A124577 -id:A124577 - cp's OEIS frontend

A103923 Triangle of partitions of n with parts of sizes 1,2,...,m, each of two different kinds, m>=1.

1, 1, 1, 2, 2, 1, 3, 4, 2, 1, 5, 7, 5, 2, 1, 7, 12, 9, 5, 2, 1, 11, 19, 17, 10, 5, 2, 1, 15, 30, 28, 19, 10, 5, 2, 1, 22, 45, 47, 33, 20, 10, 5, 2, 1, 30, 67, 73, 57, 35, 20, 10, 5, 2, 1, 42, 97, 114, 92, 62, 36, 20, 10, 5, 2, 1, 56, 139, 170, 147, 102, 64, 36, 20, 10, 5, 2, 1, 77, 195

Offset: 0

Wolfdieter Lang, Mar 24 2005

The corresponding Fine-Riordan triangle is A008951.
This is the array p_2(n,m) of Gupta et al. written as a triangle. p_2(n,m) is defined on p. x of this reference as the number of partitions of n into parts consisting of two varieties of each of the integers 1 to m and one variety of each larger integer. Therefore a(n,m) gives these numbers for the partitions of n-m.
a(n,m)= sum over partitions of n+t(m)-m of binomial(q(partition),m), with t(m):=A000217(m) and q the number of distinct parts of a given partition. m>=0.
a(n,m)= number of partitions of 2*n-m with exactly m odd parts.
a(n,m)= sum over partitions of n+t(m)-m of product(k[j],j=1..m), with t(m):=A000217(m) and k[j]=number of parts of size j (exponent of j in a given partition of n), if m>=1. If m=0 then a(n,0)=p(n):=A000041(n) (number of partitions of n). 0 is counted as a part for n=0 and only for this n.

			Triangle starts:
[1];
[1,1];
[2,2,1];
[3,4,2,1];
[5,7,5,2,1];
...
a(4,2)=5 from the partitions of 4-2=2 with two varieties of parts 1 and of 2, namely (2),(2'),(1^2),(1'^2) and (1,1').
a(4,2)=5 from the partitions of 4+t(2)-2=5 which have products of the exponents of parts 1 and 2: 0*0,1*0,0*1,2*1,1*2,5*0 and sum to 4.
a(4,2)=5 from the partitions of 4+t(2)-2=5 which have number of distinct parts (q values) 1,2,2,2,2,2,1. The corresponding binomial(q,2) values are 0,1,1,1,1,0 and sum to 4.
a(4,2)=5 from the partitions of 2*4-2=6 with exactly two odd parts, namely (1,5), (3^2), (1^2,4), (1,2,3) and (1^2,2^2), which are 5 in number.

H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958 (reprinted 1962), pp. 90-121.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.

Alois P. Heinz, Rows n = 0..140, flattened
J. Huang, A. Senger, P. Wear, T. Wu, Partition statistics equidistributed with the number of hook difference one cells, 2013. See Remark 5.7. - _N. J. A. Sloane_, May 20 2014
W. Lang: First 16 rows.

The column sequences (without leading 0's) are, for m=0..10: A000041, A000070, A000097, A000098, A000710, A103924-A103929.

Cf. A000712, A124577.

with(numtheory):
b:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*
      `if`(d<=k, 2, 1), d=divisors(j)) *b(n-j, k), j=1..n)/n)
    end:
A:= (n, k)-> b(n, k) -`if`(k=0, 0, b(n, k-1)):
seq(seq(A(n, k), k=0..n), n=0..14);  # Alois P. Heinz, Sep 14 2014

a[n_, 0] := a[n, 0] = PartitionsP[n]; a[n_, m_] /; n= m >= 0 := a[n, m] = a[n-1, m-1] + a[n-m, m]; Table[a[n, m], {n, 0, 14}, {m, 0, n}] // Flatten (* Jean-François Alcover, Dec 09 2014 *)
Flatten@Table[Length@IntegerPartitions[n-m, All, Range@n~Join~Range@m],  {n, 0, 12}, {m, 0, n}] (* Robert Price, Jul 29 2020 *)

a(n, m) = a(n-1, m-1) + a(n-m, m), n>=m>=0, with a(n, 0)= A000041(n) (partition numbers), a(n, m)=0 if n

a(n, m) = sum(a(n-1-j*m, m-1), j=0..floor((n-m)/m)), m>=1, input a(n, 0)= A000041(n).

G.f. column m: product(1/(1-x^j), j=1..m)*P(x), with P(x)= product(1/(1-x^j), j=1..infty), the o.g.f. for the partition numbers A000041.

G.f. column m>=1: (product(1/(1-x^k), k=1..m)^2)*product(1/(1-x^j), j=(m+1)..infty). For m=0 put the first product equal to 1.

A209664 T(n,k) = count of degree k monomials in the power sum symmetric polynomials p(mu,k) summed over all partitions mu of n.

