A060642 -id:A060642 - cp's OEIS frontend

A119443 Distribution of A060642 in Abramowitz and Stegun order.

1, 2, 1, 3, 4, 1, 5, 6, 4, 6, 1, 7, 10, 12, 9, 12, 8, 1, 11, 14, 20, 9, 15, 36, 8, 12, 24, 10, 1, 15, 22, 28, 30, 21, 60, 27, 36, 20, 72, 32, 15, 40, 12, 1, 22, 30, 44, 42, 25, 33, 84, 90, 60, 54, 28, 120, 54, 144, 16, 25, 120, 80, 18, 60, 14, 1

Offset: 1

Alford Arnold, May 22 2006

			1;
2,1;
3,4,1;
5,6,4,6,1;
7,10,12,9,12,8,1;

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A000041 (row lengths), A048996, A119441.

a(n,k) = A119441(n,k) * A048996(n,k)

More terms from R. J. Mathar, Jul 12 2013

A000203 a(n) = sigma(n), the sum of the divisors of n. Also called sigma_1(n).

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84, 144, 68, 126, 96, 144

Offset: 1

N. J. A. Sloane

Multiplicative: If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).
Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (this sequence) (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001
A number n is abundant if sigma(n) > 2n (cf. A005101), perfect if sigma(n) = 2n (cf. A000396), deficient if sigma(n) < 2n (cf. A005100).
a(n) is the number of sublattices of index n in a generic 2-dimensional lattice. - Avi Peretz (njk(AT)netvision.net.il), Jan 29 2001 [In the language of group theory, a(n) is the number of index-n subgroups of Z x Z. - Jianing Song, Nov 05 2022]
The sublattices of index n are in one-to-one correspondence with matrices [a b; 0 d] with a>0, ad=n, b in [0..d-1]. The number of these is Sum_{d|n} d = sigma(n), which is a(n). A sublattice is primitive if gcd(a,b,d) = 1; the number of these is n * Product_{p|n} (1+1/p), which is A001615. [Cf. Grady reference.]
Sum of number of common divisors of n and m, where m runs from 1 to n. - Naohiro Nomoto, Jan 10 2004
a(n) is the cardinality of all extensions over Q_p with degree n in the algebraic closure of Q_p, where p>n. - Volker Schmitt (clamsi(AT)gmx.net), Nov 24 2004. Cf. A100976, A100977, A100978 (p-adic extensions).
Let s(n) = a(n-1) + a(n-2) - a(n-5) - a(n-7) + a(n-12) + a(n-15) - a(n-22) - a(n-26) + ..., then a(n) = s(n) if n is not pentagonal, i.e., n != (3 j^2 +- j)/2 (cf. A001318), and a(n) is instead s(n) - ((-1)^j)*n if n is pentagonal. - Gary W. Adamson, Oct 05 2008 [corrected Apr 27 2012 by William J. Keith based on Ewell and by Andrey Zabolotskiy, Apr 08 2022]
Write n as 2^k * d, where d is odd. Then a(n) is odd if and only if d is a square. - Jon Perry, Nov 08 2012
Also total number of parts in the partitions of n into equal parts. - Omar E. Pol, Jan 16 2013
Note that sigma(3^4) = 11^2. On the other hand, Kanold (1947) shows that the equation sigma(q^(p-1)) = b^p has no solutions b > 2, q prime, p odd prime. - N. J. A. Sloane, Dec 21 2013, based on postings to the Number Theory Mailing List by Vladimir Letsko and Luis H. Gallardo
Limit_{m->infinity} (Sum_{n=1..prime(m)} a(n)) / prime(m)^2 = zeta(2)/2 = Pi^2/12 (A072691). See more at A244583. - Richard R. Forberg, Jan 04 2015
a(n) + A000005(n) is an odd number iff n = 2m^2, m>=1. - Richard R. Forberg, Jan 15 2015
a(n) = a(n+1) for n = 14, 206, 957, 1334, 1364 (A002961). - Zak Seidov, May 03 2016
Equivalent to the Riemann hypothesis: a(n) < H(n) + exp(H(n))*log(H(n)), for all n>1, where H(n) is the n-th harmonic number (Jeffrey Lagarias). See A057641 for more details. - Ilya Gutkovskiy, Jul 05 2016
a(n) is the total number of even parts in the partitions of 2*n into equal parts. More generally, a(n) is the total number of parts congruent to 0 mod k in the partitions of k*n into equal parts (the comment dated Jan 16 2013 is the case for k = 1). - Omar E. Pol, Nov 18 2019
From Jianing Song, Nov 05 2022: (Start)
a(n) is also the number of order-n subgroups of C_n X C_n, where C_n is the cyclic group of order n. Proof: by the correspondence theorem in the group theory, there is a one-to-one correspondence between the order-n subgroups of C_n X C_n = (Z x Z)/(nZ x nZ) and the index-n subgroups of Z x Z containing nZ x nZ. But an index-n normal subgroup of a (multiplicative) group G contains {g^n : n in G} automatically. The desired result follows from the comment from Naohiro Nomoto above.
The number of subgroups of C_n X C_n that are isomorphic to C_n is A001615(n). (End)

