A027349 -id:A027349 - cp's OEIS frontend

A086372 First differences of A027349.

-1, 0, 1, -1, 1, -1, 1, 0, 0, 0, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 2, 0, 1, 0, 2, 1, 1, 0, 3, 1, 1, 1, 3, 2, 1, 2, 3, 3, 2, 2, 4, 5, 1, 4, 5, 5, 3, 5, 6, 7, 4, 6, 8, 9, 5, 8, 11, 10, 8, 10, 13, 13, 11, 12, 17, 17, 13, 16, 22, 19, 19, 20, 26, 26, 23, 25, 33, 32, 29

Offset: 1

Ralf Stephan, Sep 06 2003

Sequence has the same parity as A000041, the partition numbers, i.e. a(n) is congruent A040051(n) mod 2 (proved by James Sellers using generating functions).

a(n)=polcoeff((1-x)*prod(k=1,n,1+x^(2*k+1),1+x*O(x^n)),n)

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

Original entry on oeis.org

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)

H. Gupta, An asymptotic formula in partitions. J. Indian Math. Soc., (N. S.) 10 (1946), 73-76.
H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 90.
R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 6.
D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778. - N. J. A. Sloane, Dec 30 2018
A. M. Odlyzko, Asymptotic Enumeration Methods, p. 19
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).
Stanley, R. P., Exercise 1.26 in Enumerative Combinatorics, Vol. 1. Cambridge, England: Cambridge University Press, p. 59, 1999.

T. D. Noe, Table of n, a(n) for n = 0..1000
P. A. Baikov and S. V. Mikhailov, The {beta}-expansion for Adler function, Bjorken Sum Rule, and the Crewther-Broadhurst-Kataev relation at order O(alpha_s^4), J. High Energy Phys. 09 (2022) Art. No. 185. See also arXiv:2206.14063 [hep-ph], 2022.
Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.
David Benson, Radha Kessar, and Markus Linckelmann, Hochschild cohomology of symmetric groups in low degrees, arXiv:2204.09970 [math.GR], 2022.
Philip Boalch, Counting the fission trees and nonabelian Hodge graphs, arXiv:2410.23358 [math.AG], 2024. See p. 15.
L. Bracci and L. E. Picasso, A simple iterative method to write the terms of any order of perturbation theory in quantum mechanics, The European Physical Journal Plus, 127 (2012), Article 119. - From _N. J. A. Sloane_, Dec 31 2012
Emmanuel Briand, Samuel A. Lopes, and Mercedes Rosas, Normally ordered forms of powers of differential operators and their combinatorics, arXiv:1811.00857 [math.CO], 2018.
C. C. Cadogan, On partly ordered partitions of a positive integer, Fib. Quart., 9 (1971), 329-336.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers, National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956. (Annotated scanned pages from, plus a review)
Philip Cuthbertson, Fixed hooks in arbitrary columns of partitions, Integers (2025) Vol. 25, Art. No. A28. See p. 3.
Mario De Salvo, Dario Fasino, Domenico Freni and Giovanni Lo Faro, A Family of 0-Simple Semihypergroups Related to Sequence A000070, Journal of Multiple-Valued Logic & Soft Computing, 2016, Vol. 27, Issue 5/6, pp. 553-572.
Mario De Salvo, Dario Fasino, Domenico Freni, and Giovanni Lo Faro, Semihypergroups Obtained by Merging of 0-semigroups with Groups, Filomat 32(12) (2018), 4177-4194.
P. Flajolet and B. Salvy, Euler sums and contour integral representations, Experimental Mathematics, 7(1) (1998), 15-35.
D. Frank, C. D. Savage and J. A. Sellers, On the Number of Graphical Forest Partitions, Ars Combinatoria, Vol. 65 (2002), 33-37.
D. Frank, C. D. Savage and J. A. Sellers, On the Number of Graphical Forest Partitions, preprint.
Manosij Ghosh Dastidar and Sourav Sen Gupta, Generalization of a few results in Integer Partitions, arXiv preprint arXiv:1111.0094 [cs.DM], 2011.
Petros Hadjicostas, Cyclic, Dihedral and Symmetrical Carlitz Compositions of a Positive Integer, Journal of Integer Sequences, 20 (2017), Article #17.8.5.
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
M. D. Hirschhorn, The number of 1's in the partitions of n, Fib. Quart., 51 (2013), 326-329.
M. D. Hirschhorn, The number of different parts in the partitions of n, Fib. Quart., 52 (2014), 10-15. See p. 11. - _N. J. A. Sloane_, Mar 25 2014
Nick Hobson, Solution to puzzle 56: Partition identity
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 113.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 126.
MathOverflow, Number of branches between two layers of the Young's lattice, Sep 19 2021.
Mikhailov, S. V. On a realization of beta-expansion in QCD, J. High Energy Phys. 2017, No. 4, Paper No. 169, 16 p. (2017).
M. M. Mogbonju, O. A. Ojo, and I. A. Ogunleke, Graphical Representation of Conjugacy Classes in the Order-Preserving Partial One-One Transformation Semigroup, International Journal of Science and Research (IJSR), 3(12) (2014), 711-721.
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
F. Ruskey, Combinatorial Object Server.
Maria Schuld, Kamil Brádler, Robert Israel, Daiqin Su, and Brajesh Gupt, A quantum hardware-induced graph kernel based on Gaussian Boson Sampling, arXiv:1905.12646 [quant-ph], 2019.
N. J. A. Sloane, Transforms
I. J. Ugbene, E. O. Eze, and S. O. Makanjuola, On the Number of Conjugacy Classes in the Injective Order-Decreasing Transformation Semigroup, Pacific Journal of Science and Technology, 14(1) (2013), 182-186.
Ifeanyichukwu Jeff Ugbene, Gatta Naimat Bakare, and Garba Risqot Ibrahim, Conjugacy classes of the order-preserving and order-decreasing partial one-to-one transformation semigroups, Journal of Science, Technology, Mathematics and Education (JOSTMED), 15(2) (2019), 83-88.
Joseph Vandehey, Digital problems in the theory of partitions, Integers (2024) Vol. 24A, Art. No. A18. See p. 3.
Eric Weisstein's World of Mathematics, Stanley's Theorem.

