A257463 -id:A257463 - cp's OEIS frontend

A257493 Number A(n,k) of n X n nonnegative integer matrices with all row and column sums equal to k; square array A(n,k), n >= 0, k >= 0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 6, 1, 1, 1, 4, 21, 24, 1, 1, 1, 5, 55, 282, 120, 1, 1, 1, 6, 120, 2008, 6210, 720, 1, 1, 1, 7, 231, 10147, 153040, 202410, 5040, 1, 1, 1, 8, 406, 40176, 2224955, 20933840, 9135630, 40320, 1, 1, 1, 9, 666, 132724, 22069251, 1047649905, 4662857360, 545007960, 362880, 1

Offset: 0

Alois P. Heinz, Apr 26 2015

Also the number of ordered factorizations of m^k into n factors, where m is a product of exactly n distinct primes and each factor is a product of k primes (counted with multiplicity). A(2,2) = 3: (2*3)^2 = 36 = 4*9 = 6*6 = 9*4.

			Square array A(n,k) begins:
  1,   1,      1,        1,          1,           1,            1, ...
  1,   1,      1,        1,          1,           1,            1, ...
  1,   2,      3,        4,          5,           6,            7, ...
  1,   6,     21,       55,        120,         231,          406, ...
  1,  24,    282,     2008,      10147,       40176,       132724, ...
  1, 120,   6210,   153040,    2224955,    22069251,    164176640, ...
  1, 720, 202410, 20933840, 1047649905, 30767936616, 602351808741, ...

Alois P. Heinz, Antidiagonals n = 0..20, flattened
E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce, A generalization of Eulerian numbers via rook placements, arXiv:1508.03673 [math.CO], 2015.
D. M. Jackson & G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4.4 (1975), 474-477. (Annotated scanned copy)
Richard J. Mathar, 2-regular Digraphs of the Lovelock Lagrangian, arXiv:1903.12477 [math.GM], 2019.
Dennis Pixton, Ehrhart polynomials for n = 1, ..., 9
M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements, Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970. [Annotated scanned copy]

Columns k=0-9 give: A000012, A000142, A000681, A001500, A172806, A172862, A172894, A172919, A172944, A172958.
Rows n=0+1, 2-9 give: A000012, A000027(k+1), A002817(k+1), A001496, A003438, A003439, A008552, A160318, A160319.
Main diagonal gives A110058.
Cf. A257463 (unordered factorizations), A333733 (non-isomorphic matrices), A008300 (binary matrices).

with(numtheory):
b:= proc(n, k) option remember; `if`(n=1, 1, add(
      `if`(bigomega(d)=k, b(n/d, k), 0), d=divisors(n)))
    end:
A:= (n, k)-> b(mul(ithprime(i), i=1..n)^k, k):
seq(seq(A(n, d-n), n=0..d), d=0..8);

b[n_, k_] := b[n, k] = If[n==1, 1, Sum[If[PrimeOmega[d]==k, b[n/d, k], 0], {d, Divisors[n]}]]; A[n_, k_] := b[Product[Prime[i], {i, 1, n}]^k, k]; Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 20 2016, after Alois P. Heinz *)

T(n, k)={
  local(M=Map(Mat([n, 1])));
  my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
  my(recurse(h, p, q, v, e) = if(!p, if(!e, acc(q, v)), my(i=poldegree(p), t=pollead(p)); self()(k, p-t*x^i, q+t*x^i, v, e); for(m=1, h-i, for(j=1, min(t, e\m), self()(if(j==t, k, i+m-1), p-j*x^i, q+j*x^(i+m), binomial(t, j)*v, e-j*m)))));
  for(r=1, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], recurse(k, src[i, 1], 0, src[i, 2], k))); vecsum(Mat(M)[, 2])
} \\ Andrew Howroyd, Apr 04 2020

bigomega = sloane.A001222
@cached_function
def b(n, k):
    if n == 1:
        return 1
    return sum(b(n//d, k) if bigomega(d) == k else 0 for d in n.divisors())
def A(n, k):
    return b(prod(nth_prime(i) for i in (1..n))^k, k)
[A(n, d-n) for d in (0..10) for n in (0..d)] # Freddy Barrera, Dec 27 2018, translated from Maple

from sage.combinat.integer_matrices import IntegerMatrices
[IntegerMatrices([d-n]*n, [d-n]*n).cardinality() for d in (0..10) for n in (0..d)] # Freddy Barrera, Dec 27 2018

A002135 Number of terms in a symmetrical determinant: a(n) = na(n-1) - (n-1)(n-2)*a(n-3)/2.

