A087787 -id:A087787 - 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

A182712 Number of 2's in the last section of the set of partitions of n.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 4, 3, 8, 7, 15, 15, 27, 29, 48, 53, 82, 94, 137, 160, 225, 265, 362, 430, 572, 683, 892, 1066, 1370, 1640, 2078, 2487, 3117, 3725, 4624, 5519, 6791, 8092, 9885, 11752, 14263, 16922, 20416, 24167, 29007, 34254, 40921, 48213, 57345, 67409

Offset: 0

Omar E. Pol, Nov 28 2010

Essentially the same as A087787 but here a(n) is the number of 2's in the last section of n, not n-2. See also A100818.
Note that a(1)..a(11) coincide with a(2)..a(12) of A005291.
Also number of 2's in all partitions of n that do not contain 1's as a part, if n >= 1. Also 0 together with the first differences of A024786. - Omar E. Pol, Nov 13 2011
Also number of 2's in the n-th section of the set of partitions of any integer >= n. For the definition of "section" see A135010. - Omar E. Pol, Dec 01 2013

			a(6) = 4 counts the 2's in 6 = 4+2 = 2+2+2. The 2's in 6 = 3+2+1 = 2+2+1+1 = 2+1+1+1+1 do not count. - _Omar E. Pol_, Nov 13 2011
From _Omar E. Pol_, Oct 27 2012: (Start)
----------------------------------
Last section               Number
of the set of                of
partitions of 6             2's
----------------------------------
6 .......................... 0
3 + 3 ...................... 0
4 + 2 ...................... 1
2 + 2 + 2 .................. 3
.   1 ...................... 0
.       1 .................. 0
.       1 .................. 0
.           1 .............. 0
.           1 .............. 0
.               1 .......... 0
.                   1 ...... 0
---------------------------------
.   8 - 4 =                  4
.
In the last section of the set of partitions of 6 the difference between the sum of the second column and the sum of the third column is 8 - 4 = 4, the same as the number of 2's, so a(6) = 4 (see also A024786).
(End)

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

Column 2 of A194812.

Cf. A005291, A087787, A100818, A135010, A138121, A182703, A182713, A182714.

Table[Count[Flatten@Cases[IntegerPartitions[n], x_ /; Last[x] != 1], 2], {n, 0, 49}] (* Robert Price, May 15 2020 *)

A182712 = lambda n: sum(list(p).count(2) for p in Partitions(n) if 1 not in p) # Omar E. Pol, Nov 13 2011

It appears that A000041(n) = a(n+1) + a(n+2), n >= 0. - Omar E. Pol, Feb 04 2012
G.f.: (x^2/(1 + x))*Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Jan 03 2017
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n). - Vaclav Kotesovec, Jun 02 2018

A372688 Number of integer partitions y of n whose rank Sum_i 2^(y_i-1) is prime.

Original entry on oeis.org

0, 0, 2, 2, 1, 3, 3, 6, 3, 6, 9, 20, 13, 22, 22, 45, 47, 70, 75, 100, 107, 132, 157, 202, 229, 302, 396, 495, 536, 699, 820, 962, 1193, 1507, 1699, 2064, 2455, 2945, 3408, 4026, 4691, 5749, 6670, 7614, 9127, 10930, 12329, 14370, 16955, 19961, 22950, 26574, 30941

Offset: 0

Gus Wiseman, May 16 2024

nonn

Note the function taking a set s to Sum_i 2^(s_i-1) is the inverse of A048793 (binary indices).

