A229050 -id:A229050 - cp's OEIS frontend

A034841 a(n) = (n^2)! / (n!)^n.

1, 1, 6, 1680, 63063000, 623360743125120, 2670177736637149247308800, 7363615666157189603982585462030336000, 18165723931630806756964027928179555634194028454000000, 53130688706387569792052442448845648519471103327391407016237760000000000

Offset: 0

Erich Friedman

nonn

The number of arrangements of 1,2,...,n^2 in an n X n matrix such that each row is increasing. - Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 12 2001
a(n) == 0 (mod n!). In fact (n^2)! == 0 (mod (n!)^n) by elementary combinatorics, a better result is (n^2)! == 0 (mod (n!)^(n+1)). - Amarnath Murthy, Jul 13 2005
a(n) is also the number of lattice paths from {n}^n to {0}^n using steps that decrement one component by 1. a(2) = 6: [(2,2), (1,2), (0,2), (0,1), (0,0)], [(2,2), (1,2), (1,1), (0,1), (0,0)], [(2,2), (1,2), (1,1), (1,0), (0,0)], [(2,2), (2,1), (1,1), (0,1), (0,0)], [(2,2), (2,1), (1,1), (1,0), (0,0)], [(2,2), (2,1), (2,0), (1,0), (0,0)]. - Alois P. Heinz, May 06 2013
Given n^2 distinguishable balls and n distinguishable urns, a(n) = the number of ways to place n balls in the i-th urn for all 1 <= i <= n, where n = n_1 + n_2 + ... + n_n. - Ross La Haye, Dec 28 2013

Alois P. Heinz and Tilman Piesk, Table of n, a(n) for n = 0..26 (first 20 terms from Alois P. Heinz)
Noah Lordi, Maedee Trank-Greene, Akira Kyle, and Joshua Combes, Quantum permutation puzzles with indistinguishable particles, arXiv:2410.22287 [quant-ph], 2024. See p. 8.

Cf. A000142, A039622, A229050, A229050, A340590.
Cf. A000984, A006480, A008977, A008978, A022915.
Main diagonal of A089759, A187783.

[Factorial(n^2) / Factorial(n)^n: n in [0..10]]; // Vincenzo Librandi, Oct 29 2014

a:= n-> (n^2)! / (n!)^n:
seq(a(n), n=0..10);  # Alois P. Heinz, Jul 24 2012

Prepend[Table[nn = n^2;nn! Coefficient[Series[(x^n/n!)^n, {x, 0, nn}], x^nn], {n, 1, 15}], 1] (* Geoffrey Critzer, Mar 08 2015 *)

a(n) = (n^2)! / (n!)^n; \\ Michel Marcus, Oct 28 2014

Using a higher order version of Stirling's formula (the "standard" formula appears in A000142) we have the asymptotic expression: a(n) ~ sqrt(2*Pi) * e^(-1/12) * n^(n^2 - n/2 + 1) / (2*Pi)^(n/2). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 13 2001

a(n) = Product_{k=1..n} binomial(k*n, n). - Vaclav Kotesovec, Mar 10 2019

a(0)=1 prepended by Tilman Piesk, Oct 28 2014

A082488 a(n) = Sum_{k = 0..n} C(n,k) * C(n+k,k) * C(n+2k,k) C(n+3*k,k).

Original entry on oeis.org

1, 25, 2641, 392641, 67982041, 12838867105, 2564949195985, 533008982952625, 114035552691160585, 24950692835328410305, 5557138347370070346601, 1255741805437716400557625, 287180884347761929741524361, 66343186345544102086872515761

Offset: 0

Emanuele Munarini, Apr 28 2003

Diagonal of the rational function 1/(1-(x + y + z + w + x*y*z*w)). - Gheorghe Coserea, Jul 15 2016