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

A246581 G.f.: x^((k^2 + k)/2) / (Product_{i=1..k} (1 - x^i) * Product_{r>=1} (1 + x^r)) with k = 2.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, -1, 1, -2, 1, -3, 2, -4, 3, -5, 5, -6, 7, -8, 10, -10, 13, -13, 17, -17, 21, -22, 27, -28, 33, -36, 41, -45, 50, -56, 62, -69, 75, -85, 92, -103, 111, -125, 135, -150, 162, -180, 195, -215, 232, -256, 278, -303, 329, -359, 390, -423

Offset: 0

N. J. A. Sloane, Aug 31 2014

sign

Empirical: 2*(-1)^n*a(n+1) is equal to the number of partitions mu of n such that the diagram of mu and the diagram of the transpose of mu have exactly n-1 cells in common (see below example). - John M. Campbell, Feb 01 2016

			From _John M. Campbell_, Feb 01 2016: (Start)
For example, letting n=9, there are 2*(-1)^n*a(n+1) = (-2)*(-3) = 6 partitions mu of n=9 such that the diagram of mu and the diagram of the transpose of mu have exactly n-1 cells in common: (5,2,1,1), (4,3,2), (4,3,1,1), (4,2,2,1), (4,2,1,1,1), (3,3,2,1). For example, the diagram of (3,3,2,1) is
   ooo
   ooo
   oo
   o
and the diagram of the transpose of (3,3,2,1) is
   oooo
   ooo
   oo
and these diagrams share exactly (n-1)=8 cells in common, when the diagrams are positioned so that the upper-left corners of both diagrams coincide. (End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
J. Fulman, Random matrix theory over finite fields, Bull. Amer. Math. Soc., 39 (No. 1, 2002), 51-85, MR1864086 (2002i:60012). See top of page 70, Eq. 3, with k=2.

For k=0 and 1 we get A081362, A027349 (apart from signs).

