A103923 -id:A103923 - cp's OEIS frontend

A000097 Number of partitions of n if there are two kinds of 1's and two kinds of 2's.

1, 2, 5, 9, 17, 28, 47, 73, 114, 170, 253, 365, 525, 738, 1033, 1422, 1948, 2634, 3545, 4721, 6259, 8227, 10767, 13990, 18105, 23286, 29837, 38028, 48297, 61053, 76926, 96524, 120746, 150487, 187019, 231643, 286152, 352413, 432937, 530383, 648245

Offset: 0

N. J. A. Sloane

Also number of partitions of 2*n with exactly 2 odd parts (offset 1). - Vladeta Jovovic, Jan 12 2005
Also number of transitions from one partition of n+2 to another, where a transition consists of replacing any two parts with their sum. Remove all 1' and 2' from the partition, replacing them with ((number of 2') + 1) and ((number of 1') + (number of 2') + 1); these are the two parts being summed. Number of partitions of n into parts of 2 kinds with at most 2 parts of the second kind, or of n+2 into parts of 2 kinds with exactly 2 parts of the second kind. - Franklin T. Adams-Watters, Mar 20 2006
From Christian Gutschwager (gutschwager(AT)math.uni-hannover.de), Feb 10 2010: (Start)
a(n) is also the number of pairs of partitions of n+2 which differ by only one box (for bijection see the first Gutschwager link).
a(n) is also the number of partitions of n with two parts marked.
a(n) is also the number of partitions of n+1 with two different parts marked. (End)
Convolution of A000041 and A008619. - Vaclav Kotesovec, Aug 18 2015
a(n) = P(/2,n), a particular case of P(/k,n) defined as follows: P(/0,n) = A000041(n) and P(/k,n) = P(/k-1, n) + P(/k-1,n-k) + P(/k-1, n-2k) + ... Also, P(/k,n) = the convolution of A000041 and the partitions of n with exactly k parts, and g.f. P(/k,n) = (g.f. for P(n)) * 1/(1-x)...(1-x^k). - Gregory L. Simay, Mar 22 2018
a(n) is also the sum of binomial(D(p),2) in partitions p of (n+3), where D(p)= number of different sizes of parts in p. - Emily Anible, Apr 03 2018
Also partitions of 2*(n+1) with alternating sum 2. Also partitions of 2*(n+1) with reverse-alternating sum -2 or 2. - Gus Wiseman, Jun 21 2021
Define the distance graph of the partitions of n using the distance function in A366156 as follows: two vertices (partitions) share an edge if and only if the distance between the vertices is 2. Then a(n) is the number of edges in the distance graph of the partitions of n. - Clark Kimberling, Oct 12 2023

			a(3) = 9 because we have 3, 2+1, 2+1', 2'+1, 2'+1', 1+1+1, 1+1+1', 1+1'+1' and 1'+1'+1'.
From _Gus Wiseman_, Jun 22 2021: (Start)
The a(0) = 1 through a(4) = 9 partitions of 2*(n+1) with exactly 2 odd parts:
  (1,1)  (3,1)    (3,3)      (5,3)
         (2,1,1)  (5,1)      (7,1)
                  (3,2,1)    (3,3,2)
                  (4,1,1)    (4,3,1)
                  (2,2,1,1)  (5,2,1)
                             (6,1,1)
                             (3,2,2,1)
                             (4,2,1,1)
                             (2,2,2,1,1)
The a(0) = 1 through a(4) = 9 partitions of 2*(n+1) with alternating sum 2:
  (2)  (3,1)    (4,2)        (5,3)
       (2,1,1)  (2,2,2)      (3,3,2)
                (3,2,1)      (4,3,1)
                (3,1,1,1)    (3,2,2,1)
                (2,1,1,1,1)  (4,2,1,1)
                             (2,2,2,1,1)
                             (3,2,1,1,1)
                             (3,1,1,1,1,1)
                             (2,1,1,1,1,1,1)
(End)

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.
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).

T. D. Noe and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Christian Gutschwager, The skew diagram poset and components of skew characters, arXiv:1104.0008 [math.CO], 2011.
Christian Gutschwager, Reduced Kronecker products which are multiplicity free or contain only a few components, Eur. J. Combinat. 31 (2010) 1996-2005. doi:10.1016/j.ejc.2010.05.008.
J. P. Robinson, Edges in the poset of partitions of an integer, J. Combin. Theory Ser. A, 48 (1988), 236-238.
N. J. A. Sloane, Transforms

First differences are in A024786.
Cf. A000098, A000710.
Third column of Riordan triangle A008951 and of triangle A103923.
The case of reverse-alternating sum 1 or alternating sum 0 is A000041.
The case of reverse-alternating sum -1 or alternating sum 1 is A000070.
The normal case appears to be A004526 or A065033.
The strict case is A096914.
The case of reverse-alternating sum 2 is A120452.
The case of reverse-alternating sum -2 is A344741.
A001700 counts compositions with alternating sum 2.
A035363 counts partitions into even parts.
A058696 counts partitions of 2n.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A124754 gives alternating sums of standard compositions (reverse: A344618).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
Cf. A006330, A027187, A239830, A306145, A343941, A344607, A344608, A344619, A344650, A344651, A344740.
Shift of A093695.