			The partition (3,2,1) has rank 2^(3-1) + 2^(2-1) + 2^(1-1) = 7, which is prime, so (3,2,1) is counted under a(6).
The a(2) = 2 through a(10) = 9 partitions:
(2)   (21)   (31)  (221)    (51)    (421)      (431)   (441)     (91)
(11)  (111)        (2111)   (321)   (2221)     (521)   (3321)    (631)
                   (11111)  (3111)  (4111)     (5111)  (4221)    (721)
                                    (22111)            (33111)   (3331)
                                    (211111)           (42111)   (7111)
                                    (1111111)          (411111)  (32221)
                                                                 (322111)
                                                                 (3211111)
                                                                 (31111111)

For all positive integers (not just prime) we get A000041.
For even instead of prime we have A087787, strict A025147, odd A096765.
These partitions have Heinz numbers A277319.
The strict case is A372687, ranks A372851.
The version counting only distinct parts is A372887, ranks A372850.
A014499 lists binary indices of prime numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A048793 and A272020 (reverse) list binary indices:
- length A000120
- min A001511
- sum A029931
- max A070939
A058698 counts partitions of prime numbers, strict A064688.
A372885 lists primes whose binary indices sum to a prime, indices A372886.
Cf. A000040, A005940, A023506, A029837, A035100, A038499, A096111, A372429, A372441, A372471, A372689.

Table[Length[Select[IntegerPartitions[n], PrimeQ[Total[2^#]/2]&]],{n,0,30}]

A103628 Total sum of parts of multiplicity 1 in all partitions of n.

Original entry on oeis.org

0, 1, 2, 6, 10, 21, 33, 59, 89, 145, 212, 325, 463, 680, 948, 1348, 1845, 2558, 3446, 4681, 6219, 8306, 10901, 14352, 18632, 24230, 31151, 40077, 51074, 65088, 82290, 103986, 130517, 163679, 204078, 254174, 314975, 389839, 480369, 591133, 724600, 886965

Offset: 0

Vladeta Jovovic, Mar 25 2005

Total number of parts of multiplicity 1 in all partitions of n is A024786(n+1).

Equals A000041 convolved with A026741. - Gary W. Adamson, Jun 11 2009

			Partitions of 4 are [1,1,1,1], [1,1,2], [2,2], [1,3], [4] and a(4) = 0 + 2 + 0 + (1+3) + 4 = 10.

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

Cf. A026741. - Gary W. Adamson, Jun 11 2009

Column k=1 of A222730. - Alois P. Heinz, Mar 03 2013

gf:=x*(1+x+x^2)/(1-x^2)^2/product((1-x^k), k=1..500): s:=series(gf, x, 100): for n from 0 to 60 do printf(`%d,`,coeff(s, x, n)) od: # James Sellers, Apr 22 2005
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, [0, 0], add((l->`if`(j=1, [l[1],
       l[2]+l[1]*i], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 03 2013

b[n_, p_] := b[n, p] = If[n == 0 && p == 0, {1, 0}, If[p == 0, Array[0&, n+2], Sum[Function[l, ReplacePart[l, m+2 -> p*l[[1]] + l[[m+2]]]][Join[b[n-p*m, p-1], Array[0&, p*m]]], {m, 0, n/p}]]]; a[n_] := b[n, n][[3]]; a[0] = 0; Table[a[n], {n, 0, 50}]  (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)

G.f.: x*(1+x+x^2)/(1-x^2)^2 /Product_{k>0}(1-x^k).
a(n) = A066186(n) - A194544(n). - Omar E. Pol, Nov 20 2011
a(n) = 3*A014153(n)/4 - 3*A000070(n)/4 - A270143(n+1)/4 + A087787(n)/4. - Vaclav Kotesovec, Nov 05 2016
a(n) ~ 3^(3/2) * exp(Pi*sqrt(2*n/3)) / (8*Pi^2) * (1 - Pi/(24*sqrt(6*n))). - Vaclav Kotesovec, Nov 05 2016

More terms from James Sellers, Apr 22 2005

A100818 For a given unrestricted partition pi, let P(pi)=lambda(pi), if mu(pi)=0. If mu(pi)>0 then let P(pi)=nu(pi), where nu(pi) is the number of parts of pi greater than mu(pi), mu(pi) is the number of ones in pi and lambda(pi) is the largest part of pi.