Original entry on oeis.org

1, 1, 2, 5, 17, 73, 388, 2461, 18155, 152531, 1436714, 14986879, 171453343, 2134070335, 28708008128, 415017867707, 6416208498137, 105630583492969, 1844908072865290, 34071573484225549, 663368639907213281, 13580208904207073801

Offset: 0

N. J. A. Sloane

a(n) is the number of collections of necklaces created by using exactly n different colored beads (to make the entire collection). - Geoffrey Critzer, Apr 19 2009
a(n) is the number of ways that a deck with 2 cards of each of n types may be dealt into n hands of 2 cards each, assuming that the order of the hands and the order of the cards in each hand are irrelevant. See the Art of Problem Solving link for proof. - Joel B. Lewis, Sep 30 2012
From Bruce Westbury, Jan 22 2013: (Start)
It follows from the respective exponential generating functions that A002135 is the binomial transform of A002137:
A002135(n) = Sum_{k=0..n} binomial(n,k)*A002137(k),
2 = 1.1 + 2.0 + 1.1,
5 = 1.1 + 3.0 + 3.1 + 1.1,
17 = 1.1 + 4.0 + 6.1 + 4.1 + 1.6, ...
A002137 arises from looking at the dimension of the space of invariant tensors of the r-th tensor power of the adjoint representation of the symplectic group Sp(2n) (for n large compared to r).
(End)
a(n) is the number of representations required for the symbolic central moments of order 2 for the multivariate normal distribution, that is,   E[X1^2 X2^2 .. Xn^2|mu=0, Sigma] (Phillips 2010). These representations are the upper-triangular, positive integer matrices for which for each i, the sum of the i-th row and i-th column equals 2, the power of each component. This can be shown starting from the formulation by Joel B Lewis. See "Proof for multivariate normal moments" link below for a proof. - Kem Phillips, Aug 20 2014
Equivalent to Critzer's comment, a(n) is the number of ways to cover n labeled vertices by disjoint undirected cycles, hence the exponential transform of A001710(n - 1). - Gus Wiseman, Oct 20 2018

			For n = 3, the a(3) = 5 ways to deal out the deck {1, 1, 2, 2, 3, 3} into three two-card hands are {11, 22, 33}, {12, 12, 33}, {13, 13, 22}, {11, 23, 23}, {12, 13, 23}. - _Joel B. Lewis_, Sep 30 2012

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 260, #12, a_n.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.9 and Problem 5.22.

T. D. Noe, Table of n, a(n) for n=0..100
A. C. Aitken, On the number of distinct terms in the expansion of symmetric and skew determinants, Edinburgh Math. Notes, No. 34 (1944), 1-5.
A. C. Aitken, On the number of distinct terms in the expansion of symmetric and skew determinants, Edinburgh Math. Notes, No. 34 (1944), 1-5. [Annotated scanned copy]
Art of Problem Solving, Partitioning a deck with 2 cards in n types into pairs
A. Cayley, On the number of distinct terms in a symmetrical or partially symmetrical determinant, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, pp. 185-190. [Annotated scanned copy]
Tomislav Došlic and Darko Veljan, Logarithmic behavior of some combinatorial sequences, Discrete Math. 308 (2008), no. 11, 2182--2212. MR2404544 (2009j:05019) - _N. J. A. Sloane_, May 01 2012
Rui-Li Liu and Feng-Zhen Zhao, New Sufficient Conditions for Log-Balancedness, With Applications to Combinatorial Sequences, J. Int. Seq., Vol. 21 (2018), Article 18.5.7.
P. A. MacMahon, Combinations derived from m identical sets of n different letters and their connexion with general magic squares, Proc. London Math. Soc., 17 (1917), 25-41.
Toufik Mansour, The length of the initial longest increasing sequence in a permutation, Art Disc. Appl. Math. (2023).
T. Muir, The Theory of Determinants in the Historical Order of Development, 4 vols., Macmillan, NY, 1906-1923. [Annotated scans of selected pages]
K. Phillips, R functions to symbolically compute the central moments of the multivariate normal distribution, Journal of Statistical Software, Feb 2010.
Kem Phillips, Proof for multivariate normal moments
Richard Stanley and J. Riordan, Problem E2297, Amer. Math. Monthly, 79 (1972), 519-520.

