A195820 -id:A195820 - cp's OEIS frontend

A000070 a(n) = Sum_{k=0..n} p(k) where p(k) = number of partitions of k (A000041).

1, 2, 4, 7, 12, 19, 30, 45, 67, 97, 139, 195, 272, 373, 508, 684, 915, 1212, 1597, 2087, 2714, 3506, 4508, 5763, 7338, 9296, 11732, 14742, 18460, 23025, 28629, 35471, 43820, 53963, 66273, 81156, 99133, 120770, 146785, 177970, 215308, 259891, 313065, 376326, 451501

Offset: 0

N. J. A. Sloane

Also the total number of all different integers in all partitions of n + 1. E.g., a(3) = 7 because the partitions of 4 comprise the sets {1},{1, 2},{2},{1, 3},{4} of different integers and their total number is 7. - Thomas Wieder, Apr 10 2004
With offset 1, also the number of 1's in all partitions of n. For example, 3 = 2+1 = 1+1+1, a(3) = (zero 1's) + (one 1's) + (three 1's), so a(3) = 4. - Naohiro Nomoto, Jan 09 2002. See the Riordan reference p. 184, last formula, first term, for a proof based on Fine's identity given in Riordan, p. 182 (20).
Also, number of partitions of n into parts when there are two kinds of parts of size one.
Also number of graphical forest partitions of 2n+2.
a(n) = count 2 for each partition of n and 1 for each decrement. E.g., the partitions of 4 are 4 (2), 31 (3), 22 (2), 211 (3) and 1111 (2). 2 + 3 + 2 + 3 + 2 = 12. This is related to the Ferrers representation. We can see that taking the Ferrers diagram for each partition of n and adding a new * to all available columns, we generate each partition of n+1, but with repeats (A058884). - Jon Perry, Feb 06 2004
Also the number of 1-transitions among all integer partitions of n. A 1-transition is the removal of a digit "1" from a partition containing at least one "1" and subsequent addition of that "1" to another digit in that partition. This other digit may be a "1" also, but all digits of equal amount are considered as undistinquishable (unlabeled). E.g., for n=6 one has the partition [1113] for which the following two 1-transitions are possible: [1113] --> [123] and [1113] --> [114]. The 1-transitions of n form a partial order (poset). For n=6 one has 12 1-transitions: [111111] --> [11112], [11112] --> [1113], [11112] --> [1122], [1113] --> [114], [1113] --> [123], [1122] --> [123], [1122] --> [222], [123] --> [33], [123] --> [24], [114] --> [15], [114] --> [24], [15] --> [6]. - Thomas Wieder, Mar 08 2005
Also number of partitions of 2n+1 where one of the parts is greater than n (also where there are more than n parts) and of 2n+2 where one of the parts is greater than n+1 (or with more than n+1 parts). - Henry Bottomley, Aug 01 2005
Equals left border of triangle A137633 - Gary W. Adamson, Jan 31 2008
Equals row sums of triangle A027293. - Gary W. Adamson, Oct 26 2008
Convolved with A010815 = [1,1,1,...]. n-th partial sum of A000041 convolved with A010815 = the binomial sequence starting (1, n, ...). - Gary W. Adamson, Nov 09 2008
Equals A036469 convolved with A035363. - Gary W. Adamson, Jun 09 2009
a(A004526(n)) = A025065(n). - Reinhard Zumkeller, Jan 23 2010
a(n) = if n <= 1 then A054225(1,n) else A054225(n,1). - Reinhard Zumkeller, Nov 30 2011
Also the total number of 1's among all hook-lengths in all partitions of n. E.g., a(4)=7 because hooks of the partitions of n = 4 comprise the multisets {4,3,2,1}, {4,2,1,1}, {3,2,2,1}, {4,1,2,1}, {4,3,2,1} and their total number of 1's is 7. - T. Amdeberhan, Jun 03 2012
With offset 1, a(n) is also the difference between the sum of largest and the sum of second largest elements in all partitions of n. More generally, the number of occurrences of k in all partitions of n equals the difference between the sum of k-th largest and the sum of (k+1)st largest elements in all partitions of n. And more generally, the sum of the number of occurrences of k, k+1, k+2..k+m in all partitions of n equals the difference between the sum of k-th largest and the sum of (k+m+1)st largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012
a(0) = 1 and 2*a(n-1) >= a(n) for all n > 0. Hence a(n) is a complete sequence. - Frank M Jackson, Apr 08 2013
a(n) is the number of conjugacy classes in the order-preserving, order-decreasing and (order-preserving and order-decreasing) injective transformation semigroups. - Ugbene Ifeanyichukwu, Jun 03 2015
a(n) is also the number of unlabeled subgraphs of the n-cycle C_n. For example, for n = 3, there are 3 unlabeled subgraphs of the triangle C_3 with 0 edges, 2 with 1 edge, 1 with 2 edges, and 1 with 3 edges (C_3 itself), so a(3) = 3 + 2 + 1 + 1 = 7. - John P. McSorley, Nov 21 2016
a(n) is also the number of partitions of 2n with all parts either even or equal to 1. Proof: the number of such partitions of 2n with exactly 2k 1's is p(n-k), for k = 0,..,n. Summing over k gives the formula. - Leonard Chastkofsky, Jul 24 2018
a(n) is the total number of polygamma functions that appear in the expansion of the (n+1)st derivative of x! with respect to x. More specifically, a(n) is the number of times the string "PolyGamma" appears in the expansion of D[x!, {x, n + 1}] in Mathematica. For example,  D[x!, {x, 3 + 1}] = Gamma[1 + x] PolyGamma[0, 1 + x]^4 + 6 Gamma[1 + x] PolyGamma[0, 1 + x]^2 PolyGamma[1, 1 + x] + 3 Gamma[1 + x] PolyGamma[1, 1 + x]^2 + 4 Gamma[1 + x] PolyGamma[0, 1 + x] PolyGamma[2, 1 + x] + Gamma[1 + x] PolyGamma[3, 1 + x], and we see that the string "PolyGamma" appears a total of a(3) = 7 times in this expansion. - John M. Campbell, Aug 11 2018
With offset 1, also the number of integer partitions of 2n that do not comprise the multiset of vertex-degrees of any multigraph (i.e., non-multigraphical partitions); see A209816 for multigraphical partitions. - Gus Wiseman, Oct 26 2018
Also a(n) is the number of partitions of 2n+1 with exactly one odd part.
Delete the odd part 2k+1, k=0, ..., n, to get a partition of 2n-2k into even parts. There are as many unrestricted partitions of n-k; now sum those numbers from 0 to n to get a(n). - George Beck, Jul 22 2019
In the Young's lattice, a(n) is the number of branches that connect the (n-1)-th layer to the n-th layer. - Shouvik Datta, Sep 19 2021
a(n) is the number of multiset partitions of the multiset {r^n, s^1}, equivalently, factorization patterns of any number m=p^n*q^1 where p and q are primes. - Joerg Arndt, Jan 01 2024
a(n) is the number of positive integers whose divisors are the parts of the partitions of n + 1. - Omar E. Pol, Nov 07 2024

