A144660 -id:A144660 - cp's OEIS frontend

A030662 Number of combinations of n things from 1 to n at a time, with repeats allowed.

1, 5, 19, 69, 251, 923, 3431, 12869, 48619, 184755, 705431, 2704155, 10400599, 40116599, 155117519, 601080389, 2333606219, 9075135299, 35345263799, 137846528819, 538257874439, 2104098963719, 8233430727599, 32247603683099, 126410606437751, 495918532948103

Offset: 1

Donald Mintz (djmintz(AT)home.com)

Add terms of an increasingly bigger diamond-shaped part of Pascal's triangle:
.......................... 1
............ 1 .......... 1 1
.. 1 ...... 1 1 ........ 1 2 1
. 1 1 =5 . 1 2 1 =19 .. 1 3 3 1 =69
.. 2 ...... 3 3 ........ 4 6 4
............ 6 ......... 10 10
.......................... 20
-  Ralf Stephan, May 17 2004
The prime p divides a((p-1)/2) for p in A002144 (Pythagorean primes). - Alexander Adamchuk, Jul 04 2006
Also, number of square submatrices of a square matrix. - Jono Henshaw (jjono(AT)hotmail.com), Apr 22 2008
Partial sums of A051924. - J. M. Bergot, Jun 22 2013
Number of partitions with Ferrers diagrams that fit in an n X n box (excluding the empty partition of 0). - Michael Somos, Jun 02 2014
Also number of non-descending sequences with length and last number are less or equal to n, and also the number of integer partitions (of any positive integer) with length and largest part are less or equal to n. - Zlatko Damijanic, Dec 06 2024

			G.f. = x + 5*x^2 + 19*x^3 + 69*x^4 + 251*x^5 + 923*x^6 + 3431*x^7 + ...

T. D. Noe, Table of n, a(n) for n = 1..500
Narcisse G. Bell Bogmis, Guy R. Biyogmam, Hesam Safa, and Calvin Tcheka, Upper bounds on the dimension of the Schur Lie-multiplier of Lie-nilpotent Leibniz n-algebras, arXiv:2403.14884 [math.RA], 2024. See p. 7.
Joseph D. Horton and Andrew Kurn, Counting sequences with complete increasing subsequences, Congress Numerantium, 33 (1981), 75-80. MR 681905
Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Raimundas Vidunas, Counting derangements and Nash equilibria, Ann. Comb. 21, No. 1, 131-152 (2017).
Jianqiang Zhao, Uniform Approach to Double Shuffle and Duality Relations of Various q-Analogs of Multiple Zeta Values via Rota-Baxter Algebras, arXiv preprint arXiv:1412.8044 [math.NT], 2014.

Cf. A000984, A001700, A002144, A051924, A091908, A144660, A322938.
Column k=2 of A047909.
Central column of triangle A014473.
Right-hand column 2 of triangle A102541.

[(n+1)*Catalan(n)-1: n in [1..40]]; // G. C. Greubel, Apr 07 2024

seq(sum((binomial(n,m))^2,m=1..n),n=1..23); # Zerinvary Lajos, Jun 19 2008
f:=n->add( add( binomial(i+j,i), i=0..n),j=0..n); [seq(f(n),n=0..12)]; # N. J. A. Sloane, Jan 31 2009

Table[Sum[Sum[(2n-i-j)!/(n-i)!/(n-j)!,{i,1,n}],{j,1,n}],{n,1,20}] (* Alexander Adamchuk, Jul 04 2006 *)
a[n_] := 2*(2*n-1)!/(n*(n-1)!^2)-1; Table[a[n], {n, 1, 26}] (* Jean-François Alcover, Oct 11 2012, from first formula *)

a(n)=binomial(2*n,n)-1 \\ Charles R Greathouse IV, Jun 26 2013

from math import comb
def a(n): return comb(2*n, n) - 1
print([a(n) for n in range(1, 27)]) # Michael S. Branicky, Jul 11 2023