A diagonal of A260338.
Row sums of A215771.
Column k=2 of A257463 and A333467.
Cf. A002137, A059422, A059423, A059424.
Cf. A001147, A007717, A037143, A001710, A215771, A319225, A319226, A320655, A320656.

G:=proc(n) option remember; if n <= 1 then 1 elif n=2 then
2 else n*G(n-1)-binomial(n-1,2)*G(n-3); fi; end;

a[x_]:=Log[1/(1-x)^(1/2)]+x/2+x^2/4;Range[0, 20]! CoefficientList[Series[Exp[a[x]], {x, 0, 20}], x]
RecurrenceTable[{a[0]==a[1]==1,a[2]==2,a[n]==n*a[n-1]-(n-1)(n-2)* a[n-3]/2}, a,{n,30}] (* Harvey P. Dale, Dec 16 2011 *)
Table[Sum[Binomial[k, i] Binomial[i - 1/2, n - k] (3^(k - i) n!)/(4^k k!) (-1)^(n - k - i), {k, 0, n}, {i, 0, k}], {n, 0, 12}] (* Emanuele Munarini, Aug 25 2017 *)

a(n):=sum(sum(binomial(k,i)*binomial(i-1/2,n-k)*(3^(k-i)*n!)/(4^k*k!)*(-1)^(n-k-i),i,0,k),k,0,n);
makelist(a(n),n,0,12); /* Emanuele Munarini, Aug 25 2017 */

a(n) = if(n<3, [1,1,2][n+1], n*a(n-1) - (n-1)*(n-2)*a(n-3)/2 ); /* Joerg Arndt, Apr 07 2013 */

E.g.f.: (1-x)^(-1/2)*exp(x/2+x^2/4).
D-finite with recurrence a(n+1) = (n+1)*a(n) - binomial(n, 2)*a(n-2). See Comtet.
Asymptotics: a(n) ~ sqrt(2)*exp(3/4-n)*n^n*(1+O(1/n)). - Pietro Majer, Oct 27 2009
E.g.f.: G(0)/sqrt(1-x) where G(k) = 1 + x*(x+2)/(4*(2*k+1) - 4*x*(x+2)*(2*k+1)/(x*(x+2) + 8*(k + 1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 31 2013
a(n) = Sum_{k=0..n} Sum_{i=0..k} binomial(k,i)*binomial(i-1/2,n-k)*(3^(k-i)*n!)/(4^k*k!)*(-1)^(n-k-i). - Emanuele Munarini, Aug 25 2017

A257462 Number A(n,k) of factorizations of m^n into n factors, where m is a product of exactly k distinct primes and each factor is a product of k primes (counted with multiplicity); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 2, 1, 1, 1, 1, 10, 10, 3, 1, 1, 1, 1, 26, 70, 25, 3, 1, 1, 1, 1, 71, 566, 465, 49, 4, 1, 1, 1, 1, 197, 4781, 11131, 2505, 103, 4, 1, 1, 1, 1, 554, 41357, 297381, 190131, 12652, 184, 5, 1, 1, 1, 1, 1570, 364470, 8349223, 16669641, 2928876, 57232, 331, 5, 1, 1

Offset: 0

Alois P. Heinz, Apr 24 2015

Also number of ways to partition the multiset consisting of n copies each of 1, 2, ..., k into n multisets of size k.

			A(4,2) = 3: (2*3)^4 = 1296 = 6*6*6*6 = 9*6*6*4 = 9*9*4*4.
A(3,3) = 10: (2*3*5)^3 = 2700 = 30*30*30 = 45*30*20 = 50*27*20 = 50*30*18 = 50*45*12 = 75*20*18 = 75*30*12 = 75*45*8 = 125*18*12 = 125*27*8.
A(2,4) = 10: (2*3*5*7)^2 = 44100 = 210*210 = 225*196 = 294*150 = 315*140 = 350*126 = 441*100 = 490*90 = 525*84 = 735*60 = 1225*36.
Square array A(n,k) begins:
  1, 1, 1,  1,    1,      1, ...
  1, 1, 1,  1,    1,      1, ...
  1, 1, 2,  4,   10,     26, ...
  1, 1, 2, 10,   70,    566, ...
  1, 1, 3, 25,  465,  11131, ...
  1, 1, 3, 49, 2505, 190131, ...

