 0 then 1 elif n = 1 then 12 else
5*(3*n - 2)*(3*n - 4)*(5*n - 1)*(5*n - 3)*(5*n - 7)*(5*n - 9)/(n^2*(n - 1)^2*(3*n - 1)*(3*n - 5))*a(n-2) end if:
seq(a(n), n = 0..20);

from math import factorial
from sympy import factorial2
def A352651(n): return int(factorial(5*n)*factorial2(3*n)**2//factorial(3*n)//factorial2(5*n)//factorial(n)**2//factorial2(n)) # Chai Wah Wu, Aug 08 2023

a(n) = (5*n)!*(3*n/2)!^2/( (3*n)!*(5*n/2)!*n!^2*(n/2)! ).
a(n) = 3*Sum_{k = 0..n} (-1)^(n+k)*binomial(5*n,n-k)*binomial(3*n+k-1,k)^2 for n >= 1 (this formula shows the sequence is integral).
a(n) = 3*Sum_{k = 0..n} binomial(2*n-k-2,n-k)*binomial(3*n-1,k)^2 for n >= 1.
a(n) = 3 * [x^n] ( (1 - x)^(2*n) * P(3*n-1,(1 + x)/(1 - x)) ) for n >= 1, where P(n,x) denotes the n-th Legendre polynomial.
a(n) ~ (sqrt(3)/Pi)*(5^n)^(5/2)*( 1/(2*n) - 2/(15*n^2) + 4/(225*n^3) + O(1/n^4) ).
a(n) = A008978(n)/A275652(n).
a(n) = binomial(3*n/2,n)*A262732(n).
a(n) = 3*(-1)^n*binomial(5*n,n)*hypergeom([-n, 3*n, 3*n], [1, 4*n+1], 1) for n >= 1.
a(n) = 5*(3*n-2)*(3*n-4)*(5*n-1)*(5*n-3)*(5*n-7)*(5*n-9)/(n^2*(n-1)^2*(3*n- 1)*(3*n-5)) * a(n-2) with a(0) = 1 and a(1) = 12.
a(p) == 12 (mod p^3) for prime p >= 5.
O.g.f.: A(x) = hypergeom([1/10, 3/10, 7/10, 9/10, 1/3, 2/3], [1/6, 5/6, 1/2, 1/2, 1], (5^5)*x^2) + 12*x*hypergeom([3/5, 4/5, 6/5, 7/5, 5/6, 7/6], [2/3, 4/3, 3/2, 3/2, 1], (5^5)*x^2).

A361636 Diagonal of the rational function 1/(1 - vwx*yz (1 + 1/v + 1/w + 1/x + 1/y + 1/z)).

Original entry on oeis.org

1, 1, 1, 1, 121, 721, 2521, 6721, 128521, 1277641, 7539841, 32527441, 281835841, 3031468441, 23779315561, 139431015361, 962322302761, 9034098300361, 79726215362761, 569831799431881, 3952559737085401, 32660742079719601, 289694072383115401

Offset: 0

Seiichi Manyama, Mar 19 2023

nonn

Diagonal of the rational function 1/(1 - (v^4 + w^4 + x^4 + y^4 + z^4 + v*w*x*y*z)). - Seiichi Manyama, Jul 04 2025

Winston de Greef, Table of n, a(n) for n = 0..1064

Cf. A001850, A208425, A274783.
Cf. A008978.
Cf. A082489, A361703.

Table[Sum[(n + k)!/(k!^5*(n - 4*k)!), {k, 0, n/4}], {n, 0, 25}] (* Vaclav Kotesovec, Mar 19 2023 *)

a(n) = sum(k=0, n\4, (n+k)!/(k!^5*(n-4*k)!));