			G.f.: A(x) = 1 + 25*x + 2641*x^2 + 392641*x^3 + 67982041*x^4 + 12838867105*x^5 +...
where
A(x) = 1/(1-x) + (4!/1!^4)*x/(1-x)^5 + (8!/2!^4)*x^2/(1-x)^9 + (12!/3!^4)*x^3/(1-x)^13 + (16!/4!^4)*x^4/(1-x)^17 + (20!/5!^4)*x^5/(1-x)^21 +... [Hanna]
Equivalently,
A(x) = 1/(1-x) + 24*x/(1-x)^5 + 2520*x^2/(1-x)^9 + 369600*x^3/(1-x)^13 + 63063000*x^4/(1-x)^17 + 11732745024*x^5/(1-x)^21 +...+ A008977(n)*x^n/(1-x)^(4*n+1) +...

Alois P. Heinz, Table of n, a(n) for n = 0..400
A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.
Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

Cf. A081798.
Column k = 4 of A229142.
Cf. A008977, A001850, A082489, A229049, A229050.
Related to diagonal of rational functions: A268545-A268555.

[&+[Binomial(n, k)*Binomial(n+k, k)*Binomial(n+2*k, k)*Binomial(n+3*k, k): k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Oct 16 2018

with(combinat):
a:= n-> add(multinomial(n+3*k, n-k, k$4), k=0..n):
seq(a(n), n=0..15);  # Alois P. Heinz, Sep 23 2013

Table[Sum[Binomial[n,k]*Binomial[n+k,k]*Binomial[n+2*k,k]* Binomial[n+3*k,k], {k, 0, n}],{n,0,20}] (* Vaclav Kotesovec, Sep 23 2013 *)

{a(n)=polcoeff(sum(m=0, n, (4*m)!/m!^4*x^m/(1-x+x*O(x^n))^(4*m+1)), n)}
for(n=0, 15, print1(a(n), ", "))

{a(n)=sum(k=0, n, binomial(n, k)*binomial(n+k, k)*binomial(n+2*k, k)*binomial(n+3*k, k))}
for(n=0, 15, print1(a(n), ", "))

G.f.: Sum_{n>=0} (4*n)!/n!^4 * x^n / (1-x)^(4*n+1). - Paul D. Hanna, Sep 22 2013
Recurrence: n^3*(2*n-3)*(4*n-9)*(4*n-5)*a(n) = (4*n-9)*(4*n-3)*(520*n^4 - 1820*n^3 + 2109*n^2 - 905*n + 121)*a(n-1) - (192*n^6 - 1536*n^5 + 4748*n^4 - 7050*n^3 + 5065*n^2 - 1563*n + 171)*a(n-2) + (4*n-1)*(32*n^5 - 296*n^4 + 1040*n^3 - 1689*n^2 + 1209*n - 279)*a(n-3) - (n-3)^3*(2*n-1)*(4*n-5)*(4*n-1)*a(n-4). - Vaclav Kotesovec, Sep 23 2013
a(n) ~ c*d^n/(Pi^(3/2)*n^(3/2)), where d = 65 + 46*sqrt(2) + 2*sqrt(2*(1055 + 746*sqrt(2))) = 259.976980158726979... is the maximal positive root of the equation 1 - 4*d + 6*d^2 - 260*d^3 + d^4 = 0 and c = sqrt(8 + 5*sqrt(2) + sqrt(14*(11 + 8*sqrt(2))))/8 = 0.71529801573844067904424114047445568721... - Vaclav Kotesovec, Sep 23 2013, updated Jul 16 2016
G.f.: hypergeom([1/8, 3/8],[1],256*x/(1-x)^4)^2/(1-x). - Mark van Hoeij, Sep 23 2013
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 13*x^2 + 893*x^3 + 99125*x^4 + 13706093*x^5 + ... appears to have integer coefficients. - Peter Bala, Jan 13 2016
0 = x^2*(3*x+1)^2*(1-260*x+6*x^2-4*x^3+x^4)*y''' + 3*x*(3*x+1)*(1-390*x-378*x^2+8*x^3-15*x^4+6*x^5)*y'' + (1-836*x+133*x^2+768*x^3-69*x^4-60*x^5+63*x^6)*y' + (-25+397*x-378*x^2-6*x^3+3*x^4+9*x^5)*y, where y is the g.f. - Gheorghe Coserea, Jul 15 2016

A082489 a(n) = Sum_{k = 0..n} C(n,k) * C(n+k,k) * C(n+2k,k) C(n+3k,k) C(n+4*k,k).