			G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 12*x^4 + 19*x^5 + 30*x^6 + 45*x^7 + 67*x^8 + ...
From _Omar E. Pol_, Oct 25 2012: (Start)
For n = 5 consider the partitions of n+1:
--------------------------------------
.                         Number
Partitions of 6           of 1's
--------------------------------------
6 .......................... 0
3 + 3 ...................... 0
4 + 2 ...................... 0
2 + 2 + 2 .................. 0
5 + 1 ...................... 1
3 + 2 + 1 .................. 1
4 + 1 + 1 .................. 2
2 + 2 + 1 + 1 .............. 2
3 + 1 + 1 + 1 .............. 3
2 + 1 + 1 + 1 + 1 .......... 4
1 + 1 + 1 + 1 + 1 + 1 ...... 6
------------------------------------
35-16 =                     19
.
The difference between the sum of the first column and the sum of the second column of the set of partitions of 6 is 35 - 16 = 19 and equals the number of 1's in all partitions of 6, so the 6th term of this sequence is a(5) = 19.
(End)
From _Gus Wiseman_, Oct 26 2018: (Start)
With offset 1, the a(1) = 1 through a(6) = 19 partitions of 2*n whose greatest part is > n:
  (2)  (4)   (6)    (8)     (A)      (C)
       (31)  (42)   (53)    (64)     (75)
             (51)   (62)    (73)     (84)
             (411)  (71)    (82)     (93)
                    (521)   (91)     (A2)
                    (611)   (622)    (B1)
                    (5111)  (631)    (732)
                            (721)    (741)
                            (811)    (822)
                            (6211)   (831)
                            (7111)   (921)
                            (61111)  (A11)
                                     (7221)
                                     (7311)
                                     (8211)
                                     (9111)
                                     (72111)
                                     (81111)
                                     (711111)
With offset 1, the a(1) = 1 through a(6) = 19 partitions of 2*n whose number of parts is > n:
  (11)  (211)   (2211)    (22211)     (222211)      (2222211)
        (1111)  (3111)    (32111)     (322111)      (3222111)
                (21111)   (41111)     (331111)      (3321111)
                (111111)  (221111)    (421111)      (4221111)
                          (311111)    (511111)      (4311111)
                          (2111111)   (2221111)     (5211111)
                          (11111111)  (3211111)     (6111111)
                                      (4111111)     (22221111)
                                      (22111111)    (32211111)
                                      (31111111)    (33111111)
                                      (211111111)   (42111111)
                                      (1111111111)  (51111111)
                                                    (222111111)
                                                    (321111111)
                                                    (411111111)
                                                    (2211111111)
                                                    (3111111111)
                                                    (21111111111)
                                                    (111111111111)
(End)
From _Joerg Arndt_, Jan 01 2024: (Start)
The a(5) = 19 multiset partitions of the multiset {1^5, 2^1} are:
   1:  {{1, 1, 1, 1, 1, 2}}
   2:  {{1, 1, 1, 1, 1}, {2}}
   3:  {{1, 1, 1, 1, 2}, {1}}
   4:  {{1, 1, 1, 1}, {1, 2}}
   5:  {{1, 1, 1, 1}, {1}, {2}}
   6:  {{1, 1, 1, 2}, {1, 1}}
   7:  {{1, 1, 1, 2}, {1}, {1}}
   8:  {{1, 1, 1}, {1, 1, 2}}
   9:  {{1, 1, 1}, {1, 1}, {2}}
  10:  {{1, 1, 1}, {1, 2}, {1}}
  11:  {{1, 1, 1}, {1}, {1}, {2}}
  12:  {{1, 1, 2}, {1, 1}, {1}}
  13:  {{1, 1, 2}, {1}, {1}, {1}}
  14:  {{1, 1}, {1, 1}, {1, 2}}
  15:  {{1, 1}, {1, 1}, {1}, {2}}
  16:  {{1, 1}, {1, 2}, {1}, {1}}
  17:  {{1, 1}, {1}, {1}, {1}, {2}}
  18:  {{1, 2}, {1}, {1}, {1}, {1}}
  19:  {{1}, {1}, {1}, {1}, {1}, {2}}
(End)

