A008951 -id:A008951 - 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

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.

A100824 Number of partitions of n with at most one odd part.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 3, 7, 5, 12, 7, 19, 11, 30, 15, 45, 22, 67, 30, 97, 42, 139, 56, 195, 77, 272, 101, 373, 135, 508, 176, 684, 231, 915, 297, 1212, 385, 1597, 490, 2087, 627, 2714, 792, 3506, 1002, 4508, 1255, 5763, 1575, 7338, 1958, 9296, 2436, 11732, 3010, 14742

Offset: 0

Vladeta Jovovic, Jan 13 2005

From Gus Wiseman, Jan 21 2022: (Start)
Also the number of integer partitions of n with alternating sum <= 1, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. These are the conjugates of partitions with at most one odd part. For example, the a(1) = 1 through a(9) = 12 partitions with alternating sum <= 1 are:
  1  11  21   22    32     33      43       44        54
         111  1111  221    2211    331      2222      441
                    2111   111111  2221     3311      3222
                    11111          3211     221111    3321
                                   22111    11111111  4311
                                   211111             22221
                                   1111111            33111
                                                      222111
                                                      321111
                                                      2211111
                                                      21111111
                                                      111111111
(End)

			From _Gus Wiseman_, Jan 21 2022: (Start)
The a(1) = 1 through a(9) = 12 partitions with at most one odd part:
  (1)  (2)  (3)   (4)   (5)    (6)    (7)     (8)     (9)
            (21)  (22)  (32)   (42)   (43)    (44)    (54)
                        (41)   (222)  (52)    (62)    (63)
                        (221)         (61)    (422)   (72)
                                      (322)   (2222)  (81)
                                      (421)           (432)
                                      (2221)          (441)
                                                      (522)
                                                      (621)
                                                      (3222)
                                                      (4221)
                                                      (22221)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A008951, A000070, A000097, A000098, A000710.
The case of alternating sum 0 (equality) is A000070.
A multiplicative version is A339846.
These partitions are ranked by A349150, conjugate A349151.
A000041 = integer partitions, strict A000009.
A027187 = partitions of even length, strict A067661, ranked by A028260.
A027193 = partitions of odd length, ranked by A026424.
A058695 = partitions of odd numbers.
A103919 = partitions by sum and alternating sum (reverse: A344612).
A277103 = partitions with the same number of odd parts as their conjugate.
Cf. A000984, A001791, A008549, A097805, A119620, A182616, A236559, A236913, A236914, A304620, A344607, A345958, A347443.

seq(coeff(convert(series((1+x/(1-x^2))/mul(1-x^(2*i),i=1..100),x,100),polynom),x,n),n=0..60); (C. Ronaldo)

nmax = 50; CoefficientList[Series[(1+x/(1-x^2)) * Product[1/(1-x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 07 2016 *)
Table[Length[Select[IntegerPartitions[n],Count[#,?OddQ]<=1&]],{n,0,30}] (* _Gus Wiseman, Jan 21 2022 *)

a(n) = if(n%2==0, numbpart(n/2), sum(i=1, (n+1)\2, numbpart((n-2*i+1)\2))) \\ David A. Corneth, Jan 23 2022

G.f.: (1+x/(1-x^2))/Product(1-x^(2*i), i=1..infinity). More generally, g.f. for number of partitions of n with at most k odd parts is (1+Sum(x^i/Product(1-x^(2*j), j=1..i), i=1..k))/Product(1-x^(2*i), i=1..infinity).
a(n) ~ exp(sqrt(n/3)*Pi) / (2*sqrt(3)*n) if n is even and a(n) ~ exp(sqrt(n/3)*Pi) / (2*Pi*sqrt(n)) if n is odd. - Vaclav Kotesovec, Mar 07 2016
a(2*n) = A000041(n). a(2*n + 1) = A000070(n). - David A. Corneth, Jan 23 2022

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 19 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

A090730 a(n) = 22*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 22.

Original entry on oeis.org

2, 22, 482, 10582, 232322, 5100502, 111978722, 2458431382, 53973511682, 1184958825622, 26015120652002, 571147695518422, 12539234180753282, 275292004281053782, 6043884860002429922, 132690174915772404502

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 18 2004

Indranil Ghosh, Table of n, a(n) for n = 0..743
Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for linear recurrences with constant coefficients, signature (22,-1).

Cf. A008951, A016005, A001613, A077422.

a[0] = 2; a[1] = 22; a[n_] := 22a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* Robert G. Wilson v, Jan 30 2004 *)
LinearRecurrence[{22,-1},{2,22},20] (* Harvey P. Dale, Mar 07 2018 *)

[lucas_number2(n,22,1) for n in range(0,20)] # Zerinvary Lajos, Jun 26 2008

a(n) = p^n + q^n, where p = 11 + 2*sqrt(30) and q = 11 - 2*sqrt(30). - Tanya Khovanova, Feb 06 2007
G.f.: (2-22*x)/(1-22*x+x^2). - Philippe Deléham, Nov 18 2008
a(n) = 2*A077422(n). - R. J. Mathar, Sep 27 2014

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

A100835 Number of partitions of n with at most 2 odd parts.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 8, 7, 14, 12, 24, 19, 39, 30, 62, 45, 95, 67, 144, 97, 212, 139, 309, 195, 442, 272, 626, 373, 873, 508, 1209, 684, 1653, 915, 2245, 1212, 3019, 1597, 4035, 2087, 5348, 2714, 7051, 3506, 9229, 4508, 12022, 5763, 15565, 7338, 20063, 9296, 25722

Offset: 0

Vladeta Jovovic, Jan 13 2005

			a(5) = 4 because we have [5], [4,1], [3,2] and [2,2,1] (the partitions [3,1,1], [2,1,1,1] and [1,1,1,1,1] do not qualify).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000070, A008951, A000097, A000098, A000710.

g:=(1+x/(1-x^2)+x^2/(1-x^2)/(1-x^4))/product(1-x^(2*i), i=1..40): gser:=series(g, x, 60): seq(coeff(gser, x, n), n=0..55); # Emeric Deutsch, Feb 16 2006

nmax = 50; CoefficientList[Series[(1+x/(1-x^2)+x^2/(1-x^2)/(1-x^4)) * Product[1/(1-x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 07 2016 *)

G.f.: (1+x/(1-x^2)+x^2/(1-x^2)/(1-x^4))/Product(1-x^(2*i), i=1..infinity). More generally, g.f. for number of partitions of n with at most k odd parts is (1+Sum(x^i/Product(1-x^(2*j), j=1..i), i=1..k))/Product(1-x^(2*i), i=1..infinity).

More terms from Emeric Deutsch, Feb 16 2006

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

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

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A100824 Number of partitions of n with at most one odd part.

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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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