A054688 -id:A054688 - cp's OEIS frontend

A007716 Number of polynomial symmetric functions of matrix of order n under separate row and column permutations.

1, 1, 4, 10, 33, 91, 298, 910, 3017, 9945, 34207, 119369, 429250, 1574224, 5916148, 22699830, 89003059, 356058540, 1453080087, 6044132794, 25612598436, 110503627621, 485161348047, 2166488899642, 9835209912767, 45370059225318, 212582817739535, 1011306624512711

Offset: 0

Colin Mallows

Also, the number of nonnegative integer n X n matrices with sum of elements equal to n, under row and column permutations (cf. A120733).
This is a two-dimensional generalization of the partition function (A000041), which equals the number of length n vectors of nonnegative integers with sum n, equivalent under permutations. - Franklin T. Adams-Watters, Sep 19 2011
Also number of non-isomorphic multiset partitions of weight n. - Gus Wiseman, Sep 19 2011

			The 10 non-isomorphic multiset partitions of weight 3 are {{1, 1, 1}}, {{1, 1, 2}}, {{1, 2, 3}}, {{1}, {1, 1}}, {{1}, {1, 2}}, {{1}, {2, 2}}, {{1}, {2, 3}}, {{1}, {1}, {1}}, {{1}, {1}, {2}}, {{1}, {2}, {3}}.

Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 0..30 from Seiichi Manyama)

Main diagonal of A318795.

Cf. A053307, A052365, A052366, A052367, A052372, A052373, A049311, A054688, A000041.

permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i>1 && t == v[[i-1]], k+1, 1]; m *= t*k; s += t]; s!/m];
c[p_, q_, k_] := SeriesCoefficient[1/Product[(1-x^LCM[p[[i]], q[[j]]])^GCD[ p[[i]], q[[j]]], {j, 1, Length[q]}, {i, 1, Length[p]}], {x, 0, k}];
M[m_, n_, k_] := Module[{s=0}, Do[Do[s += permcount[p]*permcount[q]*c[p, q, k], {q, IntegerPartitions[n]}], {p, IntegerPartitions[m]}]; s/(m!*n!)];
a[n_] := a[n] = M[n, n, n];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 18}] (* Jean-François Alcover, May 03 2019, after Andrew Howroyd *)

\\ See A318795
a(n) = M(n,n,n); \\ Andrew Howroyd, Sep 03 2018

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={EulerT(Vec(sum(j=1, #q, gcd(t, q[j])*x^lcm(t,q[j])) + O(x*x^k), -k))}
a(n)={my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(x*Ser(sum(t=1, n, K(q,t,n)/t))), n)); s/n!} \\ Andrew Howroyd, Mar 29 2020

a(n) is the coefficient of x^n in the cycle index Z(S_n X S_n; x_1, x_2, ...) if we replace x_i with 1+x^i+x^(2*i)+x^(3*i)+x^(4*i)+..., where S_n X S_n is the Cartesian product of symmetric groups S_n of degree n. - Vladeta Jovovic, Mar 09 2000

More terms from Vladeta Jovovic, Jun 28 2000
a(19)-a(25) from Max Alekseyev, Jan 22 2010
a(0)=1 prepended by Alois P. Heinz, Feb 03 2019
a(26)-a(27) from Seiichi Manyama, Nov 23 2019

A135860 a(n) = binomial(n*(n+1), n).

Original entry on oeis.org

1, 2, 15, 220, 4845, 142506, 5245786, 231917400, 11969016345, 706252528630, 46897636623981, 3461014728350400, 281014969393251275, 24894763097057357700, 2389461906843449885700, 247012484980695576597296, 27361230617617949782033713, 3233032526324680287912449550

Offset: 0

Paul D. Hanna, Dec 02 2007

Seiichi Manyama, Table of n, a(n) for n = 0..337
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. A014062, A014068, A054688, A107863, A135861, A135862.

