A103924 -id:A103924 - cp's OEIS frontend

A008951 Array read by columns: number of partitions of n into parts of 2 kinds.

1, 1, 1, 2, 2, 3, 4, 1, 5, 7, 2, 7, 12, 5, 11, 19, 9, 1, 15, 30, 17, 2, 22, 45, 28, 5, 30, 67, 47, 10, 42, 97, 73, 19, 1, 56, 139, 114, 33, 2, 77, 195, 170, 57, 5, 101, 272, 253, 92, 10, 135, 373, 365, 147, 20, 176, 508, 525, 227, 35, 1, 231, 684, 738, 345, 62, 2, 297

Offset: 0

N. J. A. Sloane

Fine-Riordan array S_n(m) = a(n,m) with extra row for n=0 added.
Row n of this triangle has length floor(1/2 + sqrt(2*(n+1))), n>=0. This is sequence {A002024(n+1)} = [1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,...].
Written as a triangle this becomes A103923.
a(n,m) also gives the number of partitions of n-t(m), where t(m):=A000217(m) (triangular numbers), with two kinds of parts 1,2,..m. See the column o.g.f.'s in table A103923.
In general, column m is asymptotic to exp(Pi*sqrt(2*n/3)) * 6^(m/2) * n^((m-2)/2) / (4*sqrt(3) * m! * Pi^m), equivalently to 6^(m/2) * n^(m/2) / (m! * Pi^m) * p(n), where p(n) is the partition function A000041. - Vaclav Kotesovec, Aug 28 2015

			Array begins:
m\n 0 1 2 3 4 .5 .6 .7 .8 ...
0 | 1 1 2 3 5 .7 11 15 22 ... (A000041)
1 | . 1 2 4 7 12 19 ... (A000070)
2 | . . . 1 2 .5 .9 ... (A000097)
3 | . . . . . .. .1 ... (A000098)
[1]; [1,1]; [2,2]; [3,4,1]; [5,7,2]; [7,12,5]; [11,19,9,1]...
a(3,1) = 4 because the partitions (3), (1,2) and (1^3) have q values 1,2 and 1 which sum to 4.
a(3,1) = 4 because the exponents of part 1 in the above given partitions of 3 are 0,1,3 and they sum to 4.
a(3,1) = 4 because the partitions of 3-t(1)=2 with two kinds of part 1, say 1 and 1' and one kind of part 2 are (2),(1^2), (1'^2) and (11').

H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 90.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.

Alois P. Heinz, Columns n = 0..500, flattened
William K. Keith, Restricted k-color partitions, arXiv preprint arXiv:1408.4089 [math.CO], 2014.
Wolfdieter Lang, First 20 rows and comments.

The first column (m=0) gives A000041(n). Columns m=1..10 are A000070 (partial sums of partition numbers), A000097, A000098, A000710, A103924-A103929.

a:= proc(n, m) option remember; `if`(n<0, 0,
      `if`(m=0, combinat[numbpart](n), a(n-m, m-1) +a(n-m, m)))
    end:
seq(seq(a(n,m), m=0..round(sqrt(2*n+2))-1), n=0..20);  # Alois P. Heinz, Nov 16 2012

a[n_, 0] := PartitionsP[n]; a[n_, m_] /; (n >= m*(m+1)/2) := a[n, m] = a[n-m, m-1] + a[n-m, m]; a[n_, m_] = 0; Flatten[ Table[ a[n, m], {n, 0, 18}, {m, 0, Floor[1/2 + Sqrt[2*(n+1)]] - 1}]](* Jean-François Alcover, May 02 2012, after recurrence formula *)
DeleteCases[Flatten@Transpose@Table[ConstantArray[0, m (m + 1)/2]~Join~Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@m], {n, 0, 21 - m (m + 1)/2}] , {m, 0, 6}], 0](* Robert Price, Jul 28 2020 *)

Riordan gives formula.
a(n, m) is the sum over partitions of n of Product_{j=1..m} k(j), where k(j) is the 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). O is counted as a part for n=0 and only for this n.
a(n, m) is the sum over partitions of n of binomial(q(partition), m), with q the number of distinct parts of a given partition. m>=0.
a(n, m) = a(n-m, m-1) + a(n-m, m), n >= t(m):=m*(m+1)/2 = A000217(m) (triangular numbers), otherwise 0, with input a(n, 0) = p(n):=A000041(n).