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D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778. - N. J. A. Sloane, Dec 30 2018
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T. D. Noe, Table of n, a(n) for n = 0..1000
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A diagonal of A066633.
Also second column of A126442. - George Beck, May 07 2011
Row sums of triangle A092905.
Also row sums of triangle A261555. - Omar E. Pol, Sep 14 2016
Also row sums of triangle A278427. - John P. McSorley, Nov 25 2016
Column k=2 of A292508.
Cf. A014153, A024786, A026794, A026905, A058884, A093694, A133735, A137633, A010815, A027293, A035363, A028310, A000712, A000990.
Cf. A000569, A025065, A036469, A096373, A147878, A209816, A320891, A320924.

List([0..45],n->Sum([0..n],k->NrPartitions(k))); # Muniru A Asiru, Jul 25 2018

a000070 = p a028310_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Nov 06 2012

with(combinat): a:=n->add(numbpart(j),j=0..n): seq(a(n), n=0..44); # Zerinvary Lajos, Aug 26 2008

CoefficientList[ Series[1/(1 - x)*Product[1/(1 - x^k), {k, 75}], {x, 0, 45}], x] (* Robert G. Wilson v, Jul 13 2004 *)
Table[ Count[ Flatten@ IntegerPartitions@ n, 1], {n, 45}] (* Robert G. Wilson v, Aug 06 2008 *)
Join[{1}, Accumulate[PartitionsP[Range[50]]]+1] (* _Harvey P. Dale, Mar 12 2013 *)
a[ n_] := SeriesCoefficient[ 1 / (1 - x) / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Nov 09 2013 *)
Accumulate[PartitionsP[Range[0,49]]] (* George Beck, Oct 23 2014; typo fixed by Virgile Andreani, Jul 10 2016 *)