Original entry on oeis.org

1, 2, 6, 3, 14, 39, 5, 34, 129, 356, 7, 74, 399, 1444, 4055, 11, 166, 1245, 5876, 20455, 57786, 15, 350, 3783, 23604, 102455, 347010, 983535, 22, 746, 11514, 94852, 513230, 2083902, 6887986, 19520264, 30, 1546, 34734, 379908, 2567230, 12505470, 48219486, 156167944, 441967518

Offset: 1

Wouter Meeussen, Mar 11 2012

			Table starts as:
:  1;
:  2,   6;
:  3,  14,   39;
:  5,  34,  129,  356;
:  7,  74,  399, 1444,  4055;
: 11, 166, 1245, 5876, 20455, 57786;

Alois P. Heinz, Rows n = 1..141, flattened
Wikipedia, Symmetric Polynomials

Main diagonal is A124577; row sums are A209665.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
    end:
T:= (n, k)-> b(n$2, k):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Nov 24 2016

p[n_Integer, v_] := Sum[Subscript[x, j]^n, {j, v}]; p[par_?PartitionQ, v_] := Times @@ (p[#, v] & /@ par); Table[Tr[(p[#, k] & /@ Partitions[l]) /. Subscript[x, _] -> 1], {l, 11}, {k, l}]

A298985 a(n) = [x^n] Product_{k>=1} 1/(1 - n*x^k)^k.

Original entry on oeis.org

1, 1, 8, 54, 496, 5400, 73728, 1204322, 23167808, 512093178, 12781430600, 355128859129, 10863077554224, 362572265689777, 13107541496092960, 510105773344747725, 21258690342206888192, 944467894258279964254, 44555341678790400325512, 2224158766859058600584834, 117123916650423288611260400

Offset: 0

Ilya Gutkovskiy, Jan 31 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..380

Cf. A000219, A124577, A261561, A261565, A266941, A297329, A298986, A298987, A298988.

b:= proc(n, i, k) option remember;   `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(i+j-1, j)*b(n-i*j, i-1, k)*k^j, j=0..n/i)))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 23 2018

Table[SeriesCoefficient[Product[1/(1 - n x^k)^k, {k, 1, n}], {x, 0, n}], {n, 0, 20}]

a(n) ~ n^n. - Vaclav Kotesovec, Feb 02 2018

A292132 Main diagonal of A292131.

Original entry on oeis.org

1, -1, -2, 6, 12, 45, -150, -203, -840, -1872, 6390, 8580, 30084, 69108, 165802, -494565, -514320, -1997296, -4202298, -10175526, -18908420, 49930440, 54032770, 190735688, 405256872, 948210600, 1751280726, 3624555168, -7676955468, -6724059944, -26741354430

Offset: 0

Seiichi Manyama, Sep 09 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A124577, A291698, A292131.

{a(n) = polcoeff(prod(k=1, n, 1-n*x^k+x*O(x^n)), n)}

a(n) = [x^n] Product_{k=1..n} (1 - n*x^k).

A292134 Main diagonal of A292133.

Original entry on oeis.org

1, -1, 2, -21, 220, -2705, 40926, -733537, 15124216, -352606050, 9174382490, -263533561852, 8283376452948, -282795708021411, 10420847619031710, -412243715452039440, 17425722339237083120, -783844576340696848341, 37384875796116662077194

Offset: 0

Seiichi Manyama, Sep 09 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..386

Cf. A124577, A291698.

{a(n) = polcoeff(1/prod(k=1, n, 1+n*x^k+x*O(x^n)), n)}

a(n) = [x^n] Product_{k>=1} 1/(1 + n*x^k).

a(n) ~ (-1)^n * n^n * (1 - 1/n + 2/n^2 - 3/n^3 + 5/n^4 - 7/n^5 + 11/n^6 - 15/n^7 + 22/n^8 - 30/n^9 + 42/n^10 - ...), for coefficients, see A000041. - Vaclav Kotesovec, Aug 21 2018

A303175 a(n) = [x^n] Product_{k=1..n} 1/(1 - (n - k + 1)*x^k).

Original entry on oeis.org

1, 1, 5, 34, 322, 3803, 55297, 953815, 19086057, 434477488, 11086102633, 313318606066, 9714265351819, 327788649292844, 11957321196905337, 468872400449456885, 19666225828334583690, 878560858388253803180, 41645712575272737701666, 2087686693048676581394052

Offset: 0

Ilya Gutkovskiy, Apr 19 2018

nonn

			a(0) = 1;
a(1) = [x^1] 1/(1 - x) = 1;
a(2) = [x^2] 1/((1 - 2*x)*(1 - x^2)) = 5;
a(3) = [x^3] 1/((1 - 3*x)*(1 - 2*x^2)*(1 - x^3)) = 34;
a(4) = [x^4] 1/((1 - 4*x)*(1 - 3*x^2)*(1 - 2*x^3)*(1 - x^4)) = 322;
a(5) = [x^5] 1/((1 - 5*x)*(1 - 4*x^2)*(1 - 3*x^3)*(1 - 2*x^4)*(1 - x^5)) = 3803, etc.
...
The table of coefficients of x^k in expansion of Product_{k=1..n} 1/(1 - (n - k + 1)*x^k) begins:
n = 0: (1), 0,   0,    0,    0,     0,  ...
n = 1:  1, (1),  1,    1,    1,     1,  ...
n = 2:  1,  2,  (5),  10,   21,    42,  ...
n = 3:  1,  3,  11,  (34), 106,   320,  ...
n = 4:  1,  4,  19,   78, (322), 1294,  ...
n = 5:  1,  5,  29,  148,  758, (3803), ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..384

Cf. A006906, A124577, A206228, A303188, A303189, A303190.