More terms from Robert G Bearden (nem636(AT)myrealbox.com), Apr 27 2004

Correction, comments and Riordan formulas from Wolfdieter Lang, Apr 28 2005

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

Original entry on oeis.org

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.

A103929 Number of partitions of n into parts but with two kinds of parts of sizes 1 to 10.

Original entry on oeis.org

1, 2, 5, 10, 20, 36, 65, 110, 185, 300, 481, 751, 1162, 1762, 2647, 3918, 5748, 8331, 11981, 17056, 24108, 33787, 47043, 65019, 89336, 121954, 165585, 223542, 300295, 401331, 533937, 707057, 932404, 1224376, 1601571, 2086851, 2709449, 3505228

Offset: 0

Wolfdieter Lang, Mar 24 2005

See A103923 for other combinatorial interpretations of a(n).

In general, column m of A008951 is asymptotic to exp(Pi*sqrt(2*n/3)) * 6^(m/2) * n^((m-2)/2) / (4*sqrt(3) * m! * Pi^m), equivalently to 6^(m/2) * n^(m/2) / (m! * Pi^m) * p(n), where p(n) is the partition function A000041. - Vaclav Kotesovec, Aug 28 2015

H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958 (reprinted 1962), p. 91.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.

Eleventh column (m=10) of Fine-Riordan triangle A008951 and of triangle A103923, i.e. the p_2(n, m) array of the Gupta et al. reference.

Cf. A000712 (all parts of two kinds).

nmax=60; CoefficientList[Series[Product[1/(1-x^k), {k, 1, 10}] * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 28 2015 *)
Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@10], {n,0,37}] (* Robert Price, Jul 29 2020 *)
T[n_, 0] := PartitionsP[n];
T[n_, m_] /; (n >= m (m + 1)/2) := T[n, m] = T[n - m, m - 1] + T[n - m, m];
T[, ] = 0;
a[n_] := T[n + 55, 10];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 30 2021 *)

G.f.: (product(1/(1-x^k), k=1..10)^2)*product(1/(1-x^j), j=11..infty).
a(n)=sum(A103924(n-10*j), j=0..floor(n/10)), n>=0.
a(n) ~ exp(Pi*sqrt(2*n/3)) * 6^5 * n^4 / (4*sqrt(3) * 10! * Pi^10). - Vaclav Kotesovec, Aug 28 2015

A103925 Number of partitions of n into parts but with two kinds of parts of sizes 1,2,3,4,5 and 6.

Original entry on oeis.org

1, 2, 5, 10, 20, 36, 65, 109, 182, 292, 463, 714, 1091, 1631, 2416, 3523, 5091, 7264, 10284, 14405, 20035, 27621, 37831, 51425, 69497, 93299, 124588, 165408, 218533, 287231, 375851, 489525, 634980, 820195, 1055444, 1352965, 1728326, 2200060, 2791516, 3530513

Offset: 0

Wolfdieter Lang, Mar 24 2005

See A103923 for other combinatorial interpretations of a(n).

Also the sum of binomial (D(p), 6) over partitions p of n+21, where D(p) is the number of different part sizes in p. - Emily Anible, Jun 09 2018

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

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (first 1000 terms from Alois P. Heinz)

Seventh column (m=6) of Fine-Riordan triangle A008951, of triangle A103923, i.e. the p_2(n, m) array of the Gupta et al. reference.