Andrew Howroyd, Table of n, a(n) for n = 0..377 (antidiagonals n=0..26)

Columns k=0+1, 2-5 give: A000012, A008619, A254233, A257114, A257518.
Rows n=0+1, 2-3 give: A000012, A257520, A333902.
Main diagonal gives A334286.
Cf. A257463, A333901 (ordered factorizations).

with(numtheory):
b:= proc(n, i, k) option remember; `if`(n=1, 1,
      add(`if`(d>i or bigomega(d)<>k, 0,
      b(n/d, d, k)), d=divisors(n) minus {1}))
    end:
A:= (n, k)-> b(mul(ithprime(i), i=1..k)^n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..8);

b[n_, i_, k_] := b[n, i, k] = If[n == 1, 1, Sum[If[d > i || PrimeOmega[d] != k, 0, b[n/d, d, k]], {d, Divisors[n] // Rest}]]; A[n_, k_] := Module[ {p = Product[Prime[i], {i, 1, k}]^n}, b[p, p, k]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 8}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)

A260340 Triangle read by rows: T(n,k) = number of sets of linear n-ads in k variables.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 22, 22, 1, 1, 1, 1, 130, 550, 130, 1, 1, 1, 1, 822, 16700, 16700, 822, 1, 1, 1, 1, 6202, 703297, 3330915, 703297, 6202, 1, 1, 1, 1, 52552, 38135272, 957659906, 957659906, 38135272, 52552, 1, 1

Offset: 0

N. J. A. Sloane, Jul 30 2015

T(n,k) is the number of nonequivalent n X n binary matrices with k ones in every row and column up to permutation of rows. - Andrew Howroyd, Apr 18 2020

			Triangle begins:
  1;
  1, 1;
  1, 1,    1;
  1, 1,    1,      1;
  1, 1,    6,      1,       1;
  1, 1,   22,     22,       1,      1;
  1, 1,  130,    550,     130,      1,    1;
  1, 1,  822,  16700,   16700,    822,    1, 1;
  1, 1, 6202, 703297, 3330915, 703297, 6202, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..230 (rows 0..20)
P. A. MacMahon, Combinations derived from m identical sets of n different letters and their connexion with general magic squares, Proc. London Math. Soc., 17 (1917), 25-41.

Columns k=0..4 are A000012, A000012, A002137, A333899, A333900.
Row sums are A333891.
Cf. A008300, A133687, A257463.

T(n,k) = T(n,n-k). - Andrew Howroyd, Apr 18 2020

Extended to include k=0 and more terms added by Andrew Howroyd, Apr 18 2020

A320451 Number of multiset partitions of uniform integer partitions of n in which all parts have the same length.

Original entry on oeis.org

1, 1, 3, 5, 8, 7, 19, 11, 24, 26, 38, 28, 85, 46, 89, 99, 146, 110, 246, 163, 326, 305, 416, 376, 816, 591, 903, 971, 1450, 1295, 2517, 1916, 3045, 3141, 4042, 4117, 7073, 5736, 8131, 9026, 12658, 11514, 19459, 16230, 24638, 27129, 33747, 32279, 55778, 45761, 71946

Offset: 0

Gus Wiseman, Oct 12 2018

nonn

An integer partitions is uniform if all parts appear with the same multiplicity.

Terms can be computed by the formula: Sum_{d|n} Sum_{i>=1} P(n/d,i) * Sum_{h|i*d} M(i*d/h, i, h, d) where P(n,k) is the number of partitions of n into k distinct parts and M(h,w,r,s) is the number of nonnegative integer h X w matrices up to row permutations with all row sums equal to r and all column sums equal to s. The cases of M(h,w,w,h) and M(n,n,k,k) are enumerated by the arrays A257462 and A257463. - Andrew Howroyd, Feb 04 2022

			The a(9) = 26 multiset partitions:
  {{9}}
  {{1,8}}
  {{2,7}}
  {{3,6}}
  {{4,5}}
  {{1,2,6}}
  {{1,3,5}}
  {{1},{8}}
  {{2,3,4}}
  {{2},{7}}
  {{3,3,3}}
  {{3},{6}}
  {{4},{5}}
  {{1},{2},{6}}
  {{1},{3},{5}}
  {{2},{3},{4}}
  {{3},{3},{3}}
  {{1,1,1,2,2,2}}
  {{1,1,1},{2,2,2}}
  {{1,1,2},{1,2,2}}
  {{1,1},{1,2},{2,2}}
  {{1,2},{1,2},{1,2}}
  {{1,1,1,1,1,1,1,1,1}}
  {{1,1,1},{1,1,1},{1,1,1}}
  {{1},{1},{1},{2},{2},{2}}
  {{1},{1},{1},{1},{1},{1},{1},{1},{1}}

Andrew Howroyd, Table of n, a(n) for n = 0..150
A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp. 1-12.