{a(n) = if( n<0, 0, polcoeff( 1 / prod(m=1, n, 1 - x^m, 1 + x * O(x^n)) / (1 - x), n))}; /* Michael Somos, Nov 08 2002 */

x='x+O('x^66); Vec(1/((1-x)*eta(x))) /* Joerg Arndt, May 15 2011 */

a(n) = sum(k=0, n, numbpart(k)); \\ Michel Marcus, Sep 16 2016

from itertools import accumulate
def A000070iter(n):
    L = [0]*n; L[0] = 1
    def numpart(n):
        S = 0; J = n-1; k = 2
        while 0 <= J:
            T = L[J]
            S = S+T if (k//2)%2 else S-T
            J -= k  if (k)%2 else k//2
            k += 1
        return S
    for j in range(1, n): L[j] = numpart(j)
    return accumulate(L)
print(list(A000070iter(100))) # Peter Luschny, Aug 30 2019

# Using function A365676Row. Compare also A365675.
from itertools import accumulate
def A000070List(size: int) -> list[int]:
    return [sum(accumulate(reversed(A365676Row(n)))) for n in range(size)]
print(A000070List(45))  # Peter Luschny, Sep 16 2023

def A000070_list(leng):
    p = [number_of_partitions(n) for n in range(leng)]
    return [add(p[:k+1]) for k in range(leng)]
A000070_list(45) # Peter Luschny, Sep 15 2014

Euler transform of [ 2, 1, 1, 1, 1, 1, 1, ...].
log(a(n)) ~ -3.3959 + 2.44613*sqrt(n). - Robert G. Wilson v, Jan 11 2002
a(n) = (1/n)*Sum_{k=1..n} (sigma(k)+1)*a(n-k), n > 1, a(0) = 1. - Vladeta Jovovic, Aug 22 2002
G.f.: (1/(1 - x))*Product_{m >= 1} 1/(1 - x^m).
a(n) seems to have the same parity as A027349(n+1). Comment from James Sellers, Mar 08 2006: that is true.
a(n) = A000041(2n+1) - A110618(2n+1) = A000041(2n+2) - A110618(2n+2). - Henry Bottomley, Aug 01 2005
Row sums of triangle A133735. - Gary W. Adamson, Sep 22 2007
a(n) = A092269(n+1) - A195820(n+1). - Omar E. Pol, Oct 20 2011
a(n) = A181187(n+1,1) - A181187(n+1,2). - Omar E. Pol, Oct 25 2012
From Peter Bala, Dec 23 2013: (Start)
Gupta gives the asymptotic result a(n-1) ~ sqrt(6/Pi^2)* sqrt(n)*p(n), where p(n) is the partition function A000041(n).
Let P(2,n) denote the set of partitions of n into parts k >= 2.
a(n-2) = Sum_{parts k in all partitions in P(2,n)} phi(k), where phi(k) is the Euler totient function (see A000010). Using this result and Mertens's theorem on the average order of the phi function, leads to the asymptotic result
a(n-2) ~ (6/Pi^2)*n*(p(n) - p(n-1)) = (6/Pi^2)*A138880(n) as n -> infinity. (End)
a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(3/2)*Pi*sqrt(n)) * (1 + 11*Pi/(24*sqrt(6*n)) + (73*Pi^2 - 1584)/(6912*n)). - Vaclav Kotesovec, Oct 26 2016
a(n) = A024786(n+2) + A024786(n+1). - Vaclav Kotesovec, Nov 05 2016
G.f.: exp(Sum_{k>=1} (sigma_1(k) + 1)*x^k/k). - Ilya Gutkovskiy, Aug 21 2018
a(n) = A025065(2n). - Gus Wiseman, Oct 26 2018
a(n - 1) = A000041(2n) - A209816(n). - Gus Wiseman, Oct 26 2018

A092269 Spt function: total number of smallest parts (counted with multiplicity) in all partitions of n.

Original entry on oeis.org

1, 3, 5, 10, 14, 26, 35, 57, 80, 119, 161, 238, 315, 440, 589, 801, 1048, 1407, 1820, 2399, 3087, 3998, 5092, 6545, 8263, 10486, 13165, 16562, 20630, 25773, 31897, 39546, 48692, 59960, 73423, 89937, 109553, 133439, 161840, 196168, 236843, 285816, 343667, 412950, 494702, 592063, 706671