Cf. A218907, A246582, A246583.

fSp:=proc(k) local a,i,r;
a:=x^((k^2+k)/2)/mul(1-x^i,i=1..k);
a:=a/mul(1+x^r,r=1..101);
series(a,x,101);
seriestolist(%);
end;
fSp(2);

k = 2; CoefficientList[Series[x^((k^2 + k)/2)/(Product[1 - x^i, {i, k}] Product[1 + x^r, {r, 1000}]), {x, 0, 56}], x] (* Michael De Vlieger, Feb 01 2016 *)

G.f.: x^3/((1-x)*(1-x^2)) * Product_{k>=1} 1/(1+x^k). - Vaclav Kotesovec, Mar 12 2016

a(n) ~ (-1)^(n+1) * 3^(1/4) * exp(sqrt(n/6)*Pi) / (2^(9/4)*Pi*n^(1/4)). - Vaclav Kotesovec, Mar 12 2016

A116860 Triangle read by rows: T(n,k) is the number of partitions into distinct odd parts with smallest part k (n>=1, k>=1).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 1, 0, 1, 0, 1, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Emeric Deutsch, Feb 27 2006

Row 2 has no terms, row 2n-1 has 2n-1 terms, row 4n has 2n-1 terms, row 4n+2 for n>=1 has 2n-1 terms. Row sums are A000700. T(n,1) = A027349(n). Sum_{k>=1} k*T(n,k) = A092319(n).

Within those rows, T(n, k) = 0 occurs with an odd k iff n is odd and 2*floor((n+3)/6) - 1 <= k < n. - Álvar Ibeas, Aug 03 2020

			T(25,3) = 3 because we have [17,5,3], [15,7,3] and [13,9,3].
Triangle starts:
1;
{};
0,0,1;
1;
0,0,0,0,1;
1;
0,0,0,0,0,0,1;
1,0,1;

Alois P. Heinz, Rows n = 1..200, flattened

Cf. A000700, A027349, A092319, A116857.

g:=sum(t^(2*j-1)*x^(2*j-1)*product(1+x^(2*i-1),i=j+1..30),j=1..30): gser:=simplify(series(g,x=0,52)): for n from 1 to 19 do P[n]:=sort(coeff(gser,x^n)) od: d:=proc(n) if n mod 2 = 1 then n elif n=2 then 0 elif n mod 4 = 0 then n/2-1 else n/2-2 fi end: 1; {}; for n from 3 to 19 do seq(coeff(P[n],t^j),j=1..d(n)) od; # yields sequence in triangular form

imax = 20;
s = Sum[t^(2 j - 1)*x^(2 j - 1)*Product[1 + x^(2 i - 1), {i, j + 1, imax}], {j, 1, imax}] + O[x]^imax;
Rest /@ DeleteCases[CoefficientList[#, t]& /@ CoefficientList[s, x], {}] // Flatten (* Jean-François Alcover, May 22 2018 *)

G.f.: Sum_(t^(2*j-1)*x^(2*j-1)*Product_(1+x^(2*i-1), i=j+1..infinity), j=1..infinity).

For k even, T(n, k) = 0. For k odd, T(n, n) = 1 and, if k < n, T(n, k) = Sum_{i > k} T(n - k, i). - Álvar Ibeas, Aug 03 2020

A226541 Number of unimodal compositions of n where the maximal part appears three times.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 2, 3, 5, 7, 11, 16, 24, 34, 51, 71, 102, 143, 201, 276, 384, 522, 714, 964, 1301, 1739, 2328, 3084, 4085, 5377, 7064, 9226, 12036, 15616, 20228, 26092, 33584, 43067, 55125, 70308, 89502, 113598, 143889, 181755, 229160, 288186, 361750, 453046, 566346, 706464

Offset: 0

Joerg Arndt, Jun 10 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A006330 (max part appears once), A114921 (max part appears twice).
Cf. A188674 (max part m appears m times), A001522 (max part m appears at least m times).
Cf. A001523 (max part appears any number of times).
Cf. A000009 (symmetric, max part m appears once; also symmetric, max part appears an odd number of times).
Cf. A035363 (symmetric, max part m appears twice; also symmetric, max part appears an even number of times).
Cf. A087897 (symmetric, max part m appears 3 times).
Cf. A027349 (symmetric, max part m appears m times), A189357 (symmetric, max part m appears at least m times).
Column k=3 of A247255.