Cf. A001970, A047966, A047968, A050342, A089259, A258466, A261049, A305551, A319056, A319066, A320328, A320330, A320331.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[SameQ@@Length/@Split[Sort[Join@@#]],SameQ@@Length/@#]&]],{n,10}]

Terms a(11) and beyond from Andrew Howroyd, Feb 04 2022

A254243 Number of ways to partition the multiset consisting of 3 copies each of 1, 2, ..., n into n sets of size 3.

Original entry on oeis.org

1, 1, 2, 10, 93, 1417, 32152, 1016489, 42737945, 2307295021, 155607773014, 12823004639504, 1267907392540573, 148160916629902965, 20199662575448858212, 3177820001990224608763, 571395567211112572679633, 116448309072281063992943561, 26700057600529091443246943530

Offset: 0

Tatsuru Murai, Jan 27 2015

nonn

			a(1) = 1: 111.
a(2) = 2: 111|222 and 112|122.
a(3) = 10: 111|222|333, 111|223|233, 112|122|333, 112|123|233, 112|133|223, 113|122|233, 113|123|223, 113|133|222, 122|123|133, and 123|123|123.

Andrew Howroyd, Table of n, a(n) for n = 0..100
P. A. MacMahon, Combinations derived from m identical sets of n different letters and their connexion with general magic squares, Proc. London Math. Soc., 17 (1917), 25-41. See page 40 (but there is a typo).
StackExchange, Number of Partitioning a deck with m cards in n types into n-element sets, January 2015.

Cf. A002135 (2 instead of 3), A254233 (n copies each of 1, 2, and 3).

Column k=3 of A257463.

Name and example edited by Danny Rorabaugh, Apr 22 2015
a(6)-a(10) from Alois P. Heinz, Apr 22 2015
Terms a(11) and beyond from Andrew Howroyd, Apr 18 2020

A268668 Number of factorizations of m^4 into n factors, where m is a product of exactly n distinct primes and each factor is a product of 4 primes (counted with multiplicity).

Original entry on oeis.org

1, 1, 3, 23, 465, 19834, 1532489, 193746632, 37368959742, 10437763731100, 4054349060577421, 2119978249890808761, 1452950920608600023603, 1276433147589499725385063, 1410464141866494594480406985, 1928819743142477893302566583434, 3218592064882611634798263991387049

Offset: 0

Alois P. Heinz, Feb 10 2016

nonn

Also number of ways to partition the multiset consisting of 4 copies each of 1, 2, ..., n into n multisets of size 4.

			a(2) = 3: (2*3)^4 = 1296 = 36*36 = 54*24 = 81*16.
a(3) = 23: (2*3*5)^4 = 810000 = 100*90*90 = 100*100*81 = 135*100*60 = 150*90*60 = 150*100*54 = 150*135*40 = 150*150*36 = 225*60*60 = 225*90*40 = 225*100*36 = 225*150*24 = 225*225*16 = 250*60*54 = 250*81*40 = 250*90*36 = 250*135*24 = 375*54*40 = 375*60*36 = 375*90*24 = 375*135*16 = 625*36*36 = 625*54*24 = 625*81*16.

Andrew Howroyd, Table of n, a(n) for n = 0..50

Column k=4 of A257463.

Cf. A333900.

Terms a(10) and beyond from Andrew Howroyd, Apr 18 2020

A253259 Number of factorizations of m^n into 4 factors, where m is a product of exactly 4 distinct primes and each factor is a product of n primes (counted with multiplicity).