def a(n) : return binomial(2*n,n) - 1
[a(n) for n in (1..26)] # Peter Luschny, Apr 21 2012

a(n) = A000984(n) - 1.
a(n) = 2*A001700(n-1) - 1.
a(n) = 2*(2*n-1)!/(n!*(n-1)!)-1.
a(n) = Sum_{k=1..n} binomial(n, k)^2. - Benoit Cloitre, Aug 20 2002
a(n) = Sum_{j=0..n} Sum_{i=j..n+j} binomial(i, j). - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jul 23 2003
a(n) = Sum_{i=0..n-1} Sum_{j=0..n-1} binomial(i+j, i). - N. J. A. Sloane, Jan 31 2009
Also for n>1: a(n)=(2*n)!/(n!)^2-1. - Hugo Pfoertner, Feb 10 2004
a(n) = Sum_{j=1..n} Sum_{i=1..n} (2n-i-j)!/((n-i)!*(n-j)!). - Alexander Adamchuk, Jul 04 2006
a(n) = A115112(n) + 1. - Jono Henshaw (jjono(AT)hotmail.com), Apr 22 2008
G.f.: Q(0)*(1-4*x)/x - 1/x/(1-x), where Q(k)= 1 + 4*(2*k+1)*x/( 1 - 1/(1 + 2*(k+1)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 11 2013
D-finite with recurrence: n*a(n) +2*(-3*n+2)*a(n-1) +(9*n-14)*a(n-2) +2*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Jun 25 2013
0 = a(n)*(+16*a(n+1) - 70*a(n+2) + 68*a(n+3) - 14*a(n+4)) + a(n+1)*(-2*a(n+1) + 61*a(n+2) - 96*a(n+3) + 23*a(n+4)) + a(n+2)*(-6*a(n+2) + 31*a(n+3)  - 10*a(n+4)) + a(n+3)*(-2*a(n+3) + a(n+4)) for all n in Z. - Michael Somos, Jun 02 2014
From Ilya Gutkovskiy, Jan 25 2017: (Start)
O.g.f.: (1 - x - sqrt(1 - 4*x))/((1 - x)*sqrt(1 - 4*x)).
E.g.f.: exp(x)*(exp(x)*BesselI(0,2*x) - 1). (End)
a(n) = 3*n*Sum_{k=1..n} (-1)^(k+1)/(2*n+k)*binomial(2*n+k,n-k). - Vladimir Kruchinin, Jul 29 2025
a(n) = n * binomial(2*n, n) * Sum_{k = 1..n} 1/(k*binomial(n+k, k)). - Peter Bala, Aug 05 2025

A144511 a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} (i+j+k)!/(3!i!j!*k!).

Original entry on oeis.org

0, 1, 37, 842, 18252, 405408, 9268549, 216864652, 5165454442, 124762262630, 3047235458767, 75109521108771, 1865470016184352, 46631215889276662, 1172088706950306293, 29601905040172054928, 750748513858793527974, 19110455782881086439234, 488057675380082251617235

Offset: 0

David Applegate and N. J. A. Sloane, Dec 15 2008, Jan 30 2009

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..100
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394, 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem (arXiv:0907.0513, 2009)

Column 3 of array in A144510.

Cf. A144658, A144660 (a very similar sum).

f:=n->add( add( add( (i+j+k)!/(3!*i!*j!*k!), i=1..n),j=1..n),k=1..n); [seq(f(n),n=0..20)];

Table[Sum[Sum[Sum[(i+j+k)!/i!/j!/k!/6,{i,1,n}],{j,1,n}],{k,1,n}],{n,1,30}]
Table[(5 + 3*n - 3*Binomial[2*n+2, n+1] + Sum[(1 + k + 2*n)! * HypergeometricPFQ[{1, -1 - k - n, -n}, {-1 - k - 2*n, -k - n}, 1] / ((1 + k + n)*k!*n!^2), {k, 0, n}]) / 6, {n, 0, 20}] (* Vaclav Kotesovec, Apr 04 2019 *)