N=66; x='x+O('x^N); Vec(sum(n=0,N, x^(3*n) / prod(k=1,n-1, 1-x^k )^2 ))

G.f.: sum(n>=0, x^(3*n) / prod(k=1..n-1, 1-x^k )^2 ); replace 3 by m to obtain g.f. for "... max part appears m times".

a(n) ~ Pi^2 * exp(2*Pi*sqrt(n/3)) / (16 * 3^(7/4) * n^(9/4)). - Vaclav Kotesovec, Oct 24 2018

A246582 G.f.: x^((k^2+k)/2)/(mul(1-x^i,i=1..k)*mul(1+x^r,r=1..oo)) with k = 3.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, -1, 1, -2, 1, -3, 1, -4, 2, -5, 3, -6, 5, -7, 7, -8, 10, -10, 13, -12, 17, -15, 21, -19, 26, -24, 31, -30, 38, -38, 45, -47, 54, -58, 64, -71, 77, -86, 91, -103, 109, -124, 129, -147, 154, -174, 182, -205, 216, -241, 254, -282, 300, -330, 351, -384, 412, -447, 480, -519, 560, -602, 649, -696, 753, -805

Offset: 0

N. J. A. Sloane, Aug 31 2014

sign

Fulman, Jason. Random matrix theory over finite fields. Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70, Eq. 3, with k=3.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

For k=0 and 1 we get A081362, A027349 (apart from signs). Cf. A246581, A246583.

fSp:=proc(k) local a,i,r;
a:=x^((k^2+k)/2)/mul(1-x^i,i=1..k);
a:=a/mul(1+x^r,r=1..101);
series(a,x,101);
seriestolist(%);
end;
fSp(3);

nmax = 100; CoefficientList[Series[x^6/((1-x)*(1-x^2)*(1-x^3)) * Product[1/(1+x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 11 2016 *)

a(n) ~ (-1)^n * 3^(1/4) * exp(sqrt(n/6)*Pi) / (2^(13/4)*Pi*n^(1/4)). - Vaclav Kotesovec, Mar 12 2016

A246583 G.f.: x^((k^2+k)/2)/(mul(1-x^i,i=1..k)*mul(1+x^r,r=1..oo)) with k = 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, -1, 2, -2, 3, -4, 3, -6, 5, -9, 6, -12, 10, -16, 13, -20, 20, -26, 26, -32, 37, -41, 47, -51, 63, -65, 78, -81, 101, -103, 123, -128, 155, -161, 187, -199, 232, -247, 278, -302, 341, -371, 407, -449, 495, -545, 589, -654, 711, -786, 843, -936, 1011, -1116, 1194, -1320, 1423, -1563, 1674

Offset: 0

N. J. A. Sloane, Aug 31 2014

sign

Fulman, Jason. Random matrix theory over finite fields. Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70, Eq. 3, with k=4.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

For k=0 and 1 we get A081362, A027349 (apart from signs). Cf. A246581, A246582.

fSp:=proc(k) local a,i,r;
a:=x^((k^2+k)/2)/mul(1-x^i,i=1..k);
a:=a/mul(1+x^r,r=1..101);
series(a,x,101);
seriestolist(%);
end;
fSp(4);

nmax = 100; CoefficientList[Series[x^10/((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, Mar 11 2016 *)

a(n) ~ (-1)^n * 3^(3/4) * n^(1/4) * exp(sqrt(n/6)*Pi) / (2^(15/4)*Pi^2). - Vaclav Kotesovec, Mar 12 2016

A027350 Number of partitions of n into distinct odd parts, the least being 3.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 2, 4, 3, 4, 4, 5, 5, 5, 6, 7, 8, 7, 9, 9, 11, 10, 13, 13, 15, 14, 17, 18, 21, 20, 23, 25, 27, 28, 31, 34, 36, 38, 41, 46, 48, 51, 54, 61, 63, 68, 72, 80, 83, 89, 94, 104, 109, 116

Offset: 1

Clark Kimberling

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A027349.

N=99;x='x+O('x^N);
t=sum(n=2,N,x^(n^2-1)/prod(k=1,n-2,1-x^(2*k))); /* == x^3 + x^8 + x^10 + ... */
v=Vec(t) /* show terms for n>=3 */  /* Joerg Arndt, Jan 29 2011 */

G.f. sum(n>=2, x^(n^2-1)/prod(k=1,n-2,1-x^(2*k))) - Joerg Arndt, Jan 29 2011

a(n) ~ exp(Pi*sqrt(n/6)) / (2^(15/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 09 2018

A092130 Number of partitions of n into distinct parts == 1 (mod 3), with 1 as the smallest part.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 2, 1, 1, 3, 1, 1, 3, 2, 1, 4, 3, 1, 4, 4, 2, 5, 5, 2, 5, 7, 3, 6, 8, 4, 6, 10, 6, 7, 12, 7, 8, 14, 10, 9, 16, 12, 10, 19, 16, 12, 21, 19, 14, 24, 24, 17, 27, 28, 20, 31, 35, 24, 34, 40, 29, 39, 48, 35

Offset: 1

Jon Perry, Mar 30 2004

nonn

Also number of partitions of n such that if k is the largest part, then k occurs exactly once and integers from 1 to k-1 occur a positive multiple of 3 times. Example: a(18)=2 because we have [3,2,2,2,1,1,1,1,1,1,1,1,1] and [3,2,2,2,2,2,2,1,1,1]. - Emeric Deutsch, Apr 18 2006

			For a(24), we have 19+4+1, 16+7+1, 13+10+1, so a(24)=3.

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

Cf. A027349.

g:=x*product(1+x^(1+3*k),k=1..25): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=1..51); # Emeric Deutsch, Apr 18 2006
# second Maple program
b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i<2, 0, b(n, i-3)+`if`(i>n, 0, b(n-i, i-3))))
    end:
a:= n-> b(n-1, iquo(n, 3)*3+1):
seq (a(n), n=1..100);  # Alois P. Heinz, May 01 2012

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<2, 0, b[n, i-3] + If[i>n, 0, b[n-i, i-3]]]]; a[n_] := b[n-1, Quotient[n, 3]*3+1]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, May 13 2015, after Alois P. Heinz *)

for(i=0,50,print1(","polcoeff(prod(k=1,50,(1+x^(3*k+1))),i)))

G.f.: x*Product_{k>=1} (1+x^(1+3k)). - Emeric Deutsch, Apr 18 2006
a(n) ~ exp(Pi*sqrt(n)/3) / (2^(7/3) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Aug 30 2015
G.f.: Sum_{k>=1} x^(k*(3*k - 1)/2) / Product_{j=1..k-1} (1 - x^(3*j)). - Ilya Gutkovskiy, Nov 28 2020

A298596 Expansion of Product_{k>=2} 1/(1 + x^k).

Original entry on oeis.org

1, 0, -1, -1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, -1, 1, 0, 0, -1, 1, -1, 0, -1, 2, -1, 0, -2, 2, -1, 1, -2, 3, -2, 1, -3, 4, -2, 2, -4, 5, -3, 3, -5, 6, -5, 4, -6, 9, -6, 5, -9, 10, -8, 8, -11, 13, -11, 10, -14, 17, -14, 13, -19, 21, -18, 18, -23, 26, -24, 23, -29, 34, -30, 29, -38, 41, -38, 39

Offset: 0

Ilya Gutkovskiy, Jan 22 2018

sign

The difference between the number of partitions of n into an even number of parts > 1 and the number of partitions of n into an odd number of parts > 1.

Convolution inverse of A025147.

Index entries for related partition-counting sequences

Cf. A000700, A002865, A025147, A027349, A078616, A081362.

nmax = 78; CoefficientList[Series[Product[1/(1 + x^k), {k, 2, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=2} 1/(1 + x^k).
a(n) = (-1)^n * (A000700(n) - A000700(n-1)), for n > 0. - Vaclav Kotesovec, Jun 06 2018
a(n) ~ (-1)^n * Pi * exp(Pi*sqrt(n/6)) / (2^(13/4) * 3^(3/4) * n^(5/4)). - Vaclav Kotesovec, Jun 06 2018
a(n) = A027188(n+2) - A027194(n+2). - R. J. Mathar, Jun 16 2018

cp's OEIS Frontend

A086372 First differences of A027349.

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

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A246581 G.f.: x^((k^2 + k)/2) / (Product_{i=1..k} (1 - x^i) * Product_{r>=1} (1 + x^r)) with k = 2.

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A116860 Triangle read by rows: T(n,k) is the number of partitions into distinct odd parts with smallest part k (n>=1, k>=1).

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A226541 Number of unimodal compositions of n where the maximal part appears three times.

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A246582 G.f.: x^((k^2+k)/2)/(mul(1-x^i,i=1..k)*mul(1+x^r,r=1..oo)) with k = 3.

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A246583 G.f.: x^((k^2+k)/2)/(mul(1-x^i,i=1..k)*mul(1+x^r,r=1..oo)) with k = 4.

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