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n->`if`(n<3,2,1)): seq(a(n), n=0..40); # Alois P. Heinz, Sep 08 2008

CoefficientList[Series[1/((1 - x) (1 - x^2) Product[1 - x^k, {k, 1, 100}]), {x, 0, 100}], x] (* Ben Branman, Mar 07 2012 *)
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b]; a = etr[If[# < 3, 2, 1]&]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 09 2014, after Alois P. Heinz *)
(1/((1 - x) (1 - x^2) QPochhammer[x]) + O[x]^50)[[3]] (* Vladimir Reshetnikov, Nov 22 2016 *)
Table[Length@IntegerPartitions[n,All,Join[{1,2},Range[n]]],{n,0,15}] (* Robert Price, Jul 28 2020 and Jun 21 2021 *)
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 + 3, 2];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 30 2021 *)
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];Table[Length[Select[IntegerPartitions[n],ats[#]==2&]],{n,0,30,2}] (* Gus Wiseman, Jun 21 2021 *)

my(x = 'x + O('x^66)); Vec( 1/((1-x)*(1-x^2)*eta(x)) ) \\ Joerg Arndt, Apr 29 2013

Euler transform of 2 2 1 1 1 1 1...
G.f.: 1/( (1-x) * (1-x^2) * Product_{k>=1} (1-x^k) ).
a(n) = Sum_{j=0..floor(n/2)} A000070(n-2*j), n>=0.
a(n) = A014153(n)/2 + A087787(n)/4 + A000070(n)/4. - Vaclav Kotesovec, Nov 05 2016
a(n) ~ sqrt(3) * exp(Pi*sqrt(2*n/3)) / (4*Pi^2) * (1 + 35*Pi/(24*sqrt(6*n))). - Vaclav Kotesovec, Aug 18 2015, extended Nov 05 2016
a(n) = A120452(n) + A344741(n). - Gus Wiseman, Jun 21 2021

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 04 2004
Edited by Emeric Deutsch, Mar 23 2005
More terms from Franklin T. Adams-Watters, Mar 20 2006
Edited by Charles R Greathouse IV, Apr 20 2010

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

Original entry on oeis.org

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

A000098 Number of partitions of n if there are two kinds of 1, two kinds of 2 and two kinds of 3.

Original entry on oeis.org

1, 2, 5, 10, 19, 33, 57, 92, 147, 227, 345, 512, 752, 1083, 1545, 2174, 3031, 4179, 5719, 7752, 10438, 13946, 18519, 24428, 32051, 41805, 54265, 70079, 90102, 115318, 147005, 186626, 236064, 297492, 373645, 467707

Offset: 0

N. J. A. Sloane

Also number of partitions of 2*n+1 with exactly 3 odd parts (offset 1). - Vladeta Jovovic, Jan 12 2005
Convolution of A000041 and A001399. - Vaclav Kotesovec, Aug 18 2015
Also the sum of binomial(D(p),3) over partitions p of n+6, where D(p) is the number of different sizes of parts in p. - Emily Anible, May 13 2018

			a(3)=10 because we have 3, 3', 2+1, 2+1', 2'+1, 2'+1', 1+1+1, 1+1+1', 1+1'+1' and 1'+1'+1'.

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.
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).

T. D. Noe, Table of n, a(n) for n = 0..1000
Álvaro Gutiérrez and Mercedes H. Rosas, Partial symmetries of iterated plethysms, arXiv:2201.00240 [math.CO], 2022.
N. J. A. Sloane, Transforms

Cf. A000070, A008951, A000097, A000710.
Cf. A000041, A001399.
Fourth column of Riordan triangle A008951 and of triangle A103923.

CoefficientList[1/((1-x)*(1-x^2)*(1-x^3)*QPochhammer[x]) + O[x]^40, x] (* Jean-François Alcover, Feb 04 2016 *)
Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@3], {n,0,35}] (* Robert Price, Jul 28 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 + 6, 3];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 30 2021 *)

Euler transform of 2 2 2 1 1 1 1...
G.f.: 1/((1-x)(1-x^2)(1-x^3)*Product_{k>=1} (1-x^k)).
a(n) = Sum_{j=0..floor(n/3)} A000097(n-3*j), n >= 0.
a(n) ~ sqrt(n) * exp(Pi*sqrt(2*n/3)) / (2*sqrt(2)*Pi^3). - Vaclav Kotesovec, Aug 18 2015

Edited by Emeric Deutsch, Mar 23 2005

A000710 Number of partitions of n, with two kinds of 1, 2, 3 and 4.