{a(n) = sum(i=1, n, sum(j=1, n, sum(k=1, n, (i+j+k)!/(6*i!*j!*k!))))} \\ Seiichi Manyama, Apr 03 2019

{a(n) = sum(i=3, 3*n, i!*polcoef(sum(j=1, n, x^j/j!)^3, i))/6} \\ Seiichi Manyama, May 19 2019

a(n) = (5 + 3*n - 3*binomial(2*n+2, n+1) + A144660(n))/6. - Vaclav Kotesovec, Apr 04 2019

A307318 a(n) = Sum_{i=0..n} Sum_{j=0..n} Sum_{k=0..n} (-1)^(i+j+k) * (i+j+k)!/(i!j!k!).

Original entry on oeis.org

1, -2, 37, -692, 14371, -315002, 7156969, -166785320, 3960790687, -95442311582, 2326713829837, -57260397539204, 1420295354815351, -35463581316556850, 890530353765972817, -22472131364683145552, 569507678494598796631, -14487492070374441746150

Offset: 0

Seiichi Manyama, Apr 02 2019

sign

Seiichi Manyama, Table of n, a(n) for n = 0..100

Cf. A120305, A144660, A307324.

Table[Sum[(-1)^(i + j + k) * (i + j + k)!/(i!*j!*k!), {i, 0, n}, {j, 0, n}, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 02 2019 *)

{a(n) = sum(i=0, n, sum(j=0, n, sum(k=0, n, (-1)^(i+j+k)*(i+j+k)!/(i!*j!*k!))))}

{a(n) = sum(i=0, 3*n, (-1)^i*i!*polcoef(sum(j=0, n, x^j/j!)^3, i))} \\ Seiichi Manyama, May 20 2019

From Vaclav Kotesovec, Apr 02 2019: (Start)
Recurrence: 6*(n-1)*n^2*(490*n^4 - 3948*n^3 + 11668*n^2 - 14967*n + 7027)*a(n) =  - (n-1)*(74480*n^6 - 675066*n^5 + 2399756*n^4 - 4233492*n^3 + 3852029*n^2 - 1682577*n + 272160)*a(n-1) + (131320*n^7 - 1437814*n^6 + 6472114*n^5 - 15414556*n^4 + 20770423*n^3 - 15610855*n^2 + 5939868*n - 861840)*a(n-2) - (27440*n^7 - 355838*n^6 + 1853810*n^5 - 4998800*n^4 + 7460459*n^3 - 6071312*n^2 + 2439561*n - 362880)*a(n-3) - 3*(2*n - 5)*(3*n - 8)*(3*n - 7)*(490*n^4 - 1988*n^3 + 2764*n^2 - 1515*n + 270)*a(n-4).
a(n) ~ (-1)^n * 3^(3*n + 7/2) / (128*Pi*n). (End)

A308292 A(n,k) = Sum_{i_1=0..n} Sum_{i_2=0..n} ... Sum_{i_k=0..n} multinomial(i_1 + i_2 + ... + i_k; i_1, i_2, ..., i_k), square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 16, 19, 4, 1, 1, 65, 271, 69, 5, 1, 1, 326, 7365, 5248, 251, 6, 1, 1, 1957, 326011, 1107697, 110251, 923, 7, 1, 1, 13700, 21295783, 492911196, 191448941, 2435200, 3431, 8, 1, 1, 109601, 1924223799, 396643610629, 904434761801, 35899051101, 55621567, 12869, 9, 1

Offset: 0

Seiichi Manyama, May 19 2019

For r > 1, row r is asymptotic to sqrt(2*Pi) * (r*n)^(r*n + 1/2) / ((r!)^n * exp(r*n-1)). - Vaclav Kotesovec, May 24 2020