[Binomial(n*(n+1), n): n in [0..30]]; // G. C. Greubel, Feb 20 2022

Table[Binomial[n^2 + n, n], {n, 0, 16}] (* Arkadiusz Wesolowski, Jul 18 2012 *)
(* or *)
Table[SeriesCoefficient[(1+x)^(n*(n+1)), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 06 2025 *)

a(n)=binomial(n*(n+1),n)
for(n=0,15,print1(a(n),", "))

a(n)=sum(k=0,n,binomial(n,k)*binomial(n^2,k))
for(n=0,15,print1(a(n),", "))

[binomial(n*(n+1), n) for n in (0..30)] # G. C. Greubel, Feb 20 2022

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n^2,k). - Paul D. Hanna, Nov 18 2015
a(n) is divisible by (n+1): a(n)/(n+1) = A135861(n).
a(n) is divisible by (n^2+1): a(n)/(n^2+1) = A135862(n).
a(n) = binomial(2*A000217(n),n). - Arkadiusz Wesolowski, Jul 18 2012
a(n) = [x^n] 1/(1 - x)^(n^2+1). - Ilya Gutkovskiy, Oct 03 2017
a(n) ~ exp(n + 1/2) * n^(n - 1/2) / sqrt(2*Pi). - Vaclav Kotesovec, Feb 08 2019
a(p) == p + 1 ( mod p^4 ) for prime p >= 5 and a(2*p) == (4*p + 1)*(2*p + 1) ( mod p^4 ) for all prime p. Apply Mestrovic, equation 37. - Peter Bala, Feb 27 2020
a(n) = ((n^2 + n)!)/((n^2)! * n!). - Peter Luschny, Feb 27 2020
a(n) = [x^n] (1 + x)^(n*(n+1)). - Vaclav Kotesovec, Aug 06 2025

A214398 Triangle where the g.f. of column k is 1/(1-x)^(k^2) for k>=1, as read by rows n>=1.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 10, 9, 1, 1, 20, 45, 16, 1, 1, 35, 165, 136, 25, 1, 1, 56, 495, 816, 325, 36, 1, 1, 84, 1287, 3876, 2925, 666, 49, 1, 1, 120, 3003, 15504, 20475, 8436, 1225, 64, 1, 1, 165, 6435, 54264, 118755, 82251, 20825, 2080, 81, 1, 1, 220, 12870, 170544

Offset: 1

Paul D. Hanna, Jul 15 2012

This is also the array A(n,k) read upwards antidiagonals, where the entry in row n and column k counts the vertex-labeled digraphs with n arcs and k vertices, allowing multi-edges and multi-loops (labeled analog to A138107). The binomial formula counts the weak compositions of distributing n arcs over the k^2 positions in the adjacency matrix. - R. J. Mathar, Aug 03 2017

			Triangle begins:
1;
1, 1;
1, 4, 1;
1, 10, 9, 1;
1, 20, 45, 16, 1;
1, 35, 165, 136, 25, 1;
1, 56, 495, 816, 325, 36, 1;
1, 84, 1287, 3876, 2925, 666, 49, 1;
1, 120, 3003, 15504, 20475, 8436, 1225, 64, 1;
1, 165, 6435, 54264, 118755, 82251, 20825, 2080, 81, 1;
1, 220, 12870, 170544, 593775, 658008, 270725, 45760, 3321, 100, 1; ...

Paul D. Hanna, Rows n = 0..45, flattened.
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 (2017) Table 80.

Cf. A214400 (central terms), A178325 (row sums), A054688, A000290 (1st subdiagonal), A037270 (2nd subdiagonal).

Cf. A230049.

A214398 := proc(n,k)
    binomial(k^2+n-k-1,n-k) ;
end proc:
seq(seq(A214398(n,k),k=1..n),n=1..10) ; # R. J. Mathar, Aug 03 2017

nmax = 11;
T[n_, k_] := SeriesCoefficient[1/(1-x)^(k^2), {x, 0, n-k}];
Table[T[n, k], {n, 1, nmax}, {k, 1, n}] // Flatten

T(n,k)=binomial(k^2+n-k-1,n-k)
for(n=1,11,for(k=1,n,print1(T(n,k),", "));print(""))

T(n,k) = binomial(k^2+n-k-1, n-k).
Row sums form A178325.
Central terms form A214400.
T(n,n-2) = A037270(n-2). - R. J. Mathar, Aug 03 2017
T(n,n-3) = (n^2-6*n+11)*(n^2-6*n+10)*(n-3)^2 /6. - R. J. Mathar, Aug 03 2017

A214400 a(n) = binomial(n^2 + 3*n, n).