			For example, 6 is divisible by 1, 2, 3 and 6, so sigma(6) = 1 + 2 + 3 + 6 = 12.
Let L = <V,W> be a 2-dimensional lattice. The 7 sublattices of index 4 are generated by <4V,W>, <V,4W>, <4V,W+-V>, <2V,2W>, <2V+W,2W>, <2V,2W+V>. Compare A001615.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 116ff.
Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 407.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16, (6), 2nd formula.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, pp. 141, 166.
H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Clarendon Press, Oxford, 2003.
Ross Honsberger, "Mathematical Gems, Number One," The Dolciani Mathematical Expositions, Published and Distributed by The Mathematical Association of America, page 116.
Kanold, Hans Joachim, Kreisteilungspolynome und ungerade vollkommene Zahlen. (German), Ber. Math.-Tagung Tübingen 1946, (1947). pp. 84-87.
M. Krasner, Le nombre des surcorps primitifs d'un degré donné et le nombre des surcorps métagaloisiens d'un degré donné d'un corps de nombres p-adiques. Comptes Rendus Hebdomadaires, Académie des Sciences, Paris 254, 255, 1962.
A. Lubotzky, Counting subgroups of finite index, Proceedings of the St. Andrews/Galway 93 group theory meeting, Th. 2.1. LMS Lecture Notes Series no. 212 Cambridge University Press 1995.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section III.1, page 77.
G. Pólya, Induction and Analogy in Mathematics, vol. 1 of Mathematics and Plausible Reasoning, Princeton Univ Press 1954, page 92.
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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 91, 395.
Robert M. Young, Excursions in Calculus, The Mathematical Association of America, 1992 p. 361.