			For (n,k) = (3,2), (Sum_{i=0..3} x^i/i!)^2 = (1 + x + x^2/2 + x^3/6)^2 = 1 + 2*x + 4*x^2/2 + 8*x^3/6 + 14*x^4/24 + 20*x^5/120 + 20*x^6/720. So A(3,2) = 1 + 2 + 4 + 8 + 14 + 20 + 20 = 69.
Square array begins:
   1, 1,    1,        1,             1,                   1, ...
   1, 2,    5,       16,            65,                 326, ...
   1, 3,   19,      271,          7365,              326011, ...
   1, 4,   69,     5248,       1107697,           492911196, ...
   1, 5,  251,   110251,     191448941,        904434761801, ...
   1, 6,  923,  2435200,   35899051101,    1856296498826906, ...
   1, 7, 3431, 55621567, 7101534312685, 4098746255797339511, ...

Seiichi Manyama, Antidiagonals n = 0..50, flattened

Columns k=0..4 give A000012, A000027(n+1), A030662(n+1), A144660, A144661.
Rows n=0..4 give A000012, A000522, A003011, A308294, A308295.
Main diagonal gives A274762.
Cf. A144510.

A(n,k) = Sum_{i=0..k*n} b(i) where Sum_{i=0..k*n} b(i) * x^i/i! = (Sum_{i=0..n} x^i/i!)^k.

A144661 a(n) = Sum_{i=0..n} Sum_{j=0..n} Sum_{k=0..n} Sum_{l=0..n} (i+j+k+l)!/(i!j!k!*l!).

Original entry on oeis.org

1, 65, 7365, 1107697, 191448941, 35899051101, 7101534312685, 1458965717496881, 308290573348183629, 66577182435768923245, 14629025943480502591445, 3260160391173522631759533, 735119604833362632050789701, 167408468505328518543519208949, 38448088693846486556578015883325

Offset: 0

N. J. A. Sloane, Feb 01 2009

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..400 (terms 0..50 from Seiichi Manyama)
Vaclav Kotesovec, Recurrence (of order 4)
Vidunas, Raimundas Counting derangements and Nash equilibria Ann. Comb. 21, No. 1, 131-152 (2017).

Cf. A030662, A144660, A144662, A307324.

f:=n->add( add( add( add( (i+j+k+l)!/(i!*j!*k!*l!), i=0..n),j=0..n),k=0..n),l=0..n); [seq(f(n),n=0..20)];

Table[Sum[(i + j + k + l)! / (i!*j!*k!*l!), {i, 0, n}, {j, 0, n}, {k, 0, n}, {l, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 02 2019 *)
Table[Sum[(1 + j + k + l + n)!/((1 + j + k + l)*j!*k!*l!), {j, 0, n}, {k, 0, n}, {l, 0, n}] / n!, {n, 0, 20}] (* Vaclav Kotesovec, Apr 03 2019 *)
Table[Sum[(1 + k + l + 2*n)! * HypergeometricPFQ[{1, -1 - k - l - n, -n}, {-1 - k - l - 2*n, -k - l - n}, 1] / ((1 + k + l + n)*k!*l!*n!), {k, 0, n}, {l, 0, n}]/n!, {n, 0, 20}] (* Vaclav Kotesovec, Apr 03 2019 *)

{a(n) = sum(i=0, n, sum(j=0, n, sum(k=0, n, sum(l=0, n, (i+j+k+l)!/(i!*j!*k!*l!)))))} \\ Seiichi Manyama, Apr 02 2019

a(n) ~ 2^(8*n + 15/2) / (81 * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Apr 02 2019

A144662 a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} Sum_{l=1..n} (i+j+k+l)!/(4!i!j!k!l!).