Original entry on oeis.org

1, 121, 114121, 169417921, 308238414121, 629799991355641, 1387152264043496161, 3220175519103433952161, 7771784978946238318454761, 19326687177288750280293146161, 49215884415076728067274047737961, 127771596843320597524806463425540481

Offset: 0

Emanuele Munarini, Apr 28 2003

Diagonal of the rational function 1 / (1 - x - y - z - u - v - x*y*z*u*v). - Ilya Gutkovskiy, Apr 22 2025

			G.f.: A(x) = 1 + 121*x + 114121*x^2 + 169417921*x^3 + 308238414121*x^4 +...
where
A(x) = 1/(1-x) + (5!/1!^5)*x/(1-x)^6 + (10!/2!^5)*x^2/(1-x)^11 + (15!/3!^5)*x^3/(1-x)^16 + (20!/4!^5)*x^4/(1-x)^21 + (25!/5!^5)*x^5/(1-x)^26 +... [Hanna]
Equivalently,
A(x) = 1/(1-x) + 120*x/(1-x)^6 + 113400*x^2/(1-x)^11 + 168168000*x^3/(1-x)^16 + 305540235000*x^4/(1-x)^21 + 623360743125120*x^5/(1-x)^26 +...+ A008978(n)*x^n/(1-x)^(5*n+1) +...

Alois P. Heinz, Table of n, a(n) for n = 0..200

Column k = 5 of A229142.

Cf. A008978, A001850, A081798, A082488, A229049, A229050.

with(combinat):
a:= n-> add(multinomial(n+4*k, n-k, k$5), k=0..n):
seq(a(n), n=0..15);  # Alois P. Heinz, Sep 23 2013

Table[Sum[Binomial[n,k]*Binomial[n+k,k]*Binomial[n+2*k,k]* Binomial[n+3*k,k]*Binomial[n+4*k,k], {k, 0, n}],{n,0,20}] (* Vaclav Kotesovec, Sep 23 2013 *)

{a(n)=polcoeff(sum(m=0, n, (5*m)!/m!^5*x^m/(1-x+x*O(x^n))^(5*m+1)), n)}
for(n=0, 15, print1(a(n), ", ")) \\ Paul D. Hanna, Sep 22 2013

{a(n)=sum(k=0, n, binomial(n, k)*binomial(n+k, k)*binomial(n+2*k, k) *binomial(n+3*k, k)*binomial(n+4*k, k))}
for(n=0, 15, print1(a(n), ", ")) \\ Paul D. Hanna, Sep 22 2013

G.f.: Sum_{n>=0} (5*n)!/n!^5 * x^n / (1-x)^(5*n+1). - Paul D. Hanna, Sep 22 2013
Recurrence: n^4*(5*n-16)*(5*n-12)*(5*n-11)*(5*n-8)*(5*n-7)*(5*n-6)*a(n) = (5*n-16)*(5*n-12)*(5*n-11)*(5*n-4)*(78250*n^6 - 422550*n^5 + 885665*n^4 - 906704*n^3 + 468906*n^2 - 114379*n + 10086)*a(n-1) - (5*n-16)*(31250*n^9 - 400000*n^8 + 2154375*n^7 - 6337750*n^6 + 11073100*n^5 - 11721380*n^4 + 7379043*n^3 - 2629646*n^2 + 489456*n - 36000)*a(n-2) + (5*n-1)*(31250*n^9 - 556250*n^8 + 4241875*n^7 - 18056500*n^6 + 46858025*n^5 - 76033760*n^4 + 76116292*n^3 - 44628880*n^2 + 13702848*n - 1693440)*a(n-3) - (5*n-6)*(5*n-2)*(5*n-1)*(625*n^7 - 11375*n^6 + 86025*n^5 - 347305*n^4 + 798274*n^3 - 1025292*n^2 + 661408*n - 156480)*a(n-4) + (n-4)^4*(5*n-11)*(5*n-7)*(5*n-6)*(5*n-3)*(5*n-2)*(5*n-1)*a(n-5). - Vaclav Kotesovec, Sep 23 2013
a(n) ~ c*d^n/n^2, where d = 3129.996806129131084... is the root of the equation -1 + 5*d - 10*d^2 + 10*d^3 - 3130*d^4 +d^5 = 0 and c = 0.05674890286773483081841276583916042181... - Vaclav Kotesovec, Sep 23 2013
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 61*x^2 + 38101*x^3 + 42394381*x^4 + ... appears to have integer coefficients. - Peter Bala, Jan 13 2016