Original entry on oeis.org

1, 4, 45, 816, 20475, 658008, 25827165, 1198774720, 64276915527, 3911395881900, 266401260897200, 20082459351180240, 1660305826125766950, 149389005978091284720, 14533945899753270066525, 1520398315196482557890304, 170190601112537814791748255

Offset: 0

Paul D. Hanna, Jul 15 2012

nonn

Equals the central terms of triangle A214398.

Robert Israel, Table of n, a(n) for n = 0..336

Cf. A054688, A178325, A214398.

seq(binomial(n^2+3*n,n),n=0..30); # Robert Israel, Mar 04 2022

PARI
```
a(n)=binomial(n^2+3*n, n)
```

a(n) = [x^n] 1/(1 - x)^((n+1)^2). - Ilya Gutkovskiy, Oct 04 2017

a(n) ~ n^(n-1/2)*exp(n+5/2)/sqrt(2*Pi). - Robert Israel, Mar 04 2022

A262030 Partition array in Abramowitz-Stegun order: Schur functions evaluated at 1.

Original entry on oeis.org

1, 3, 1, 10, 8, 1, 35, 45, 20, 15, 1, 126, 224, 175, 126, 75, 24, 1, 462, 1050, 1134, 490, 840, 896, 175, 280, 189, 35, 1, 1716, 4752, 6468, 4704, 4950, 7350, 3528, 2646, 2400, 2940, 784, 540, 392, 48, 1, 6435, 21021, 34320, 33264, 13860, 27027, 50688, 41580, 25872, 15876, 17325, 29700, 15120, 14700, 1764, 5775, 7680, 2352, 945, 720, 63, 1

Offset: 1

Wolfdieter Lang, Oct 15 2015

The length of row n >= 1 of this irregular triangle is A000041(n) (partition numbers).
The Abramowitz-Stegun (A-St) order of the partitions is used.
For Schur functions (or polynomials) s^(lambda) = s(lambda(n, k);x[1], ..., x[n]) defined for the k-th partition of n (here in A-St order) see, e.g., the Macdonald reference, ch. I, 3. S-functions, p. 23, and Wikipedia reference. For the combinatorial interpretation of Schur functions see the Stanley reference.
The partition lambda(n,k) has m = m(n,k) nonincreasing parts lamda_j, j = 1..m, (the reverse of the partitions given in A-St) and n-m 0's are appended to obtain a length n partition. E.g., lambda(4, 3) = (2, 2, 0, 0) with m = 2.
The Schur function s(lambda(n, k),1,...,1) (n 1's) gives the number of semistandard Young tableaux (SSYT) for the Young (Ferrers) diagram of lambda(n, k) (forgetting about trailing 0's) with the box numbers taken out of the set {1, 2, ..., n} where the rows increase weakly and the columns increase strictly. See the Stanley reference pp. 309 and 310, and the example below.
The sum of the row numbers give A209673: 1, 4, 19, 116, 751, 5552, 43219, 366088, ...
Conjecture: The sum of the squares of row numbers give A054688: 1, 10, 165, 3876, ... = binomial(n^2+n-1, n). - Wouter Meeussen, Sep 25 2016