Original entry on oeis.org

1, 2, 5, 10, 20, 35, 62, 102, 167, 262, 407, 614, 919, 1345, 1952, 2788, 3950, 5524, 7671, 10540, 14388, 19470, 26190, 34968, 46439, 61275, 80455, 105047, 136541, 176593, 227460, 291673, 372605, 474085, 601105, 759380, 956249, 1200143, 1501749, 1873407

Offset: 0

N. J. A. Sloane

Also number of partitions of 2*n+4 with exactly 4 odd parts. - Vladeta Jovovic, Jan 12 2005
Convolution of A000041 and A001400. - Vaclav Kotesovec, Aug 18 2015
Also the sum of binomial (D(p), 4) over partitions p of n+10, where D(p) is the number of different part sizes in p. - Emily Anible, Jun 09 2018

			a(2) = 5 because we have 2, 2', 1+1, 1+1', 1'+1'.

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.
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).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Cf. A000712.
Cf. A000070, A008951, A000097, A000098.
Fifth column of Riordan triangle A008951 and of triangle A103923.

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> `if`(n<5,2,1)): seq(a(n), n=0..40); # Alois P. Heinz, Sep 08 2008

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]; b]; a = etr[If[#<5, 2, 1]&]; Table[a[n], {n, 0, 39}] (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)
nmax = 50; CoefficientList[Series[1/((1-x)(1-x^2)(1-x^3)(1-x^4))*Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 18 2015 *)
Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@4], {n,0,39}] (* Robert Price, Jul 28 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 + 10, 4];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 30 2021 *)

Euler transform of 2 2 2 2 1 1 1...
G.f.: 1/((1-x)(1-x^2)(1-x^3)(1-x^4)*Product_{k>=1} (1-x^k)).
a(n) = Sum_{j=0..floor(n/4)} A000098(n-4*j), n >= 0.
a(n) ~ sqrt(3)*n * exp(Pi*sqrt(2*n/3)) / (8*Pi^4). - Vaclav Kotesovec, Aug 18 2015

Edited by Emeric Deutsch, Mar 22 2005

A103924 Number of partitions of n into parts but with two kinds of parts of sizes 1,2,3,4 and 5.

Original entry on oeis.org

1, 2, 5, 10, 20, 36, 64, 107, 177, 282, 443, 678, 1026, 1522, 2234, 3231, 4628, 6550, 9193, 12774, 17619, 24098, 32740, 44161, 59213, 78894, 104553, 137787, 180702, 235806, 306354, 396226, 510392, 654787, 836911, 1065734, 1352475, 1710535, 2156536, 2710318

Offset: 0

Wolfdieter Lang, Mar 24 2005

See A103923 for other combinatorial interpretations of a(n).
Convolution of A001401 and A000041. - Vaclav Kotesovec, Aug 28 2015
Also the sum of binomial (D(p), 5) over partitions p of n+15, 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, Table of n, a(n) for n = 0..10000

Sixth column (m=5) 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).

with(numtheory): a:= proc(n) a(n):=`if`(n=0, 1, add(add(d*`if`(d<6, 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

a[n_] := a[n] = If[n==0, 1, Sum[Sum[d*If[d<6, 2, 1], {d, Divisors[j]}] * a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Aug 28 2015, after Alois P. Heinz *)
nmax=60; CoefficientList[Series[Product[1/(1-x^k), {k, 1, 5}] * 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@5],  {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+15, 5];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 30 2021 *)

G.f.: (product(1/(1-x^k), k=1..5)^2)*product(1/(1-x^j), j=6..infty).
a(n) = sum(A000710(n-5*j), j=0..floor(n/5)), n>=0.
a(n) ~ 3*n^(3/2) * exp(Pi*sqrt(2*n/3)) / (20*sqrt(2)*Pi^5). - Vaclav Kotesovec, Aug 28 2015

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

cp's OEIS Frontend

A000097 Number of partitions of n if there are two kinds of 1's and two kinds of 2's.

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

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A000098 Number of partitions of n if there are two kinds of 1, two kinds of 2 and two kinds of 3.

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A000710 Number of partitions of n, with two kinds of 1, 2, 3 and 4.

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A103924 Number of partitions of n into parts but with two kinds of parts of sizes 1,2,3,4 and 5.

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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.