A229049 G.f.: Sum_{n >= 0} (6n)!/n!^6 x^n / (1-x)^(6*n+1).

Original entry on oeis.org

1, 721, 7489441, 137322405361, 3249278494922881, 88913838899388373921, 2672932604015450235911761, 85821373873727899097881489201, 2892941442791065880984595547275841, 101239016762863657789924022384195015281, 3649717311362112161867915447690005522733041

Offset: 0

Paul D. Hanna, Sep 22 2013

Diagonal of the rational function 1 / (1 - x - y - z - u - v - w - x*y*z*u*v*w). - Ilya Gutkovskiy, Apr 22 2025

			G.f.: A(x) = 1 + 721*x + 7489441*x^2 + 137322405361*x^3 + 3249278494922881*x^4 +...
where
A(x) = 1/(1-x) + (6!/1!^6)*x/(1-x)^7 + (12!/2!^6)*x^2/(1-x)^13 + (18!/3!^6)*x^3/(1-x)^19 + (24!/4!^6)*x^4/(1-x)^25 + (30!/5!^6)*x^5/(1-x)^31 +...
Equivalently,
A(x) = 1/(1-x) + 720*x/(1-x)^7 + 7484400*x^2/(1-x)^13 + 137225088000*x^3/(1-x)^19 + 3246670537110000*x^4/(1-x)^25 + 88832646059788350720*x^5/(1-x)^31 +...+ A008979(n)*x^n/(1-x)^(6*n+1) +...

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A008979, A001850, A081798, A082488, A082489, A229050.

Column k = 6 of A229142.

with(combinat):
a:= n-> add(multinomial(n+5*k, n-k, k$6), k=0..n):
seq(a(n), n=0..15);  # Alois P. Heinz, Sep 23 2013

Table[Sum[Binomial[n,k]*Binomial[n+k,k]*Binomial[n+2*k,k] *Binomial[n+3*k,k] *Binomial[n+4*k,k]*Binomial[n+5*k,k], {k, 0, n}],{n,0,20}] (* Vaclav Kotesovec, Sep 23 2013 *)

{a(n)=polcoeff(sum(m=0, n, (6*m)!/m!^6*x^m/(1-x+x*O(x^n))^(6*m+1)), n)}
for(n=0,15,print1(a(n),", "))

{a(n)=sum(k=0,n,binomial(n,k)*binomial(n+k,k) *binomial(n+2*k,k) *binomial(n+3*k,k) *binomial(n+4*k,k)*binomial(n+5*k,k))}
for(n=0,15,print1(a(n),", "))