Offset: 1

Vladeta Jovovic, Feb 16 2004

nonn

Row sums of triangle A220504. - Omar E. Pol, Jan 19 2013

			Partitions of 4 are [1,1,1,1], [1,1,2], [2,2], [1,3], [4]. 1 appears 4 times in the first, 1 twice in the second, 2 twice in the third, etc.; thus a(4)=4+2+2+1+1=10.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 550 terms from Joerg Arndt)
F. G. Garvan, Table of a(n) for n = 1..10000 (Coefficients of Andrews spt-function)
G. E. Andrews, The number of smallest parts in the partitions of n, Journal für die reine und angewandte Mathematik, Volume 2008 Issue 624, 133-142.
Scott Ahlgren, Nickolas Andersen, Euler-like recurrences for smallest parts functions, arXiv:1402.5366
George E. Andrews, Song Heng Chan and Byungchan Kim, The Odd Moments of Ranks and Cranks, Journal of Combinatorial Theory, Series A, Volume 120, Issue 1, January 2013, Pages 77-91. - From _N. J. A. Sloane_, Sep 04 2012
G. E. Andrews, F. G. Garvan, and J. Liang, Combinatorial interpretation of congruences for the spt-function
G. E. Andrews, F. G. Garvan, and J. Liang, Self-conjugate vector partitions and the parity of the spt-function
William Y.C. Chen, The spt-Function of Andrews, arXiv:1707.04369 [math.CO], Jul 14 2017.
A. Folsom and K. Ono, The spt-function of Andrews, PNAS, December 23 2008, 105 (51) 20152-20156.
F. G. Garvan, Congruences for Andrews' smallest parts partition function and new congruences for Dyson's rank
F. G. Garvan, Congruences for Andrews' spt-function modulo powers of 5, 7 and 13, arXiv:1011.1955 [math.NT], Nov 9 2010.
F. G. Garvan, Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences, arXiv:1011.1957 [math.NT], Nov 9 2010.
F. G. Garvan, Higher Order Spt-functions, Adv. Math. 228 (2011), no. 1, 241-265. - From _N. J. A. Sloane_, Jan 02 2013
F. G. Garvan, The smallest parts partition function, 2012.
F. G. Garvan, Dyson's rank function and Andrews's SPT-function, slides 11, 12.
M. H. Mertens, K. Ono, L. Rolen, Mock modular Eisenstein series with Nebentypus, arXiv:1906.07410 [math.NT], 2019.
K. Ono, Congruences for the Andrews spt-function, PNAS, December 21 2010, 108 (2) 473-476.
Omar E. Pol, Illustration of initial terms
Wikipedia, Spt function

Cf. A092314, A092322, A092309, A092321, A092313, A092310, A092311, A092268, A006128, A195053.

For higher-order spt functions see A221140-A221144.

b:= proc(n, i) option remember; `if`(n=0 or i=1, n,
      `if`(irem(n, i, 'r')=0, r, 0)+add(b(n-i*j, i-1), j=0..n/i))
    end:
a:= n-> b(n, n):
seq(a(n), n=1..60);  # Alois P. Heinz, Jan 16 2013

terms = 47; gf = Sum[x^n/(1 - x^n)*Product[1/(1 - x^k), {k, n, terms}], {n, 1, terms}]; CoefficientList[ Series[gf, {x, 0, terms}], x] // Rest (* Jean-François Alcover, Jan 17 2013 *)
b[n_, i_] := b[n, i] = If[n==0 || i==1, n, {q, r} = QuotientRemainder[n, i]; If[r==0, q, 0] + Sum[b[n-i*j, i-1], {j, 0, n/i}]]; a[n_] := b[n, n]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Nov 23 2015, after Alois P. Heinz *)

N = 66;  x = 'x + O('x^N);
gf = sum(n=1,N, x^n/(1-x^n) * prod(k=n,N, 1/(1-x^k) )  );
v = Vec(gf)
/* Joerg Arndt, Jan 12 2013 */

G.f.: Sum_{n>=1} x^n/(1-x^n) * Product_{k>=n} 1/(1-x^k).
a(n) = A000070(n-1) + A195820(n). - Omar E. Pol, Oct 19 2011
a(n) = n*p(n) - N_2(n)/2 = n*A000041(n) - A220908(n)/2 = A066186(n) - A220907(n) = (A220909(n) - A220908(n))/2 = A211982(n)/2 (from Andrews's paper and Garvan's paper). - Omar E. Pol, Jan 03 2013
a(n) = A000041(n) + A000070(n-2) + A220479(n), n >= 2. - Omar E. Pol, Feb 16 2013
Asymptotics (Bringmann-Mahlburg, 2009): a(n) ~ exp(Pi*sqrt(2*n/3)) / (Pi*sqrt(8*n)) ~ sqrt(6*n)*A000041(n)/Pi. - Vaclav Kotesovec, Jul 30 2017

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004

A220479 Total number of smallest parts that are also emergent parts in all partitions of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 3, 1, 5, 5, 10, 8, 22, 19, 33, 40, 62, 67, 107, 118, 175, 208, 282, 331, 462, 542, 712, 859, 1112, 1323, 1709, 2030, 2568, 3078, 3830, 4577, 5687, 6760, 8291, 9885, 12045, 14290, 17334, 20515, 24710, 29242, 35004, 41282, 49283, 57963, 68836

Offset: 1

Omar E. Pol, Jan 12 2013

nonn

For the definition of emergent parts see A182699.