			The irregular triangle begins (commas separate entries for partitions of like numbers of parts in A-St order):
n\k  1     2    3    4    5   6   7    8   9 10 11
1:   1
2:   3,    1
3:  10,    8,   1
4:  35,   45   20,  15,   1
5: 126,  224  175, 126   75, 24,  1
6: 462, 1050 1134  490, 840 896 175, 280 189,35, 1
...
Row 7: 1716, 4752 6468 4704, 4950 7350 3528 2646, 2400 2940 784, 540 392, 48, 1;
Row 8: 6435, 21021 34320 33264 13860, 27027 50688 41580 25872 15876, 17325 29700 15120 14700 1764, 5775 7680 2352, 945 720, 63, 1.
...
n = 4, k = 4: lambda(4, 4) = (2,1,1,0) (m=3), SSYT (we use semicolons to separate the three rows): [1,1;2;3], [1,1;2;4], [1,1;3;4],
  [1,2;2;3], [1,2;2;4], [1,2;3;4],
  [1,3;2;3], [1,3;2;4], [1,3;3;4],
  [1,4;2;3], [1,4;2;4], [1,4;3;4],
  [2,2;3;4], [2,3;3;4], [2,4;3;4], hence a(4, 4) = 15. The three tableaux with distinct numbers are standard Young tableaux and give A117506(4, 4) = 3.

Macdonald, I. G., Symmetric functions and Hall polynomials, Oxford University Press, 1979.
Stanley, R. P., Enumerative Combinatorics, Vol. 2, Cambridge University Press 1999, sect. 7.30, pp. 308-316.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 831-2.
Wikipedia, Schur functions.

Cf. A054688, A117506, A209673, A210391 (in the diagonal).

a(n, k) = Det_{i,j=1..n} x[i]^(lambda_j + n-j) / Det_{i,j=1..n} x[i]^(n-j), evaluated at x[i] = 1 for i = 1..n (after division). The denominator is the Vandermonde determinant, the numerator an alternant. See, e.g., the Macdonald reference p. 24.

A055007 Number of nonnegative integer 4 X 4 matrices with no zero rows or columns and with sum of elements equal to n.

Original entry on oeis.org

1, 0, 0, 0, 24, 528, 4648, 26224, 112666, 401424, 1246000, 3476368, 8905432, 21266208, 47875272, 102482048, 210000931, 414160240, 789572072, 1460372624, 2628456428, 4615495808, 7924479264, 13328517504, 21997272036, 35674700896, 56926058920, 89477437120

Offset: 0

Vladeta Jovovic, May 30 2000

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1).

Cf. A054688, A054974, A054975, A052366.

Number of nonnegative integer p X q matrices with no zero rows or columns and with sum of elements equal to n is Sum_{k=0...q} (-1)^k*C(q, k)*m(p, q-k, n) where m(p, q, n)=Sum_{k=0..p} (-1)^k*C(p, k)*C((p-k)*q+n-1, n).
For p = q = 4 we get a(n) = (1/15!)*(n^15 + 120*n^14 + 6580*n^13 + 218400*n^12 + 4637542*n^11 + 61261200*n^10 + 423591740*n^9 + 164392800*n^8 - 17247717487*n^7 - 47940252360*n^6 + 346941238280*n^5 + 557885764800*n^4 - 4897231459056*n^3 + 8643549191040*n^2 - 5894285241600*n + 1307674368000).
G.f.: -(16*x^15 -192*x^14 +1040*x^13 -3356*x^12 +7200*x^11 -10952*x^10 +12544*x^9 -11712*x^8 +9664*x^7 -7088*x^6 +4224*x^5 -1844*x^4 +560*x^3 -120*x^2 +16*x -1) / (x -1)^16. - Colin Barker, Jul 11 2013

More terms from James Sellers, May 31 2000

A386879 a(n) = [x^n] 1/(1 - x)^(n*(n-1)/2).

Original entry on oeis.org

1, 0, 1, 10, 126, 2002, 38760, 888030, 23535820, 708930508, 23930713170, 895068996640, 36749279048405, 1643385429346680, 79515468511191440, 4139207762053520646, 230672804560960311000, 13703037308872895467960, 864424422377992704918690, 57711135174726478041405270, 4065392394346039279040037520

Offset: 0

Vaclav Kotesovec, Aug 06 2025

nonn

Cf. A014068, A054688, A116508, A177234, A386880.

Table[SeriesCoefficient[1/(1-x)^(n*(n-1)/2), {x, 0, n}], {n, 0, 25}]
Join[{1}, Table[Binomial[n*(n+1)/2, n] * (n-1) / (n+1), {n, 1, 25}]]

a(n) ~ exp(n) * n^(n - 1/2) / (sqrt(Pi) * 2^(n + 1/2)).