a(n) = Sum_{k=0..n} C(n,k) * C(n+k,k) * C(n+2*k,k) * C(n+3*k,k) * C(n+4*k,k) * C(n+5*k,k).
Recurrence: n^5*(2*n - 5)*(2*n - 3)*(3*n - 10)*(3*n - 7)*(3*n - 5)*(3*n - 4)*(6*n - 25)*(6*n - 19)*(6*n - 13)*(6*n - 7)*a(n) = (2*n - 5)*(3*n - 10)*(3*n - 7)*(6*n - 25)*(6*n - 19)*(6*n - 13)*(6*n - 5)*(839916*n^8 - 6159384*n^7 + 18804591*n^6 - 30967129*n^5 + 29803190*n^4 - 16984623*n^3 + 5534242*n^2 - 929843*n + 60482)*a(n-1) - (3*n - 10)*(6*n - 25)*(6*n - 19)*(58320*n^12 - 1030320*n^11 + 7973640*n^10 - 35550360*n^9 + 101096973*n^8 - 191892891*n^7 + 247426961*n^6 - 216687345*n^5 + 127127767*n^4 - 48662719*n^3 + 11593839*n^2 - 1535715*n + 84350)*a(n-2) + 10*(6*n - 25)*(6*n - 1)*(23328*n^13 - 618192*n^12 + 7344432*n^11 - 51616440*n^10 + 238504338*n^9 - 761904909*n^8 + 1722993100*n^7 - 2778206390*n^6 + 3175831572*n^5 - 2526793076*n^4 + 1352618106*n^3 - 459806772*n^2 + 89082095*n - 7435050)*a(n-3) - 5*(3*n - 1)*(6*n - 7)*(6*n - 1)*(11664*n^12 - 369360*n^11 + 5235192*n^10 - 43777800*n^9 + 239670873*n^8 - 901183065*n^7 + 2374616540*n^6 - 4392523494*n^5 + 5622136222*n^4 - 4816276070*n^3 + 2596763070*n^2 - 784074950*n + 100205500)*a(n-4) + (2*n - 1)*(3*n - 4)*(3*n - 1)*(6*n - 13)*(6*n - 7)*(6*n - 1)*(648*n^9 - 19548*n^8 + 256338*n^7 - 1909293*n^6 + 8851093*n^5 - 26285080*n^4 + 49492875*n^3 - 56141750*n^2 + 34024625*n - 8063750)*a(n-5) - (n-5)^5*(2*n - 3)*(2*n - 1)*(3*n - 7)*(3*n - 4)*(3*n - 2)*(3*n - 1)*(6*n - 19)*(6*n - 13)*(6*n - 7)*(6*n - 1)*a(n-6). - Vaclav Kotesovec, Sep 23 2013
a(n) ~ c*d^n/(n^(5/2)), where d = 46661.9996785484656481246... is the root of the equation 1 - 6*d + 15*d^2 - 20*d^3 + 15*d^4 - 46662*d^5 + d^6 = 0 and c = 0.024758197509539176365175770882978221... - Vaclav Kotesovec, Sep 23 2013
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 361*x^2 + 2496841*x^3 + 34333162981*x^4 + ... appears to have integer coefficients. - Peter Bala, Jan 13 2016

A229051 G.f.: Sum_{n>=0} (n^2)!/n!^n * (2*x)^n / (1-x)^(n^2+1).

Original entry on oeis.org

1, 3, 29, 13567, 1009142769, 19947560933879891, 170891375663413844489045533, 942542805274443250197129297402029958879, 4650425326497533656923054675764068523027405525255181377, 27202912617670436035808496756146798219927348043651854025145948950565355779