Original entry on oeis.org

1, 2, 1, 4, 3, 8, 7, 15, 15, 27, 29, 48, 53, 82, 94, 137, 160, 225, 265, 362, 430, 572, 683, 892, 1066, 1370, 1640, 2078, 2487, 3117, 3725, 4624, 5519, 6791, 8092, 9885, 11752, 14263, 16922, 20416, 24167, 29007, 34254, 40921, 48213, 57345, 67409, 79864

Offset: 1

David S. Newman, Jan 13 2005

nonn

Note that this is very similar to the "crank" of Andrews and Garvan. The number of partitions pi with P(pi) odd is the given sequence.
The sequence is the same as A087787 except for the value of a(1) (this was established by George Andrews, Jan 18 2005). If "even" is replace by "odd" in the definition of the sequence, the new sequence is almost identical except for two values and a shift to the right.
Also, positive integers of A182712. a(n) is also the number of 2's in the n-th row that contain a 2 as a part in the triangle of A138121 (note that rows 1 and 3 do not contain a 2 as a part). - Omar E. Pol, Nov 28 2010

			a(3)=1 because P(3)=3, P(2 1)=1 and P(1 1 1)=0.

G. C. Greubel, Table of n, a(n) for n = 1..1000
G. E. Andrews and F. Garvan, Dyson's crank of a partition, Bull. Amer. Math. Soc., 18 (1988), 167-171.

Cf. A000041, A087787, A135010, A138121, A182712.

Rest[ CoefficientList[ Series[x + 1/(1 + x) Product[1/(1 - x^n), {n, 50}], {x, 0, 50}], x]] (* Robert G. Wilson v, Feb 11 2005 *)

G.f.: x+(1/(1+x))* Product_{n>=1}(1/(1-x^n)). [corrected by Vaclav Kotesovec, Aug 29 2019]
a(n) = A000041(n) - a(n-1), for n>2. - Jon Maiga, Aug 29 2019 [corrected by Vaclav Kotesovec, Aug 29 2019]
a(n) = a(n-2) + A000041(n-1) - A000041(n-2), for n>=3. - Vaclav Kotesovec, Aug 29 2019

More terms from Robert G. Wilson v, Feb 11 2005

A372687 Number of prime numbers whose binary indices sum to n. Number of strict integer partitions y of n such that Sum_i 2^(y_i-1) is prime.

Original entry on oeis.org

0, 0, 1, 1, 1, 0, 2, 1, 2, 0, 3, 3, 1, 4, 1, 6, 5, 8, 4, 12, 8, 12, 7, 20, 8, 16, 17, 27, 19, 38, 19, 46, 33, 38, 49, 65, 47, 67, 83, 92, 94, 113, 103, 130, 146, 127, 215, 224, 176, 234, 306, 270, 357, 383, 339, 393, 537, 540, 597, 683, 576, 798, 1026, 830, 1157

Offset: 0

Gus Wiseman, May 15 2024

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

Note the inverse of A048793 (binary indices) takes a set s to Sum_i 2^(s_i-1).

			The a(2) = 1 through a(17) = 8 prime numbers:
  2  3  5  .  17  11  19  .  257  131  73  137  97  521  4099  1031
              7       13     67   41       71       263  2053  523
                             37   23       43       139  1033  269
                                           29       83   193   163
                                                    53   47    149
                                                    31         101
                                                               89
                                                               79
The a(2) = 1 through a(11) = 3 strict partitions:
  (2)  (2,1)  (3,1)  .  (5,1)    (4,2,1)  (4,3,1)  .  (9,1)    (6,4,1)
                        (3,2,1)           (5,2,1)     (6,3,1)  (8,2,1)
                                                      (7,2,1)  (5,3,2,1)

For all positive integers (not just prime) we get A000009.
Number of prime numbers p with A029931(p) = n.
For odd instead of prime we have A096765, even A025147, non-strict A087787
Number of times n appears in A372429.
Number of rows of A372471 with sum n.
The non-strict version is A372688 (or A372887), ranks A277319 (or A372850).
These (strict) partitions have Heinz numbers A372851.
A014499 lists binary indices of prime numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A038499 counts partitions of prime length, strict A085756.
A048793 lists binary indices:
- length A000120
- min A001511
- sum A029931
- max A070939
- reverse A272020
A058698 counts partitions of prime numbers, strict A064688.
A096111 gives product of binary indices.
A372689 lists numbers whose binary indices sum to a prime.
A372885 lists primes whose binary indices sum to a prime, indices A372886.
Cf. A000040, A005940, A023506, A029837, A035100, A071814, A230877, A231204, A359359, A372436, A372441.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&PrimeQ[Total[2^#]/2]&]],{n,0,30}]

A092295 Number of partitions of n with even number (or 0) 2's.