Original entry on oeis.org

0, 1, 266, 45296, 7958726, 1495388159, 295887993624, 60790021361170, 12845435390707724, 2774049143394729653, 609542744597785306189, 135840016223787254538508, 30629983532857972983331740, 6975352854342057056747327899, 1602003695575764851150428242804, 370631496919828403109950449644134

Offset: 0

N. J. A. Sloane, Feb 01 2009

nonn

Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394, 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem (arXiv:0907.0513, 2009)

Column 4 of A144512. Cf. A144660, A144661.

f:=n->add( add( add( add( (i+j+k+l)!/(4!*i!*j!*k!*l!), i=1..n),j=1..n),k=1..n),l=1..n); [seq(f(n),n=0..16)];

a[n_] := Sum[(i+j+k+l)!/(4! i! j! k! l!), {i, n}, {j, n}, {k, n}, {l, n}];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Sep 05 2018 *)
Table[(Binomial[2*n + 2, n + 1] - 2*(1 + n) + Sum[(1 + k + l + 2*n)! HypergeometricPFQ[{1, -1 - k - l - n, -n}, {-1 - k - l - 2*n, -k - l - n}, 1]/((1 + k + l + n) k! l! (n!)^2) - (2*(1 + k + l + n)!)/((1 + k + l) k! l! n!), {k, 1, n}, {l, 1, n}])/24, {n, 0, 15}] (* Vaclav Kotesovec, Apr 04 2019 *)

A307352 a(n) = Sum_{0<=i<=j<=k<=n} (i+j+k)!/(i!j!k!).

Original entry on oeis.org

1, 10, 152, 2857, 59258, 1299434, 29540536, 688792297, 16365424655, 394524030964, 9621387028097, 236859068544553, 5876752849424588, 146774130990116924, 3686474939260449666, 93044751867820344115, 2358431594464812420404, 60004708149086107604240

Offset: 0

Seiichi Manyama, Apr 03 2019

nonn

Cf. A079309, A144660, A307353.

Table[Sum[Sum[Sum[(i+j+k)!/(i!*j!*k!), {i, 0, j}], {j, 0, k}], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 04 2019 *)

{a(n) = sum(i=0, n, sum(j=i, n, sum(k=j, n, (i+j+k)!/(i!*j!*k!))))}

From Vaclav Kotesovec, Apr 04 2019: (Start)
Recurrence: 3*(n-1)*n^2*(2*n + 1)*(15680*n^7 - 198268*n^6 + 1049184*n^5 - 3003295*n^4 + 5004388*n^3 - 4830736*n^2 + 2483598*n - 518661)*a(n)=(n-1)*(2*n - 1)*(1238720*n^9 - 15631812*n^8 + 82366948*n^7 - 233558317*n^6 + 380532743*n^5 - 345530522*n^4 + 141797620*n^3 + 8081106*n^2 - 23913486*n + 4762800)*a(n-1) + 2*(909440*n^11 - 14525784*n^10 + 100260068*n^9 - 390684898*n^8 + 940603537*n^7 - 1433395699*n^6 + 1346188538*n^5 - 691297162*n^4 + 97138838*n^3 + 77570673*n^2 - 37619991*n + 4762800)*a(n-2) - 2*(1662080*n^11 - 24324888*n^10 + 154076996*n^9 - 552269110*n^8 + 1226963821*n^7 - 1732162636*n^6 + 1512829217*n^5 - 721942210*n^4 + 86052929*n^3 + 81957789*n^2 - 37651608*n + 4762800)*a(n-3) - (2*n - 3)*(862400*n^10 - 12613860*n^9 + 77917844*n^8 - 263521873*n^7 + 527376397*n^6 - 624837256*n^5 + 401742338*n^4 - 90648379*n^3 - 35886325*n^2 + 22963194*n - 3175200)*a(n-4) + 3*(n-3)*(2*n - 5)*(3*n - 8)*(3*n - 7)*(15680*n^7 - 88508*n^6 + 188856*n^5 - 182595*n^4 + 66488*n^3 + 9758*n^2 - 11818*n + 1890)*a(n-5).
a(n) ~ 3^(3*n + 13/2) / (832*Pi*n).
(End)

A307353 a(n) = Sum_{1<=i<=j<=k<=n} (i+j+k)!/(i!j!k!).