a(n) = Sum_{k=0..floor(n/4)} (n+k)!/(k!^5 * (n-4*k)!).
G.f.: Sum_{k>=0} (5*k)!/k!^5 * x^(4*k)/(1-x)^(5*k+1).
Recurrence: n^4*(5*n - 19)*(5*n - 18)*(5*n - 17)*(5*n - 14)*(5*n - 13)*(5*n - 9)*a(n) = (5*n - 19)*(5*n - 18)*(5*n - 14)*(625*n^7 - 6125*n^6 + 23025*n^5 - 43195*n^4 + 45394*n^3 - 28716*n^2 + 10144*n - 1536)*a(n-1) - (5*n - 19)*(31250*n^9 - 568750*n^8 + 4441875*n^7 - 19516000*n^6 + 53172025*n^5 - 93366740*n^4 + 106140132*n^3 - 75781664*n^2 + 30987264*n - 5529600)*a(n-2) + (5*n - 4)*(31250*n^9 - 725000*n^8 + 7354375*n^7 - 42784750*n^6 + 157237100*n^5 - 378480620*n^4 + 596812963*n^3 - 594970390*n^2 + 340845072*n - 85743360)*a(n-3) + (5*n - 16)*(5*n - 9)*(5*n - 8)*(5*n - 4)*(78000*n^6 - 1450800*n^5 + 11179085*n^4 - 45672814*n^3 + 104341702*n^2 - 126378083*n + 63400710)*a(n-4) + (n-4)^4*(5*n - 14)*(5*n - 13)*(5*n - 12)*(5*n - 9)*(5*n - 8)*(5*n - 4)*a(n-5). - Vaclav Kotesovec, Mar 19 2023

A370295 G.f.: exp(Sum_{k>=1} (5k)!/(5!k!^5) * x^k/k).

Original entry on oeis.org

1, 1, 473, 467606, 637121154, 1039792179805, 1905441263652576, 3785382599457953517, 7981116324798212651066, 17613760342120835610374245, 40303877398793645855018120732, 94970269248783993542201925505548, 229287077006842005926064077532676555, 565001770629439341048001870559581136157

Offset: 0

Vaclav Kotesovec, Feb 14 2024

nonn

In general, for m>=2, if g.f. = exp(Sum_{k>=1} (m*k)!/(m!*k!^m) * x^k/k), then a(n,m) ~ c(m) * m^(m*n) / n^((m+1)/2), where c(m) = exp(HypergeometricPFQ[{1, 1, (m+1)/m, (m+2)/m, ... , (2*m-1)/m}, {2, 2, ...m-times... 2, 2}, 1] / m^m) / (m! * (2*Pi)^((m-1)/2) / sqrt(m)).

Limit_{m->oo} c(m) / (exp(m)/(m^m*(2*Pi)^(m/2))) = 1.

Cf. A008978, A322252, A229452, A370294, A333043.

CoefficientList[Series[Exp[Sum[(5*k)!/(5!*k!^5)*x^k/k, {k, 1, 20}]], {x, 0, 20}], x]
CoefficientList[Series[Exp[x*HypergeometricPFQ[{1, 1, 6/5, 7/5, 8/5, 9/5}, {2, 2, 2, 2, 2}, 3125*x]], {x, 0, 20}], x]

G.f. A(x) = G(x)^(1/120), where G(x) is the g.f. for A333043.

a(n) ~ c * 5^(5*n)/n^3, where c = exp(HypergeometricPFQ[{1, 1, 6/5, 7/5, 8/5, 9/5}, {2, 2, 2, 2, 2}, 1] / 3125) / (96*sqrt(5)*Pi^2) = 0.00047219161473962545263459216995582653262467228952818554361164671183728...

A001460 a(n) = (5n)!/((2n)!*(n!)^3).

Original entry on oeis.org

1, 60, 18900, 8408400, 4364860500, 2473653742560, 1483630051503600, 925833064837824000, 594927307937311420500, 391004487919622186610000, 261614105944603801295306400, 177601637048592673099585584000, 122027661025630720013771117910000

Offset: 0

N. J. A. Sloane

James Spahlinger, Table of n, a(n) for n = 0..400
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.

Cf. A000984, A008978.

Cf. A268545 - A268555.

[Factorial(5*n)/(Factorial(2*n)*Factorial(n)^3):n in [0..15]]; // Marius A. Burtea, Feb 17 2020

f := n->(5*n)!/((2*n)!*(n!)^3);
seq((5*n)!/(n!)^5/binomial(2*n,n), n=0..15); # Zerinvary Lajos, Jun 28 2007