Table[SeriesCoefficient[Product[1/(1 - (n - k + 1) x^k), {k, 1, n}], {x, 0, n}], {n, 0, 19}]

a(n) ~ n^n * (1 + 1/n + 1/n^2 - 1/n^3 - 3/n^4 - 8/n^5 - 7/n^6 - 13/n^7 + 2/n^8 - 3/n^9 + 31/n^10 + 21/n^11 + 81/n^12 + 2/n^13 + 152/n^14 - 114/n^15 + 173/n^16 - 341/n^17 + 260/n^18 - 936/n^19 + 861/n^20 - 2187/n^21 + 2630/n^22 - 4551/n^23 + 6211/n^24 - 8866/n^25 + 14889/n^26 - 22374/n^27 + 38490/n^28 - 55911/n^29 + 87688/n^30 - ...). - Vaclav Kotesovec, Aug 21 2018

A338697 a(n) = [x^n] Product_{k>=1} 1 / (1 - n^(k-1)*x^k).

Original entry on oeis.org

1, 1, 3, 13, 101, 931, 12391, 178809, 3331721, 66288142, 1589753211, 40104031166, 1183380156013, 36187564837217, 1262524447510383, 45533370885563716, 1834219414937219601, 76016894083755947753, 3479900167920331954531, 162982921698852088968886, 8341707623665223127224821

Offset: 0

Ilya Gutkovskiy, Apr 24 2021

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..385

Cf. A008284, A075900, A124577, A300579, A338673, A338674, A338675, A338676, A338677, A338678, A338679, A344095.

Table[SeriesCoefficient[Product[1/(1 - n^(k - 1) x^k), {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Join[{1}, Table[Sum[Length[IntegerPartitions[n, {k}]] n^(n - k), {k, 0, n}], {n, 1, 20}]]
Join[{1}, Table[SeriesCoefficient[x + (n-1)/(n*QPochhammer[1/n, n*x]), {x, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, May 09 2021 *)

a(n) = Sum_{k=0..n} p(n,k) * n^(n-k), where p(n,k) is the number of partitions of n into k parts.

a(n) ~ c * n^(n-1), where c = BesselI(1,2) = A096789 = 1.590636854637329... - Vaclav Kotesovec, May 09 2021

A292417 a(n) = [x^n] Product_{k>=1} 1/(1 - n^2*x^k).

Original entry on oeis.org

1, 1, 20, 819, 70160, 10188775, 2240751636, 692647082799, 286013768613952, 151994274055319070, 101020305070908050100, 82086758986568812837856, 80056656965795630400382608, 92282612223268812357487227077, 124113156850218393012451734737460

Offset: 0

Vaclav Kotesovec, Sep 16 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..214

Cf. A077335, A124577, A292304.

nmax = 20; Table[SeriesCoefficient[Product[1/(1-n^2*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]

{a(n)= polcoef(prod(k=1, n, 1/(1-n^2*x^k +x*O(x^n))), n)};
for(n=0,20, print1(a(n), ", ")) \\ G. C. Greubel, Feb 02 2019

a(n) ~ n^(2*n) * (1 + 1/n^2 + 2/n^4 + 3/n^6 + 5/n^8 + 7/n^10), for coefficients see A000041.

Showing 1-10 of 15 results. Next

In general, column k > 1 is asymptotic to c * k^n, where c = Product_{j>=1} 1/(1-1/k^j) = 1/QPochhammer[1/k,1/k]. - Vaclav Kotesovec, Mar 19 2015

When k is a prime power greater than 1, A(n,k) is the number of conjugacy classes of n X n matrices over a field of size k. - Geoffrey Critzer, Nov 11 2022

The number of partitions of n into distinct parts where each part can be colored in n different ways. For example, there are 4 partitions of 6 into distinct parts, namely 6, 5 + 1, 4 + 2 and 3 + 2 + 1; allowing for the colorings gives a(6) = 6 + 6*6 + 6*6 + 6*6*6 = 294. - Peter Bala, Aug 31 2017

cp's OEIS Frontend

A246935 Number A(n,k) of partitions of n into k sorts of parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A291698 a(n) = [x^n] Product_{k>=1} (1 + n*x^k).

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A103923 Triangle of partitions of n with parts of sizes 1,2,...,m, each of two different kinds, m>=1.

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A209664 T(n,k) = count of degree k monomials in the power sum symmetric polynomials p(mu,k) summed over all partitions mu of n.

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A298985 a(n) = [x^n] Product_{k>=1} 1/(1 - n*x^k)^k.

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A292132 Main diagonal of A292131.

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A292134 Main diagonal of A292133.

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A303175 a(n) = [x^n] Product_{k=1..n} 1/(1 - (n - k + 1)*x^k).

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