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000

Cf. A000041, A000070, A002865, A006128, A092269, A120452, A182699, A182709, A195820, A206437, A215513.

b[n_, i_] := b[n, i] = If[n==0 || i==1, n, {q, r} = QuotientRemainder[n, i]; If[r==0, q, 0] + Sum[b[n-i*j, i-1], {j, 0, n/i}]];
c[n_] := b[n, n];
d[n_] := Total[PartitionsP[Range[0, n-3]]] + PartitionsP[n-1];
a[n_] := c[n] - d[n+1];
Array[a, 50] (* Jean-François Alcover, Jun 05 2021, using Alois P. Heinz's code for A092269 *)

a(n) = A092269(n) - A000070(n-1) - A002865(n) = A092269(n) - A120452(n+1) = A195820(n) - A002865(n).
a(n) = A092269(n) - A000041(n) - A000070(n-2), n >= 2.
a(n) = A215513(n) - A000070(n-2), n >= 2.
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n). - Vaclav Kotesovec, Jul 31 2017

a(43) corrected by Vaclav Kotesovec, Jul 31 2017

A195011 Triangle read by rows: T(n,m) = number of vector partitions of n in S with crank m counted according to the weight w_1.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 2, 1, 1, 0, 2, 2, 2, 1, 1, 0, 4, 4, 3, 2, 1, 1, 0, 5, 4, 4, 3, 2, 1, 1, 0, 7, 7, 6, 5, 3, 2, 1, 1, 0, 10, 9, 8, 6, 5, 3, 2, 1, 1, 0, 13, 13, 11, 10, 7, 5, 3, 2, 1, 1, 0, 17, 16, 15, 12, 10, 7, 5, 3, 2, 1, 1

Offset: 0

Omar E. Pol, Jan 10 2012

This is the essential triangle in the square array of the Andrews-Garvan-Liang paper, see chapter 7 and page 3 for a precise definition.

Apparently columns (their positive terms) converge to A000041.

			Triangle begins:
1;
1, 0;
1, 1, 0;
1, 1, 1, 0;
2, 2, 1, 1, 0;
2, 2, 2, 1, 1, 0;
4, 4, 3, 2, 1, 1, 0;
5, 4, 4, 3, 2, 1, 1, 0;
7, 7, 6, 5, 3, 2, 1, 1, 0;
10,9, 8, 6, 5, 3, 2, 1, 1, 0;

G. E. Andrews, The number of smallest parts in the partitions of n
G. E. Andrews, F. G. Garvan, and J. Liang, Combinatorial interpretation of congruences for the spt-function
Frank Garvan, Dyson's rank function and Andrews's SPT-function, slide 17.

Cf. A092269, A195012, A195820.

A284827 Expansion of Sum_{i>=1} x^prime(i)/(1 - x^prime(i)) * Product_{j>=i} 1/(1 - x^prime(j)).

Original entry on oeis.org

0, 1, 1, 2, 2, 5, 4, 6, 9, 11, 13, 18, 20, 26, 34, 37, 47, 55, 66, 80, 96, 111, 130, 150, 180, 206, 240, 278, 318, 366, 419, 483, 549, 626, 716, 803, 913, 1034, 1167, 1314, 1477, 1659, 1861, 2085, 2332, 2605, 2902, 3232, 3602, 3999, 4442, 4930, 5454, 6034, 6675, 7375, 8133, 8967, 9870, 10855

Offset: 1

Ilya Gutkovskiy, Apr 03 2017

nonn

Total number of smallest parts in all partitions of n into prime parts.

			a(10) = 11 because we have [7, 3], [5, 5], [5, 3, 2], [3, 3, 2, 2], [2, 2, 2, 2, 2] and 1 + 2 + 1 + 2 + 5 = 11.