Original entry on oeis.org

1, 1, 17, 93, 465, 1746, 5741, 16238, 41650, 97407, 212412, 434767, 845366, 1569344, 2801696, 4828140, 8069053, 13114785, 20796651, 32242621, 48986553, 73052382, 107114645, 154621230, 220021932, 308940815, 428492880, 587520315, 797019526, 1070458096

Offset: 0

Alois P. Heinz, Apr 30 2015

			a(2) = 17: (2*3*5*7)^2 = 44100 = 15*15*14*14 = 21*15*14*10 = 21*21*10*10 = 25*14*14*9 = 25*21*14*6 = 25*21*21*4 = 35*14*10*9 = 35*15*14*6 = 35*21*10*6 = 35*21*15*4 = 35*35*6*6 = 35*35*9*4 = 49*10*10*9 = 49*15*10*6 = 49*15*15*4 = 49*25*6*6 = 49*25*9*4.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
P. A. MacMahon, Combinations derived from m identical sets of n different letters and their connexion with general magic squares, Proc. London Math. Soc., 17 (1917), 25-41. This sequence is column 4 of table f(m,n) on page 40.
Math StackExchange, Number of ways to partition 40 balls with 4 colors into 4 baskets
Marko Riedel, Maple program to compute sequence from cycle indices

Row n=4 of A257463.

[A^n B^n C^n D^n] Z(S_4)(Z(S_n)(A+B+C+D)) with Z(S_q) the cycle index of the symmetric group; parenthesis denote the canonical substitution of the argument into the cycle index. - Marko Riedel, Feb 06 2016

G.f.: (x^18 +6*x^17 +58*x^16 +213*x^15 +646*x^14 +1415*x^13 +2515*x^12 +3554*x^11 +4296*x^10 +4248*x^9 +3578*x^8 +2452*x^7 +1421*x^6 +628*x^5 +240*x^4 +61*x^3 +12*x^2-x+1) /((1-x)^10 *(1+x)^5 *(1+x+x^2)^3 *(1+x^2)). [This was found by Will Orrick and confirmed by Marko Riedel, see the StackExchange link above.] - Alois P. Heinz, Feb 09 2016

A253263 Number of factorizations of m^n into 5 factors, where m is a product of exactly 5 distinct primes and each factor is a product of n primes (counted with multiplicity).

Original entry on oeis.org

1, 1, 73, 1417, 19834, 190131, 1398547, 8246011, 40837569, 174901563, 664006236, 2274999093, 7139338769, 20758868781, 56466073587, 144806582536, 352420554194, 818441723112, 1822255658908, 3905392549491, 8084697432576, 16214886608495, 31590986308810, 59926495673780

Offset: 0

Alois P. Heinz, Apr 30 2015

nonn

			a(2) = 73: (2*3*5*7*11)^2 = 5336100 = 25*22*22*21*21 = 33*25*22*21*14 = 33*33*25*14*14 = 35*22*22*21*15 = 35*33*22*15*14 = 35*33*22*21*10 = 35*33*33*14*10 = 35*35*22*22*9 = 35*35*33*22*6 = 35*35*33*33*4 = 49*22*22*15*15 = 49*25*22*22*9 = 49*33*22*15*10 = 49*33*25*22*6 = 49*33*33*10*10 = 49*33*33*25*4 = 55*22*21*15*14 = 55*22*21*21*10 = 55*33*15*14*14 = 55*33*21*14*10 = 55*35*22*14*9 = 55*35*22*21*6 = 55*35*33*14*6 = 55*35*33*21*4 = 55*49*22*10*9 = 55*49*22*15*6 = 55*49*33*10*6 = 55*49*33*15*4 = 55*55*14*14*9 = 55*55*21*14*6 = 55*55*21*21*4 = 55*55*49*6*6 = 55*55*49*9*4 = 77*22*15*15*14 = 77*22*21*15*10 = 77*25*22*14*9 = 77*25*22*21*6 = 77*33*15*14*10 = 77*33*21*10*10 = 77*33*25*14*6 = 77*33*25*21*4 = 77*35*22*10*9 = 77*35*22*15*6 = 77*35*33*10*6 = 77*35*33*15*4 = 77*55*14*10*9 = 77*55*15*14*6 = 77*55*21*10*6 = 77*55*21*15*4 = 77*55*35*6*6 = 77*55*35*9*4 = 77*77*10*10*9 = 77*77*15*10*6 = 77*77*15*15*4 = 77*77*25*6*6 = 77*77*25*9*4 = 121*15*15*14*14 = 121*21*15*14*10 = 121*21*21*10*10 = 121*25*14*14*9 = 121*25*21*14*6 = 121*25*21*21*4 = 121*35*14*10*9 = 121*35*15*14*6 = 121*35*21*10*6 = 121*35*21*15*4 = 121*35*35*6*6 = 121*35*35*9*4 = 121*49*10*10*9 = 121*49*15*10*6 = 121*49*15*15*4 = 121*49*25*6*6 = 121*49*25*9*4.