For n > 0, a(n) = binomial(n*(n+1)/2, n) * (n-1)/(n+1).

A386880 a(n) = [x^n] 1/(1 - x)^(n*(n+1)/2).

Original entry on oeis.org

1, 1, 6, 56, 715, 11628, 230230, 5379616, 145008513, 4431613550, 151473214816, 5727160371180, 237377895350076, 10704005376506540, 521748877569771510, 27338999059076777600, 1532576541123942256285, 91527291781199227579626, 5801648509628587739612170, 389032765009190361630625600

Offset: 0

Vaclav Kotesovec, Aug 06 2025

nonn

Cf. A014068, A054688, A116508, A177234, A386879.

Table[SeriesCoefficient[1/(1-x)^(n*(n+1)/2), {x, 0, n}], {n, 0, 25}]
Join[{1}, Table[Binomial[n*(n + 3)/2, n]*(n + 1)/(n + 3), {n, 1, 25}]]

a(n) ~ exp(n+2) * n^(n - 1/2) / (sqrt(Pi) * 2^(n + 1/2)).

For n > 0, a(n) = binomial(n*(n+3)/2, n) * (n+1)/(n+3).

A055005 Number of nonnegative integer 3 X 3 matrices with no zero rows or columns and with sum of elements equal to n.

Original entry on oeis.org

1, 0, 0, 6, 63, 306, 1038, 2844, 6750, 14437, 28521, 52911, 93258, 157509, 256581, 405171, 622719, 934542, 1373158, 1979820, 2806281, 3916812, 5390496, 7323822, 9833604, 13060251, 17171415, 22366045, 28878876, 36985383, 47007231

Offset: 0

Vladeta Jovovic, May 30 2000

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A054688, A054974, A054975.

a(n)=([0,1,0,0,0,0,0,0,0; 0,0,1,0,0,0,0,0,0; 0,0,0,1,0,0,0,0,0; 0,0,0,0,1,0,0,0,0; 0,0,0,0,0,1,0,0,0; 0,0,0,0,0,0,1,0,0; 0,0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,0,1; 1,-9,36,-84,126,-126,84,-36,9]^n*[1;0;0;6;63;306;1038;2844;6750])[1,1] \\ Charles R Greathouse IV, Aug 14 2023

Number of nonnegative integer p X q matrices with no zero rows or columns and with sum of elements equal to n is Sum_{k=0...q} (-1)^k*C(q, k)*m(p, q-k, n) where m(p, q, n)=Sum_{k=0..p} (-1)^k*C(p, k)*C((p-k)*q+n-1, n).
For p=q=3 we get a(n)=C(n + 8, 8) - 6*C(n + 5, 5) + 9*C(n + 3, 3) + 6*C(n + 2, 2) - 18*C(n + 1, 1) + 9=(1/8!)*(n^8 + 36*n^7 + 546*n^6 + 2520*n^5 - 7791*n^4 - 43596*n^3 + 148364*n^2 - 140400*n + 40320).
G.f.: -(9*x^8-54*x^7+132*x^6-171*x^5+135*x^4-78*x^3+36*x^2-9*x+1) / (x-1)^9. - Colin Barker, Jul 13 2013

A165984 Number of ways to put n indistinguishable balls into n^3 distinguishable boxes.

Original entry on oeis.org

1, 1, 36, 3654, 766480, 275234400, 151111164204, 117774526188844, 123672890985095232, 168324948170849366820, 288216356245328994082600, 606320062786763763996747618, 1537230010624231669678572481296, 4622745700243196227504110670860680

Offset: 0

Thomas Wieder, Oct 03 2009

nonn

See A165817 for the case n indistinguishable balls into 2*n distinguishable boxes.
See A054688 for the case n indistinguishable balls into n^2 distinguishable boxes.
a(n) is the number of (weak) compositions of n into n^3 parts. - Joerg Arndt, Oct 04 2017