Table[(5 n)!/((2 n)! (n!)^3), {n, 0, 15}] (* or *)
Table[(5 n)!/(n!)^5/Binomial[2 n, n], {n, 0, 15}] (* Michael De Vlieger, Jul 18 2016 *)

a(n) = (5*n)!/((2*n)!*n!^3);  \\ Gheorghe Coserea, Jul 18 2016

f=factorial; [f(5*n)/(f(2*n)*f(n)^3) for n in range(16)] # G. C. Greubel, Sep 03 2023

a(n) = A008978(n)/A000984(n). - Zerinvary Lajos, Jun 28 2007
From Gheorghe Coserea, Jul 18 2016: (Start)
a(n) = [(xyzw)^(3n)] 1/(1-(w*x*y+w*z+x*z+y*z)).
a(n) ~ sqrt(5)/(4*Pi^(3/2)) * n^(-3/2) * (3125/4)^n.
0 = (-4*x^3+3125*x^6)*y'''' + (-18*x^2+37500*x^5)*y''' + (-10*x+117500*x^4)*y'' + (2+95000*x^3)*y' + (9720*x^2)*y, where y(x) = A(x^3). (End)
From Peter Bala, Dec 30 2019: (Start)
a(n) = binomial(3*n,n)*binomial(4*n,n)*binomial(5*n,n).
a(n) = ( [x^n](1 + x)^(3*n) ) * ( [x^n](1 + x)^(4*n) ) * ( [x^n](1 + x)^(5*n) ).
a(n) = [x^n]( F(x)^(60*n) ), where [x^n] is the coefficient extraction operator and where F(x) = 1 + x + 98*x^2 + 23861*x^3 + 7987534*x^4 + 3169655645*x^5 + 1398711076599*x^6 + ... appears to have integer coefficients. Cf. A008978. (End)
From Peter Bala, Feb 16 2020: (Start)
Congruences: a(m*p^k) == a(m*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers m and k - apply Mestrovic, equation 39, p. 12.
a(n) = [(x*y*z)^n] (1 + x + y + z)^(5*n). (End)
a(n) = a(n-1)*5*(5*n - 1)*(5*n - 2)*(5*n - 3)*(5*n - 4)/(2*n^3*(2*n - 1)). - Neven Sajko, Jul 22 2023

A333043 G.f.: exp(Sum_{k>=1} (5k)!/k!^5 x^k/k).

Original entry on oeis.org

1, 120, 63900, 63148000, 85136103750, 137629764435024, 250331826090382280, 494436455370401985600, 1037731227148399567352625, 2281874234819846601146115000, 5205960892339635531670022801628, 12237148815599682784939438806708960, 29483782935554473122496294160376815950

Offset: 0

Vaclav Kotesovec, Mar 06 2020

nonn

In general, if r>=2, m>0 and g.f. = exp(m * Sum_{k>=1} (r*k)!/k!^r * x^k/k), then a(n) ~ c(r,m) * m * r^(r*n + 1/2) / ((2*Pi)^((r-1)/2) * n^((r+1)/2)) , where c(r,m) = exp((m * r! / r^r) * HypergeometricPFQ[{1, 1, (r+1)/r, (r+2)/r, ... , (2*r-1)/r}, {2, 2, ...r-times... 2, 2}, 1]). - Vaclav Kotesovec, Feb 16 2024

Cf. A000108, A008978, A229451, A333042, A370295.

CoefficientList[Series[Exp[Sum[(5*k)!/k!^5*x^k/k, {k, 1, 20}]], {x, 0, 20}], x]
CoefficientList[Series[Exp[120*x*HypergeometricPFQ[{1, 1, 6/5, 7/5, 8/5, 9/5}, {2, 2, 2, 2, 2}, 3125*x]], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 09 2024 *)

a(n) ~ c * 5^(5*n)/n^3, where c = sqrt(5) * exp(24*HypergeometricPFQ[{1, 1, 6/5, 7/5, 8/5, 9/5}, {2, 2, 2, 2, 2}, 1] / 625) / (4*Pi^2) = 0.05943406... - Vaclav Kotesovec, Mar 06 2020, updated Feb 16 2024

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} A008978(k) * a(n-k). - Seiichi Manyama, Feb 09 2024

A367569 a(n) = Product_{k=0..n} (5*k)! / k!^5.

Original entry on oeis.org

1, 120, 13608000, 2288430144000000, 699207483978843840000000000, 435858496811697532778806061260800000000000, 597507154003470929939550139366865942134606725120000000000000, 1898554530971015145216561379837863419725314413457243266261094236160000000000000000

Offset: 0

Vaclav Kotesovec, Nov 23 2023

nonn

Cf. A000178, A008978, A268506.

Cf. A007685, A367567, A367568, A367570, A367571.