Index entries for related partition-counting sequences

Cf. A000607, A084993, A092268, A092269, A195820, A284828.

nmax = 60; Rest[CoefficientList[Series[Sum[x^Prime[i]/(1 - x^Prime[i]) Product[1/(1 - x^Prime[j]), {j, i, nmax}], {i, 1, nmax}], {x, 0, nmax}], x]]

G.f.: Sum_{i>=1} x^prime(i)/(1 - x^prime(i)) * Product_{j>=i} 1/(1 - x^prime(j)).

A284828 Expansion of Sum_{i>=2} x^prime(i)/(1 - x^prime(i)) * Product_{j>=i} 1/(1 - x^prime(j)).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 1, 3, 3, 3, 5, 4, 6, 9, 7, 10, 11, 12, 17, 19, 22, 23, 26, 33, 36, 41, 48, 52, 59, 66, 78, 85, 97, 112, 117, 134, 151, 169, 187, 207, 230, 255, 284, 313, 348, 379, 418, 465, 508, 561, 620, 674, 737, 812, 892, 972, 1064, 1157, 1257, 1379, 1503, 1639, 1776, 1935, 2101, 2279, 2483

Offset: 1

Ilya Gutkovskiy, Apr 03 2017

nonn

Total number of smallest parts in all partitions of n into odd prime parts (A065091).

			a(16) = 7 because we have [13, 3], [11, 5], [7, 3, 3, 3], [5, 5, 3, 3] and 1 + 1 + 3 + 2 = 7.

Index entries for related partition-counting sequences

Cf. A065091, A084993, A092268, A092269, A099773, A195820, A281544, A284827.

nmax = 68; Rest[CoefficientList[Series[Sum[x^Prime[i]/(1 - x^Prime[i]) Product[1/(1 - x^Prime[j]), {j, i, nmax}], {i, 2, nmax}], {x, 0, nmax}], x]]

x = 'x + O('x ^ 70); concat([0, 0], Vec(sum(i=2, 70, x^prime(i)/(1 - x^prime(i)) * prod(j=i, 70, 1/(1 - x^prime(j)))))) \\ Indranil Ghosh, Apr 05 2017

G.f.: Sum_{i>=2} x^prime(i)/(1 - x^prime(i)) * Product_{j>=i} 1/(1 - x^prime(j)).

A195821 Total number of parts that are not the smallest part in all partitions of n that do not contain 1 as a part.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 5, 7, 12, 19, 25, 37, 56, 72, 102, 138, 187, 246, 330, 422, 563, 721, 931, 1177, 1523, 1903, 2421, 3020, 3797, 4700, 5875, 7218, 8956, 10954, 13474, 16401, 20083, 24316, 29576, 35685, 43179, 51870, 62490, 74757, 89666, 106927, 127687

Offset: 1

Omar E. Pol, Oct 19 2011

nonn

Total number of parts that are not the smallest part in all partitions of the head of the last section of the set of partitions of n. For more information see A195820.

			For n = 8 the seven partitions of 8 that do not contain 1 as a part are:
.   8
.   4  +  4
.  (5) +  3
.  (6) +  2
.  (3) + (3) +  2
.  (4) +  2  +  2
.   2  +  2  +  2  +  2
Note that in every partition the parts that are not the smallest part are shown between parentheses. The total number of parts that are not the smallest part is 0+0+1+1+2+1+0 = 5, so a(8) = 5.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000041, A000070, A002865, A092269, A135010, A138121, A138135, A138137, A182984, A195820.

a(n) = A138135(n) - A195820(n) = A138137(n) - A195820(n) - A000041(n-1).

A220483 Total number of smallest parts that are also emergent parts in all partitions of n with at least one distinct part: a(n) = n + d(n) + p(n-1) + spt(n) - A000070(n) - sigma(n) - 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 3, 5, 8, 11, 19, 26, 34, 51, 67, 91, 118, 158, 200, 271, 331, 433, 538, 699, 849, 1089, 1323, 1674, 2030, 2542, 3066, 3813, 4567, 5640, 6760, 8272, 9871, 12002, 14290, 17287, 20515, 24675, 29214, 34981, 41282, 49216, 57957, 68798

Offset: 1

Omar E. Pol, Jan 16 2013

nonn

For the definition of "emergent part" see A182699, A182709.