			For n = 2 the a(2) = 36 solutions are
[0, 0, 0, 0, 0, 0, 0, 2]
[0, 0, 0, 0, 0, 0, 1, 1]
[0, 0, 0, 0, 0, 0, 2, 0]
[0, 0, 0, 0, 0, 1, 0, 1]
[0, 0, 0, 0, 0, 1, 1, 0]
[0, 0, 0, 0, 0, 2, 0, 0]
[0, 0, 0, 0, 1, 0, 0, 1]
[0, 0, 0, 0, 1, 0, 1, 0]
[0, 0, 0, 0, 1, 1, 0, 0]
[0, 0, 0, 0, 2, 0, 0, 0]
[0, 0, 0, 1, 0, 0, 0, 1]
[0, 0, 0, 1, 0, 0, 1, 0]
[0, 0, 0, 1, 0, 1, 0, 0]
[0, 0, 0, 1, 1, 0, 0, 0]
[0, 0, 0, 2, 0, 0, 0, 0]
[0, 0, 1, 0, 0, 0, 0, 1]
[0, 0, 1, 0, 0, 0, 1, 0]
[0, 0, 1, 0, 0, 1, 0, 0]
[0, 0, 1, 0, 1, 0, 0, 0]
[0, 0, 1, 1, 0, 0, 0, 0]
[0, 0, 2, 0, 0, 0, 0, 0]
[0, 1, 0, 0, 0, 0, 0, 1]
[0, 1, 0, 0, 0, 0, 1, 0]
[0, 1, 0, 0, 0, 1, 0, 0]
[0, 1, 0, 0, 1, 0, 0, 0]
[0, 1, 0, 1, 0, 0, 0, 0]
[0, 1, 1, 0, 0, 0, 0, 0]
[0, 2, 0, 0, 0, 0, 0, 0]
[1, 0, 0, 0, 0, 0, 0, 1]
[1, 0, 0, 0, 0, 0, 1, 0]
[1, 0, 0, 0, 0, 1, 0, 0]
[1, 0, 0, 0, 1, 0, 0, 0]
[1, 0, 0, 1, 0, 0, 0, 0]
[1, 0, 1, 0, 0, 0, 0, 0]
[1, 1, 0, 0, 0, 0, 0, 0]
[2, 0, 0, 0, 0, 0, 0, 0]

Cf. A001700, A054688, A060690, A165817.

a:= n-> binomial(n^3+n-1, n): seq(a(n), n=0..16);

Table[Binomial[n^3 + n - 1, n], {n, 0, 13}] (* Michael De Vlieger, Oct 05 2017 *)

a(n) = binomial(n^3+n-1, n); \\ Altug Alkan, Oct 03 2017

a(n) = binomial(n^3+n-1, n).
Let denote P(n) = the number of integer partitions of n,
p(i) = the number of parts of the i-th partition of n,
d(i) = the number of different parts of the i-th partition of n,
m(i,j) = multiplicity of the j-th part of the i-th partition of n.
Then one has:
a(n) = Sum_{i=1..P(n)} (n^3)!/((n^3-p(i))!*(Product_{j=1..d(i)} m(i,j)!)).
a(n) = [x^n] 1/(1 - x)^(n^3). - Ilya Gutkovskiy, Oct 03 2017

cp's OEIS Frontend

A007716 Number of polynomial symmetric functions of matrix of order n under separate row and column permutations.

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A135860 a(n) = binomial(n*(n+1), n).

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A214398 Triangle where the g.f. of column k is 1/(1-x)^(k^2) for k>=1, as read by rows n>=1.

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Maple

Mathematica

PARI

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A214400 a(n) = binomial(n^2 + 3*n, n).

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Maple

PARI

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A262030 Partition array in Abramowitz-Stegun order: Schur functions evaluated at 1.

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A055007 Number of nonnegative integer 4 X 4 matrices with no zero rows or columns and with sum of elements equal to n.

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Extensions

A386879 a(n) = [x^n] 1/(1 - x)^(n*(n-1)/2).

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Mathematica

Formula

A386880 a(n) = [x^n] 1/(1 - x)^(n*(n+1)/2).

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