Table[Product[(5*k)!/k!^5, {k, 0, n}], {n, 0, 10}]
Table[Product[Binomial[5*k,k] * Binomial[4*k,k] * Binomial[3*k,k] * Binomial[2*k,k], {k, 0, n}], {n, 0, 10}]

a(n) = Product_{k=0..n} binomial(5*k,k) * binomial(4*k,k) * binomial(3*k,k) * binomial(2*k,k).
a(n) = A268506(n) / A000178(n)^5.
a(n) ~ A^(24/5) * Gamma(1/5)^(3/5) * Gamma(2/5)^(2/5) * Gamma(3/5)^(1/5) * 5^(5*n^2/2 + 3*n + 23/60) * exp(2*n - 2/5) / (n^(2*n + 7/5) * (2*Pi)^(2*n + 13/5)), where A is the Glaisher-Kinkelin constant A074962.
Equivalently, a(n) ~ A^(24/5) * Gamma(1/5)^(3/5) * Gamma(2/5)^(1/5) * 5^(5*n^2/2 + 3*n + 1/3) * exp(2*n - 2/5) / ((1 + sqrt(5))^(1/10) * 2^(2*n + 23/10) * Pi^(2*n + 12/5) * n^(2*n + 7/5)).

A381164 a(n) = Sum_{k=0..n} binomial(n,k)(5k)!/(k!)^5.

Original entry on oeis.org

1, 121, 113641, 168508561, 306213587881, 624890127114721, 1374618918516663841, 3187068298971939367561, 7682172545187676630759081, 19079663136489248380982551201, 48525227073661262262248690661841, 125818607409307965748858681991235961, 331488456546076036761442657285875590881

Offset: 0

Stefano Spezia, Feb 15 2025

nonn

Calabi-Yau series number 79.

S. Hassani, J.-M. Maillard, and N. Zenine, On the diagonals of rational functions: the minimal number of variables (unabridged version), arXiv:2502.05543 [math-ph], 2025. See p. 16.

Cf. A007318, A008978, A100734.

Cf. A381161, A381162, A381163, A381165.

a[n_]:=Sum[Binomial[n, k](5k)!/k!^5, {k, 0, n}]; Array[a, 13, 0]

G.f.: hypergeom([1/5, 2/5, 3/5, 4/5], [1, 1, 1], 5^5*x/(1-x))/(1-x).
a(n) = hypergeom([1/5, 2/5, 3/5, 4/5, -n], [1, 1, 1, 1], -5^5).
a(n) == 1 (mod 120).
a(n) ~ 2^n * 3^(n+2) * 521^(n+2) / (5^(19/2) * Pi^2 * n^2). - Vaclav Kotesovec, May 29 2025

cp's OEIS Frontend

A071552 a(n) = (10n)!/n!^10.

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A089659 a(n) = S1(n,2), where S1(n, t) = Sum_{k=0..n} (k^t * Sum_{j=0..k} binomial(n,j)).

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A322252 a(0) = 1 and a(n) = (5*n)!/(5!*n!^5) for n > 0.

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A352651 a(n) = ( binomial(5*n,2*n)*binomial(5*n/2,2*n)*binomial(2*n,n)^2 ) / binomial(5*n/2,n)^2.

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A361636 Diagonal of the rational function 1/(1 - v*w*x*y*z * (1 + 1/v + 1/w + 1/x + 1/y + 1/z)).

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A370295 G.f.: exp(Sum_{k>=1} (5*k)!/(5!*k!^5) * x^k/k).

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A001460 a(n) = (5*n)!/((2*n)!*(n!)^3).

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SageMath

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A333043 G.f.: exp(Sum_{k>=1} (5*k)!/k!^5 * x^k/k).

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A322252 a(0) = 1 and a(n) = (5n)!/(5!n!^5) for n > 0.

A352651 a(n) = ( binomial(5n,2n)binomial(5n/2,2n)binomial(2n,n)^2 ) / binomial(5n/2,n)^2.

A361636 Diagonal of the rational function 1/(1 - vwx*yz (1 + 1/v + 1/w + 1/x + 1/y + 1/z)).

A370295 G.f.: exp(Sum_{k>=1} (5k)!/(5!k!^5) * x^k/k).

A001460 a(n) = (5n)!/((2n)!*(n!)^3).

A333043 G.f.: exp(Sum_{k>=1} (5k)!/k!^5 x^k/k).

A381164 a(n) = Sum_{k=0..n} binomial(n,k)(5k)!/(k!)^5.