Original entry on oeis.org

1, 1, 1, 2, 4, 5, 7, 10, 15, 20, 27, 36, 50, 65, 85, 111, 146, 186, 239, 304, 388, 488, 614, 767, 961, 1191, 1475, 1819, 2243, 2746, 3361, 4096, 4988, 6047, 7322, 8836, 10655, 12801, 15360, 18384, 21978, 26199, 31196, 37062, 43979, 52072, 61579, 72682

Offset: 0

Vladeta Jovovic, Feb 06 2004

			a(5)=5 because the partitions [5],[4,1],[3,1,1],[2,2,1] and [1,1,1,1,1] of 5 have an even number of 2's ([3,2] and [2,1,1,1] do not qualify).

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A087787.

g:=1/(1+x^2)/product(1-x^j,j=1..70): gser:=series(g,x=0,50): seq(coeff(gser,x,n),n=0..47); # Emeric Deutsch, Mar 30 2006

nmax = 50; CoefficientList[Series[1/((1+x^2) * Product[1-x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 30 2015 *)

a(n) = A000041(n)-a(n-2).
G.f.=1/[(1+x^2)*product(1-x^j, j=1..infinity)]. - Emeric Deutsch, Mar 30 2006
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)). - Vaclav Kotesovec, Oct 30 2015

More terms from Benoit Cloitre, Feb 08 2004

A220504 Triangle read by rows: T(n,k) is the total number of appearances of k as the smallest part in all partitions of n.

Original entry on oeis.org

1, 2, 1, 4, 0, 1, 7, 2, 0, 1, 12, 1, 0, 0, 1, 19, 4, 2, 0, 0, 1, 30, 3, 1, 0, 0, 0, 1, 45, 8, 1, 2, 0, 0, 0, 1, 67, 7, 4, 1, 0, 0, 0, 0, 1, 97, 15, 3, 1, 2, 0, 0, 0, 0, 1, 139, 15, 4, 1, 1, 0, 0, 0, 0, 0, 1, 195, 27, 8, 4, 1, 2, 0, 0, 0, 0, 0, 1, 272, 29, 8, 3, 1, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Omar E. Pol, Jan 19 2013

In other words, T(n,k) is the total number of appearances of k in all partitions of n whose smallest part is k.
The sum of row n equals spt(n), the smallest part partition function (see A092269).
T(n,k) is also the sum of row k in the slice n of tetrahedron A209314.

			Triangle begins:
    1;
    2,  1;
    4,  0, 1;
    7,  2, 0, 1;
   12,  1, 0, 0, 1;
   19,  4, 2, 0, 0, 1;
   30,  3, 1, 0, 0, 0, 1;
   45,  8, 1, 2, 0, 0, 0, 1;
   67,  7, 4, 1, 0, 0, 0, 0, 1;
   97, 15, 3, 1, 2, 0, 0, 0, 0, 1;
  139, 15, 4, 1, 1, 0, 0, 0, 0, 0, 1;
  195, 27, 8, 4, 1, 2, 0, 0, 0, 0, 0, 1;
  272, 29, 8, 3, 1, 1, 0, 0, 0, 0, 0, 0, 1;
  ...
The partitions of 6 with the smallest part in brackets are
..........................
.                      [6]
..........................
.                  [3]+[3]
..........................
.                   4 +[2]
.              [2]+[2]+[2]
..........................
.                   5 +[1]
.               3 + 2 +[1]
.               4 +[1]+[1]
.           2 + 2 +[1]+[1]
.           3 +[1]+[1]+[1]
.       2 +[1]+[1]+[1]+[1]
.  [1]+[1]+[1]+[1]+[1]+[1]
..........................
There are 19 smallest parts of size 1. Also there are four smallest parts of size 2. Also there are two smallest parts of size 3. There are no smallest part of size 4 or 5. Finally there is only one smallest part of size 6. So row 6 gives 19, 4, 2, 0, 0, 1. The sum of row 6 is 19+4+2+0+0+1 = A092269(6) = 26.