Andrew Howroyd, Table of n, a(n) for n = 0..50

Row n=5 of A257463.

A257464 Number of factorizations of m^n into 3 factors, where m is a product of exactly 3 distinct primes and each factor is a product of n primes (counted with multiplicity).

Original entry on oeis.org

1, 1, 5, 10, 23, 40, 73, 114, 180, 262, 379, 521, 712, 938, 1228, 1567, 1986, 2469, 3052, 3715, 4499, 5383, 6410, 7558, 8875, 10335, 11991, 13816, 15865, 18110, 20611, 23336, 26350, 29620, 33213, 37095, 41338, 45904, 50870, 56197, 61964, 68131, 74782, 81873

Offset: 0

Alois P. Heinz, Apr 24 2015

			a(2) = 5: (2*3*5)^2 = 900 = 10*10*9 = 15*10*6 = 15*15*4 = 25*6*6 = 25*9*4.
a(4) = 23: (2*3*5)^4 = 810000 = 100*90*90 = 100*100*81 = 135*100*60 = 150*90*60 = 150*100*54 = 150*135*40 = 150*150*36 = 225*60*60 = 225*90*40 = 225*100*36 = 225*150*24 = 225*225*16 = 250*60*54 = 250*81*40 = 250*90*36 = 250*135*24 = 375*54*40 = 375*60*36 = 375*90*24 = 375*135*16 = 625*36*36 = 625*54*24 = 625*81*16.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Alexander V. Karpov, An Informational Basis for Voting Rules, NRU Higher School of Economics. Series WP BRP "Economics/EC". 2018. No. 188.
P. A. MacMahon, Combinations derived from m identical sets of n different letters and their connexion with general magic squares, Proc. London Math. Soc., 17 (1917), 25-41. This sequence is column 3 of table f(m,n) on page 40.
Index entries for linear recurrences with constant coefficients, signature (2,1,-3,-1,1,3,-1,-2,1).

Row n=3 of A257463.

a:= n-> coeff(series(-(x^6-x^5+2*x^4+2*x^3+2*x^2-x+1)/
        ((x^2+x+1)*(x+1)^2*(x-1)^5), x, n+1), x, n):
seq(a(n), n=0..60);

CoefficientList[Series[-(x^6 - x^5 + 2 x^4 + 2 x^3 + 2 x^2 - x + 1)/((x^2 + x + 1) (x + 1)^2*(x - 1)^5), {x, 0, 43}], x] (* Michael De Vlieger, Jul 02 2018 *)
LinearRecurrence[{2,1,-3,-1,1,3,-1,-2,1},{1,1,5,10,23,40,73,114,180},50] (* Harvey P. Dale, Jan 08 2022 *)

G.f.: -(x^6-x^5+2*x^4+2*x^3+2*x^2-x+1)/((x^2+x+1)*(x+1)^2*(x-1)^5).

cp's OEIS Frontend

A257493 Number A(n,k) of n X n nonnegative integer matrices with all row and column sums equal to k; square array A(n,k), n >= 0, k >= 0, read by antidiagonals.

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A002135 Number of terms in a symmetrical determinant: a(n) = n*a(n-1) - (n-1)*(n-2)*a(n-3)/2.

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A257462 Number A(n,k) of factorizations of m^n into n factors, where m is a product of exactly k distinct primes and each factor is a product of k primes (counted with multiplicity); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A260340 Triangle read by rows: T(n,k) = number of sets of linear n-ads in k variables.

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A320451 Number of multiset partitions of uniform integer partitions of n in which all parts have the same length.

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A254243 Number of ways to partition the multiset consisting of 3 copies each of 1, 2, ..., n into n sets of size 3.

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A268668 Number of factorizations of m^4 into n factors, where m is a product of exactly n distinct primes and each factor is a product of 4 primes (counted with multiplicity).

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A253259 Number of factorizations of m^n into 4 factors, where m is a product of exactly 4 distinct primes and each factor is a product of n primes (counted with multiplicity).

A002135 Number of terms in a symmetrical determinant: a(n) = na(n-1) - (n-1)(n-2)*a(n-3)/2.