Offset: 0

Paul D. Hanna, Sep 22 2013

nonn

			G.f.: A(x) = 1 + 3*x + 29*x^2 + 13567*x^3 + 1009142769*x^4 +...
where
A(x) = 1/(1-x) + (2*x)/(1-x)^2 + (4!/2!^2)*(2*x)^2/(1-x)^5 + (9!/3!^3)*(2*x)^3/(1-x)^10 + (16!/4!^4)*(2*x)^4/(1-x)^17 + (25!/5!^5)*(2*x)^5/(1-x)^26 +...
Equivalently,
A(x) = 1/(1-x) + (2*x)/(1-x)^2 + 6*(2*x)^2/(1-x)^5 + 1680*(2*x)^3/(1-x)^10 + 63063000*(2*x)^4/(1-x)^17 + 623360743125120*(2*x)^5/(1-x)^26 +...+ A034841(n)*(2*x)^n/(1-x)^(n^2+1) +...
Illustrate formula a(n) = Sum_{k=0..n} 2^k * Product_{j=0..k-1} C(n+j*k,k) for initial terms:
a(0) = 1;
a(1) = 1 + 2*C(1,1);
a(2) = 1 + 2*C(2,1) + 4*C(2,2)*C(4,2);
a(3) = 1 + 2*C(3,1) + 4*C(3,2)*C(5,2) + 8*C(3,3)*C(6,3)*C(9,3);
a(4) = 1 + 2*C(4,1) + 4*C(4,2)*C(6,2) + 8*C(4,3)*C(7,3)*C(10,3) + 16*C(4,4)*C(8,4)*C(12,4)*C(16,4);
a(5) = 1 + 2*C(5,1) + 4*C(5,2)*C(7,2) + 8*C(5,3)*C(8,3)*C(11,3) + 16*C(5,4)*C(9,4)*C(13,4)*C(17,4) + 32*C(5,5)*C(10,5)*C(15,5)*C(20,5)*C(25,5); ...
which numerically equals:
a(0) = 1;
a(1) = 1 + 2*1 = 3;
a(2) = 1 + 2*2 + 4*1*6 = 29;
a(3) = 1 + 2*3 + 4*3*10 + 8*1*20*84 = 13567;
a(4) = 1 + 2*4 + 4*6*15 + 8*4*35*120 + 16*1*70*495*1820 = 1009142769;
a(5) = 1 + 2*5 + 4*10*21 + 8*10*56*165 + 16*5*126*715*2380 + 32*1*252*3003*15504*53130 = 19947560933879891; ...

Alois P. Heinz, Table of n, a(n) for n = 0..26

Cf. A229050, A034841; A001850, A081798, A082488, A082489, A229049.

with(combinat):
a:= n-> add(2^k*multinomial(n+(k-1)*k, n-k, k$k), k=0..n):
seq(a(n), n=0..10);  # Alois P. Heinz, Sep 23 2013

Table[Sum[2^k*Product[Binomial[n+j*k,k],{j,0,k-1}],{k,0,n}],{n,0,10}] (* Vaclav Kotesovec, Sep 23 2013 *)

{a(n)=polcoeff(sum(m=0, n, (m^2)!/m!^m*(2*x)^m/(1-x+x*O(x^n))^(m^2+1)), n)}
for(n=0,15,print1(a(n),", "))

{a(n)=sum(k=0,n,2^k*prod(j=0,k-1,binomial(n+j*k,k)))}
for(n=0,15,print1(a(n),", "))

a(n) = Sum_{k=0..n} 2^k * Product_{j=0..k-1} binomial(n+j*k,k).

a(n) ~ exp(-1/12) * n^(n^2-n/2+1) * 2^n / (2*Pi)^((n-1)/2). - Vaclav Kotesovec, Sep 23 2013

A264410 G.f.: Sum_{n>=0} (n^2)!/n!^(n+1) * x^n / (1-x)^(n^2+1).

Original entry on oeis.org

1, 2, 6, 299, 2630475, 5194717544512, 3708580324835714831258, 1461034854533485247412937306733067, 450538787986948219326155652866541933427447505845, 146413934927214452212855330835382443952802497537220927026261910086, 64954656894649578286072291957497800888821547620744164639686415856667115998603764278502