Alois P. Heinz, Rows n = 1..141, flattened

Columns 1-3: A000070, A087787, A174455.
Row sums give A092269.
Cf. A026794, A182703, A209314.

b:= proc(n, i) option remember; local j, r; if n=0 or i<1 then 0
      else `if`(irem(n, i, 'r')=0, [0$(i-1), r], []); for j from 0
      to n/i do zip((x, y)->x+y, %, [b(n-i*j, i-1)], 0) od; %[] fi
    end:
T:= n-> b(n, n):
seq(T(n), n=1..20);  # Alois P. Heinz, Jan 20 2013

b[n_, i_] := b[n, i] = Module[{j, q, r, pc}, If [n == 0 || i<1, 0, {q, r} = QuotientRemainder[n, i]; pc = If[r == 0, Append[Array[0&, i-1], q], {}]; For[j = 0, j <= n/i, j++, pc = Plus @@ PadRight[{pc, b[n-i*j, i-1]}]]; pc]]; T[n_] := b[n, n]; Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Jan 30 2014, after Alois P. Heinz *)

A265257 Number of odd singletons in all partitions of n (n>=0).

Original entry on oeis.org

0, 1, 0, 2, 2, 5, 5, 11, 13, 23, 28, 45, 57, 86, 108, 156, 199, 276, 350, 475, 601, 798, 1005, 1312, 1646, 2120, 2643, 3365, 4178, 5264, 6500, 8122, 9981, 12375, 15136, 18638, 22697, 27779, 33679, 40993, 49504, 59947, 72109

Offset: 0

Emeric Deutsch, Jan 01 2016

nonn

			a(6) = 5 because in [1,1,1,3], [1,2,3], [1,5] we have 1+2+2 odd singletons, while the other 8 partitions of 6 have no odd singletons.

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

Cf. A265255.

g := x*(1-x+x^2)/((1-x^4)*mul(1-x^j, j = 1 .. 80)): gser := series(g, x = 0, 55): seq(coeff(gser, x, m), m = 0 .. 50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, 0, add((p-> `if`(j=1 and i::odd, p+
      [0, p[1]], p))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..80);  # Alois P. Heinz, Jan 01 2016

nmax = 50; CoefficientList[Series[x*(1-x+x^2)/(1-x^4) * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 01 2016 *)

a(n) = Sum(k*A265255(n,k), k>=0).
G.f.: g(x) = x(1 - x + x^2)/((1-x^4)*Product_{j>=1}(1-x^j)).
From Vaclav Kotesovec, Jan 01 2016: (Start)
a(n) = 1/4 * A000070(n) - 3/4 * A087787(n) + 1/2 * A092295(n).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*Pi*sqrt(2*n)).
(End)

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A182712 Number of 2's in the last section of the set of partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

A372688 Number of integer partitions y of n whose rank Sum_i 2^(y_i-1) is prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A103628 Total sum of parts of multiplicity 1 in all partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A100818 For a given unrestricted partition pi, let P(pi)=lambda(pi), if mu(pi)=0. If mu(pi)>0 then let P(pi)=nu(pi), where nu(pi) is the number of parts of pi greater than mu(pi), mu(pi) is the number of ones in pi and lambda(pi) is the largest part of pi.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A372687 Number of prime numbers whose binary indices sum to n. Number of strict integer partitions y of n such that Sum_i 2^(y_i-1) is prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A092295 Number of partitions of n with even number (or 0) 2's.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple