

A000070


a(n) = Sum_{k=0..n} p(k) where p(k) = number of partitions of k (A000041).
(Formerly M1054 N0396)


433



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
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OFFSET

0,2


COMMENTS

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 1transitions among all integer partitions of n. A 1transition 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 1transitions are possible: [1113] > [123] and [1113] > [114]. The 1transitions of n form a partial order (poset). For n=6 one has 12 1transitions: [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
Also the total number of 1's among all hooklengths 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 kth 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 kth 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(n1) >= 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 orderpreserving, orderdecreasing and (orderpreserving and orderdecreasing) injective transformation semigroups.  Ugbene Ifeanyichukwu, Jun 03 2015
a(n) is also the number of unlabeled subgraphs of the ncycle 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(nk), 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 vertexdegrees of any multigraph (i.e., nonmultigraphical 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 2n2k into even parts. There are as many unrestricted partitions of nk; 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 (n1)th layer to the nth 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


REFERENCES

H. Gupta, An asymptotic formula in partitions. J. Indian Math. Soc., (N. S.) 10 (1946), 7376.
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 A1, 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.


LINKS



FORMULA

Euler transform of [ 2, 1, 1, 1, 1, 1, 1, ...].
a(n) = (1/n)*Sum_{k=1..n} (sigma(k)+1)*a(nk), n > 1, a(0) = 1.  Vladeta Jovovic, Aug 22 2002
G.f.: (1/(1  x))*Product_{m >= 1} 1/(1  x^m).
Gupta gives the asymptotic result a(n1) ~ 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(n2) = 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(n2) ~ (6/Pi^2)*n*(p(n)  p(n1)) = (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
G.f.: exp(Sum_{k>=1} (sigma_1(k) + 1)*x^k/k).  Ilya Gutkovskiy, Aug 21 2018


EXAMPLE

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 + ...
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

3516 = 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)
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)
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)


MAPLE

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


MATHEMATICA

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 *)


PROG

(PARI) {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 */
(PARI) x='x+O('x^66); Vec(1/((1x)*eta(x))) /* Joerg Arndt, May 15 2011 */
(PARI) a(n) = sum(k=0, n, numbpart(k)); \\ Michel Marcus, Sep 16 2016
(Haskell)
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
(Sage)
p = [number_of_partitions(n) for n in range(leng)]
return [add(p[:k+1]) for k in range(leng)]
(GAP) List([0..45], n>Sum([0..n], k>NrPartitions(k))); # Muniru A Asiru, Jul 25 2018
(Python)
from itertools import accumulate
def A000070iter(n):
L = [0]*n; L[0] = 1
def numpart(n):
S = 0; J = n1; k = 2
while 0 <= J:
T = L[J]
S = S+T if (k//2)%2 else ST
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)
(Python) # 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)]


CROSSREFS

Cf. A014153, A024786, A026794, A026905, A058884, A093694, A133735, A137633, A010815, A027293, A035363, A028310, A000712, A000990.


KEYWORD

nonn,easy,nice


AUTHOR



STATUS

approved