Offset: 0

Paul D. Hanna, Nov 26 2015

nonn

			G.f.: A(x) = 1 + 2*x + 9*x^2 + 1714*x^3 + 63079895*x^4 + 623361815288736*x^5 +...
where
A(x) = 1/(1-x) + x/(1-x)^2 + (4!/2!^3)*x^2/(1-x)^5 + (9!/3!^4)*x^3/(1-x)^10 + (16!/4!^5)*x^4/(1-x)^17 + (25!/5!^6)*x^5/(1-x)^26 +...
Equivalently,
A(x) = 1/(1-x) + x/(1-x)^2 + 3*x^2/(1-x)^5 + 280*x^3/(1-x)^10 + 2627625*x^4/(1-x)^17 + 5194672859376*x^5/(1-x)^26 +...+ A057599(n)*x^n/(1-x)^(n^2+1) +...
Illustrate formula a(n) = Sum_{k=0..n} (1/k!)*Product_{j=0..k-1} C(n+j*k,k) for initial terms:
a(0) = 1;
a(1) = 1 + C(1,1);
a(2) = 1 + C(2,1) + C(2,2)*C(4,2)/2!;
a(3) = 1 + C(3,1) + C(3,2)*C(5,2)/2! + C(3,3)*C(6,3)*C(9,3)/3!;
a(4) = 1 + C(4,1) + C(4,2)*C(6,2)/2! + C(4,3)*C(7,3)*C(10,3)/3! + C(4,4)*C(8,4)*C(12,4)*C(16,4)/4!;
a(5) = 1 + C(5,1) + C(5,2)*C(7,2)/2! + C(5,3)*C(8,3)*C(11,3)/3! + C(5,4)*C(9,4)*C(13,4)*C(17,4)/4! + C(5,5)*C(10,5)*C(15,5)*C(20,5)*C(25,5)/5!; ...
which numerically equals:
a(0) = 1;
a(1) = 1 + 1 = 2;
a(2) = 1 + 2 + 1*6/2! = 6;
a(3) = 1 + 3 + 3*10/2! + 1*20*84/3! = 299;
a(4) = 1 + 4 + 6*15/2! + 4*35*120/3! + 1*70*495*1820/4! = 2630475;
a(5) = 1 + 5 + 10*21/2! + 10*56*165/3! + 5*126*715*2380/4! + 1*252*3003*15504*53130/5! = 5194717544512; ...

Paul D. Hanna, Table of n, a(n) for n = 0..30

Cf. A229050, A057599.

Table[Sum[Product[Binomial[n+j*k, k], {j, 0, k-1}]/k!, {k, 0, n}], {n, 0, 10}] (* Vaclav Kotesovec, Dec 09 2015 *)

{a(n)=polcoeff(sum(m=0, n, (m^2)!/m!^(m+1)*x^m/(1-x +x*O(x^n))^(m^2+1)), n)}
for(n=0, 15, print1(a(n), ", "))

{a(n) = sum(k=0, n, (1/k!) * prod(j=0, k-1, binomial(n+j*k, k)) )}
for(n=0, 15, print1(a(n), ", "))

a(n) = Sum_{k=0..n} (1/k!) * Product_{j=0..k-1} binomial(n+j*k, k).

a(n) ~ exp(n - 1/12) * n^(n^2 - 3*n/2 + 1/2) / (2*Pi)^(n/2). - Vaclav Kotesovec, Dec 09 2015

cp's OEIS Frontend

A034841 a(n) = (n^2)! / (n!)^n.

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A082488 a(n) = Sum_{k = 0..n} C(n,k) * C(n+k,k) * C(n+2*k,k) * C(n+3*k,k).

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A082489 a(n) = Sum_{k = 0..n} C(n,k) * C(n+k,k) * C(n+2*k,k) * C(n+3*k,k) * C(n+4*k,k).

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A229049 G.f.: Sum_{n >= 0} (6*n)!/n!^6 * x^n / (1-x)^(6*n+1).

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A229051 G.f.: Sum_{n>=0} (n^2)!/n!^n * (2*x)^n / (1-x)^(n^2+1).

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A264410 G.f.: Sum_{n>=0} (n^2)!/n!^(n+1) * x^n / (1-x)^(n^2+1).

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A082488 a(n) = Sum_{k = 0..n} C(n,k) * C(n+k,k) * C(n+2k,k) C(n+3*k,k).

A082489 a(n) = Sum_{k = 0..n} C(n,k) * C(n+k,k) * C(n+2k,k) C(n+3k,k) C(n+4*k,k).

A229049 G.f.: Sum_{n >= 0} (6n)!/n!^6 x^n / (1-x)^(6*n+1).