Cf. A000005, A000041, A000070, A000203, A002865, A092269, A182699, A182709, A183152, A193827, A195820, A206437, A215513, A220479, A220489.

b[n_, i_] := b[n, i] = If[n==0 || i==1, n, {q, r} = QuotientRemainder[n, i]; If[r == 0, q, 0] + Sum[b[n - i*j, i - 1], {j, 0, n/i}]];
a[n_] := n + DivisorSigma[0, n] + PartitionsP[n - 1] + b[n, n] -
  Total[PartitionsP[Range[0, n]]] - DivisorSigma[1, n] - 1;
Array[a, 50] (* Jean-François Alcover, Jun 05 2021, using Alois P. Heinz's code for A092269 *)

a(n) = n + A000005(n) + A000041(n-1) + A092269(n) - A000070(n) - A000203(n) - 1.

a(49) corrected by Jean-François Alcover, Jun 05 2021

A220489 Total number of smallest parts in all partitions of n minus the total number of smallest parts that are also emergent parts in all partitions of n with at least one distinct part.

Original entry on oeis.org

1, 3, 5, 10, 14, 26, 34, 56, 77, 114, 153, 227, 296, 414, 555, 750, 981, 1316, 1702, 2241, 2887, 3727, 4761, 6112, 7725, 9787, 12316, 15473, 19307, 24099, 29867, 37004, 45626, 56147, 68856, 84297, 102793, 125167, 151969, 184166, 222553, 268529, 323152

Offset: 1

Omar E. Pol, Feb 24 2013

nonn

First differs from A092269 at a(7).

For the definition of "emergent part" see A182699, A182709.

Cf. A000005, A000041, A000070, A000203, A002865, A092269, A120452, A195820, A220479, A220483.

a(n) = spt(n) - A220483(n) = 1 + sigma(n) + A000070(n) - p(n-1) - d(n) - n.

a(n) = A092269(n) - A220483(n) = 1 + A000203(n) + A000070(n) - A000041(n-1) - A000005(n) - n.

A284829 Expansion of Sum_{i>=1} mu(i)^2x^i/(1 - x^i) Product_{j>=i} 1/(1 - mu(j)^2*x^j), where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 3, 5, 9, 13, 23, 30, 45, 64, 89, 118, 165, 211, 282, 369, 475, 606, 779, 978, 1236, 1547, 1922, 2375, 2936, 3602, 4403, 5362, 6506, 7864, 9493, 11399, 13661, 16317, 19443, 23122, 27415, 32418, 38268, 45065, 52968, 62125, 72742, 84969, 99112, 115409, 134139, 155665, 180368, 208658, 241051

Offset: 1

Ilya Gutkovskiy, Apr 03 2017

nonn

Total number of smallest parts in all partitions of n into squarefree parts (A005117).

			a(5) = 13 because we have [5], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1], [1, 1, 1, 1, 1] and 1 + 1 + 2 + 1 + 3 + 5 = 13.

Index entries for related partition-counting sequences

Cf. A005117, A008683, A073576, A092268, A092269, A195820, A281572.

nmax = 50; Rest[CoefficientList[Series[Sum[MoebiusMu[i]^2 x^i/(1 - x^i) Product[1/(1 - MoebiusMu[j]^2 x^j), {j, i, nmax}], {i, 1, nmax}], {x, 0, nmax}], x]]

x='x+O('x^50); Vec(sum(i=1, 50, moebius(i)^2*x^i/(1 - x^i) * prod(j=i, 50, 1/(1 - moebius(j)^2*x^j)))) \\ Indranil Ghosh, Apr 04 2017

G.f.: Sum_{i>=1} mu(i)^2*x^i/(1 - x^i) * Product_{j>=i} 1/(1 - mu(j)^2*x^j).

cp's OEIS Frontend

A000070 a(n) = Sum_{k=0..n} p(k) where p(k) = number of partitions of k (A000041).

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A092269 Spt function: total number of smallest parts (counted with multiplicity) in all partitions of n.

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A220479 Total number of smallest parts that are also emergent parts in all partitions of n.

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A195011 Triangle read by rows: T(n,m) = number of vector partitions of n in S with crank m counted according to the weight w_1.

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A284827 Expansion of Sum_{i>=1} x^prime(i)/(1 - x^prime(i)) * Product_{j>=i} 1/(1 - x^prime(j)).

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A284828 Expansion of Sum_{i>=2} x^prime(i)/(1 - x^prime(i)) * Product_{j>=i} 1/(1 - x^prime(j)).

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A195821 Total number of parts that are not the smallest part in all partitions of n that do not contain 1 as a part.

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A284829 Expansion of Sum_{i>=1} mu(i)^2x^i/(1 - x^i) Product_{j>=i} 1/(1 - mu(j)^2*x^j), where mu() is the Moebius function (A008683).