Cf. A000712 (all parts of two kinds).

with(numtheory): a:= proc(n) option remember; `if`(n=0, 1, add(add(d*`if`(d<7, 2, 1), d=divisors(j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..40);  # Alois P. Heinz, Sep 14 2014

nmax=60; CoefficientList[Series[Product[1/(1-x^k), {k, 1, 6}] * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 28 2015 *)
Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@6],  {n,0,39}] (* Robert Price, Jul 29 2020 *)
T[n_, 0] := PartitionsP[n];
T[n_, m_] /; (n >= m(m+1)/2) := T[n, m] = T[n-m, m-1] + T[n-m, m];
T[, ] = 0;
a[n_] := T[n+21, 6];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 30 2021 *)

G.f.: (product(1/(1-x^k), k=1..6)^2)*product(1/(1-x^j), j=7..infty).
a(n) = sum(A103924(n-6*j), j=0..floor(n/6)), n>=0.
a(n) ~ exp(Pi*sqrt(2*n/3)) * 6^3 * n^2 / (4*sqrt(3) * 6! * Pi^6) = exp(Pi*sqrt(2*n/3)) * sqrt(3) * n^2 / (40*Pi^6). - Vaclav Kotesovec, Aug 28 2015

A103926 Number of partitions of n into parts but with two kinds of parts of sizes 1 to 7.

Original entry on oeis.org

1, 2, 5, 10, 20, 36, 65, 110, 184, 297, 473, 734, 1127, 1696, 2526, 3707, 5388, 7737, 11018, 15532, 21731, 30147, 41538, 56813, 77234, 104317, 140120, 187139, 248680, 328769, 432664, 566759, 739297, 960315, 1242583, 1601645, 2057095, 2632724

Offset: 0

Wolfdieter Lang, Mar 24 2005

See A103923 for other combinatorial interpretations of a(n).

Also the sum of binomial (D(p), 7) over partitions p of n+28, where D(p) is the number of different part sizes in p. - Emily Anible, Jun 09 2018

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

Eighth column (m=7) of Fine-Riordan triangle A008951 and of triangle A103923, i.e. the p_2(n, m) array of the Gupta et al. reference.

Cf. A000712 (all parts of two kinds).

nmax=60; CoefficientList[Series[Product[1/(1-x^k), {k, 1, 7}] * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 28 2015 *)
Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@7], {n,0,37}] (* Robert Price, Jul 29 2020 *)
T[n_, 0] := PartitionsP[n];
T[n_, m_] /; (n >= m (m + 1)/2) := T[n, m] = T[n - m, m - 1] + T[n - m, m];
T[, ] = 0;
a[n_] := T[n + 28, 7];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 30 2021 *)

G.f.: (product(1/(1-x^k), k=1..7)^2)*product(1/(1-x^j), j=8..infty).
a(n) = sum(A103924(n-7*j), j=0..floor(n/7)), n>=0.
a(n) ~ exp(Pi*sqrt(2*n/3)) * 6^(7/2) * n^(5/2) / (4*sqrt(3) * 7! * Pi^7). - Vaclav Kotesovec, Aug 28 2015

A103927 Number of partitions of n into parts but with two kinds of parts of sizes 1 to 8.

Original entry on oeis.org

1, 2, 5, 10, 20, 36, 65, 110, 185, 299, 478, 744, 1147, 1732, 2591, 3817, 5573, 8036, 11496, 16276, 22878, 31879, 44129, 60630, 82807, 112353, 151616, 203415, 271558, 360648, 476793, 627389, 822104, 1072668, 1394199, 1805060, 2328653, 2993372, 3835068, 4897199

Offset: 0

Wolfdieter Lang, Mar 24 2005

See A103923 for other combinatorial interpretations of a(n).

Also the sum of binomial (D(p), 8) over partitions p of n+36, where D(p) is the number of different part sizes in p. - Emily Anible, Jun 09 2018

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

Ninth column (m=8) of Fine-Riordan triangle A008951 and of triangle A103923, i.e., the p_2(n, m) array of the Gupta et al. reference.

Cf. A000712 (all parts of two kinds).

a:= proc(n) option remember; `if`(n=0, 1, add(add(d+
     `if`(d<9, d, 0), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 11 2018

nmax=60; CoefficientList[Series[Product[1/(1-x^k), {k, 1, 8}] * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 28 2015 *)
Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@8], {n,0,39}] (* Robert Price, Jul 29 2020 *)
T[n_, 0] := PartitionsP[n];
T[n_, m_] /; (n >= m (m + 1)/2) := T[n, m] = T[n - m, m - 1] + T[n - m, m];
T[, ] = 0;
a[n_] := T[n + 36, 8];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 30 2021 *)

G.f.: (Product_{k=1..8} 1/(1-x^k))^2*Product_{j>=9} 1/(1-x^j).
a(n) = Sum_{j=0..floor(n/8)} A103924(n-8*j), n >= 0.
a(n) ~ exp(Pi*sqrt(2*n/3)) * 6^4 * n^3 / (4*sqrt(3) * 8! * Pi^8). - Vaclav Kotesovec, Aug 28 2015

A103928 Number of partitions of n into parts but with two kinds of parts of sizes 1 to 9.

Original entry on oeis.org

1, 2, 5, 10, 20, 36, 65, 110, 185, 300, 480, 749, 1157, 1752, 2627, 3882, 5683, 8221, 11796, 16756, 23627, 33036, 45881, 63257, 86689, 118036, 159837, 215211, 288314, 384275, 509829, 673270, 885361, 1159357, 1512235, 1964897, 2543864, 3281686

Offset: 0

Wolfdieter Lang, Mar 24 2005

See A103923 for other combinatorial interpretations of a(n).

Also the sum of binomial (D(p), 9) over partitions p of n+45, where D(p) is the number of different part sizes in p. - Emily Anible, Jun 09 2018

H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958 (reprinted 1962), p. 91.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.

Tenth column (m=9) of Fine-Riordan triangle A008951 and of triangle A103923, i.e., the p_2(n, m) array of the Gupta et al. reference.

Cf. A000712 (all parts of two kinds).

nmax=60; CoefficientList[Series[Product[1/(1-x^k), {k, 1, 9}] * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 28 2015 *)
Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@9], {n,0,37}] (* Robert Price, Jul 29 2020 *)
T[n_, 0] := PartitionsP[n];
T[n_, m_] /; (n >= m (m + 1)/2) := T[n, m] = T[n - m, m - 1] + T[n - m, m];
T[, ] = 0;
a[n_] := T[n + 45, 9];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 30 2021 *)

G.f.: (Product_{k=1..9} 1/(1-x^k))^2 * Product_{j>=10} 1/(1-x^j).
a(n) = Sum_{j=0..floor(n/9)} A103924(n-9*j), n >= 0.
a(n) ~ exp(Pi*sqrt(2*n/3)) * 6^(9/2) * n^(7/2) / (4*sqrt(3) * 9! * Pi^9). - Vaclav Kotesovec, Aug 28 2015

Showing 1-8 of 8 results.

Suppose that p, with parts x(1) >= x(2) >= ... >= x(k), is a partition of n. Define AS(p), the alternating sum of p, by x(1) - x(2) + x(3) - ... + ((-1)^(k-1))*x(k); note that AS(p) has the same parity as n. Column 1 is given by T(n,1) = (number of partitions of 2n-1 having AS(p) = 1) = A000070(n) for n >= 1. Columns 2 and 3 are essentially A000098 and A103924, and the limiting column (after deleting initial 0's), A000712. The sum of numbers in row n is A000041(2n-1). The corresponding array for partitions into distinct parts is given by A152157 (defined as the number of partitions of 2n+1 into 2k+1 odd parts).

cp's OEIS Frontend

A239829 Triangular array: T(n,k) = number of partitions of 2n - 1 that have alternating sum 2k - 1.

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A008951 Array read by columns: number of partitions of n into parts of 2 kinds.

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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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A103929 Number of partitions of n into parts but with two kinds of parts of sizes 1 to 10.

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A103925 Number of partitions of n into parts but with two kinds of parts of sizes 1,2,3,4,5 and 6.

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A103926 Number of partitions of n into parts but with two kinds of parts of sizes 1 to 7.

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A103927 Number of partitions of n into parts but with two kinds of parts of sizes 1 to 8.

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