Original entry on oeis.org

0, 6, 138, 2808, 59083, 1298797, 29538183, 688783509, 16365391557, 394523905488, 9621386549905, 236859066714283, 5876752842394018, 146774130963028054, 3686474939155802036, 93044751867415156290, 2358431594463240429469

Offset: 0

Seiichi Manyama, Apr 03 2019

nonn

Cf. A120279, A144511, A144660, A307352.

Table[Sum[Sum[Sum[(i+j+k)!/(i!*j!*k!), {i, 1, j}], {j, 1, k}], {k, 1, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 04 2019 *)

{a(n) = sum(i=1, n, sum(j=i, n, sum(k=j, n, (i+j+k)!/(i!*j!*k!))))}

a(n) ~ 3^(3*n + 13/2) / (832*Pi*n). - Vaclav Kotesovec, Apr 04 2019

A307358 a(n) = Sum_{0<=i<=j<=k<=n} (-1)^(i+j+k) * (i+j+k)!/(i!j!k!).

Original entry on oeis.org

1, -4, 72, -1345, 27886, -610558, 13861334, -322838475, 7663363513, -184598740512, 4498935186891, -110693299767349, 2745124008220296, -68532209858173364, 1720678086867077832, -43415209670536390089, 1100146390869600888470

Offset: 0

Seiichi Manyama, Apr 04 2019

sign

Cf. A144660, A307318, A307352, A307354.

Table[Sum[Sum[Sum[(-1)^(i+j+k) * (i+j+k)!/(i!*j!*k!), {i, 0, j}], {j, 0, k}], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 04 2019 *)

{a(n) = sum(i=0, n, sum(j=i, n, sum(k=j, n, (-1)^(i+j+k)*(i+j+k)!/(i!*j!*k!))))}

a(n) ~ (-1)^n * 3^(3*n + 13/2) / (1792*Pi*n). - Vaclav Kotesovec, Apr 04 2019

cp's OEIS Frontend

A030662 Number of combinations of n things from 1 to n at a time, with repeats allowed.

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A144511 a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} (i+j+k)!/(3!*i!*j!*k!).

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A307318 a(n) = Sum_{i=0..n} Sum_{j=0..n} Sum_{k=0..n} (-1)^(i+j+k) * (i+j+k)!/(i!*j!*k!).

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A308292 A(n,k) = Sum_{i_1=0..n} Sum_{i_2=0..n} ... Sum_{i_k=0..n} multinomial(i_1 + i_2 + ... + i_k; i_1, i_2, ..., i_k), square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.

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A144661 a(n) = Sum_{i=0..n} Sum_{j=0..n} Sum_{k=0..n} Sum_{l=0..n} (i+j+k+l)!/(i!*j!*k!*l!).

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A144662 a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} Sum_{l=1..n} (i+j+k+l)!/(4!*i!*j!*k!*l!).

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A307352 a(n) = Sum_{0<=i<=j<=k<=n} (i+j+k)!/(i!*j!*k!).

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Formula

A307353 a(n) = Sum_{1<=i<=j<=k<=n} (i+j+k)!/(i!*j!*k!).

Views

A144511 a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} (i+j+k)!/(3!i!j!*k!).

A307318 a(n) = Sum_{i=0..n} Sum_{j=0..n} Sum_{k=0..n} (-1)^(i+j+k) * (i+j+k)!/(i!j!k!).

A144661 a(n) = Sum_{i=0..n} Sum_{j=0..n} Sum_{k=0..n} Sum_{l=0..n} (i+j+k+l)!/(i!j!k!*l!).

A144662 a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} Sum_{l=1..n} (i+j+k+l)!/(4!i!j!k!l!).

A307352 a(n) = Sum_{0<=i<=j<=k<=n} (i+j+k)!/(i!j!k!).

A307353 a(n) = Sum_{1<=i<=j<=k<=n} (i+j+k)!/(i!j!k!).

A307358 a(n) = Sum_{0<=i<=j<=k<=n} (-1)^(i+j+k) * (i+j+k)!/(i!j!k!).