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 20000 terms from N. J. A. Sloane)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
B. Apostol and L. Petrescu, Extremal Orders of Certain Functions Associated with Regular Integers (mod n), Journal of Integer Sequences, 2013, # 13.7.5.
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
M. Baake and U. Grimm, Quasicrystalline combinatorics
Henry Bottomley, Illustration of initial terms
C. K. Caldwell, The Prime Glossary, sigma function
Imanuel Chen and Michael Z. Spivey, Integral Generalized Binomial Coefficients of Multiplicative Functions, Preprint 2015; Summer Research Paper 238, Univ. Puget Sound.
D. Christopher and T. Nadu, Partitions with Fixed Number of Sizes, Journal of Integer Sequences, 15 (2015), #15.11.5.
J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, 2012. - _N. J. A. Sloane_, Dec 25 2012
R. L. Duncan, Note on the Divisors of a Number, The American Mathematical Monthly, 68(4), 356-359, (1961).
Jason Earls, The Smarandache sum of composites between factors function, in Smarandache Notions Journal (2004), Vol. 14.1, page 243.
L. Euler, Discovery of a most extraordinary law of numbers, relating to the sum of their divisors
L. Euler, Observatio de summis divisorum
L. Euler, An observation on the sums of divisors, arXiv:math/0411587 [math.HO], 2004-2009.
J. A. Ewell, Recurrences for the sum of divisors, Proc. Amer. Math. Soc. 64 (2) 1977.
Farideh Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors ..., J. Integer Seqs., Vol. 6, 2003.
Johan Gielis and Ilia Tavkhelidze, The general case of cutting of GML surfaces and bodies, arXiv:1904.01414 [math.GM], 2019.
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167.
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167. [Annotated scanned copy]
M. J. Grady, A group theoretic approach to a famous partition formula, Amer. Math. Monthly, 112 (No. 7, 2005), 645-651.
Masazumi Honda and Takuya Yoda, String theory, N = 4 SYM and Riemann hypothesis, arXiv:2203.17091 [hep-th], 2022.
Douglas E. Iannucci, On sums of the small divisors of a natural number, arXiv:1910.11835 [math.NT], 2019.
Antti Karttunen, Scheme program for computing this sequence.
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1921), 75-113.
M. Maia and M. Mendez, On the arithmetic product of combinatorial species, arXiv:math.CO/0503436, 2005.
K. Matthews, Factorizing n and calculating phi(n),omega(n),d(n),sigma(n) and mu(n)
Walter Nissen, Abundancy : Some Resources
P. Pollack and C. Pomerance, Some problems of Erdős on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B 3 (2016), 1-26; errata.
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Math. Comp. 83 (2014), 1903-1913.
John S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices, Act. Cryst. (1992) A48, 500-508. - _N. J. A. Sloane_, Mar 14 2009
John S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices II, Acta Cryst. A49 (1993), 293-300. - _N. J. A. Sloane_, Mar 14 2009
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 1]. - _N. J. A. Sloane_, Feb 23 2009
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 3.
Eric Weisstein's World of Mathematics, Divisor Function
Wikipedia, Divisor function
Index entries for sequences related to sublattices
Index entries for sequences related to sigma(n)
Index entries for "core" sequences

See A034885, A002093 for records. Bisections give A008438, A062731. Values taken are listed in A007609. A054973 is an inverse function.
For partial sums see A024916.
Row sums of A127093.
sigma_i (i=0..25): A000005, A000203, A001157, A001158, A001159, A001160, A013954, A013955, A013956, A013957, A013958, A013959, A013960, A013961, A013962, A013963, A013964, A013965, A013966, A013967, A013968, A013969, A013970, A013971, A013972, A281959.
Cf. A144736, A158951, A158902, A174740, A147843, A001158, A001160, A001065, A002192, A001001, A001615 (primitive sublattices), A039653, A088580, A074400, A083728, A006352, A002659, A083238, A000593, A050449, A050452, A051731, A027748, A124010, A069192, A057641, A001318.
Cf. A009194, A082062 (gcd(a(n),n) and its largest prime factor), A179931, A192795 (gcd(a(n),A001157(n)) and largest prime factor).
Cf. also A034448 (sum of unitary divisors).
Cf. A007955 (products of divisors).
Cf. A144613, A144614, A144615, A146076.
A001227, A000593 and this sequence have the same parity: A053866. - Omar E. Pol, May 14 2016
Cf. A001221, A054533.

A000203:=List([1..10^2],n->Sigma(n)); # Muniru A Asiru, Oct 01 2017

a000203 n = product $ zipWith (\p e -> (p^(e+1)-1) `div` (p-1)) (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, May 07 2012

Magma
```
[SumOfDivisors(n): n in [1..70]];
```

[DivisorSigma(1,n): n in [1..70]]; // Bruno Berselli, Sep 09 2015

with(numtheory): A000203 := n->sigma(n); seq(A000203(n), n=1..100);

Table[ DivisorSigma[1, n], {n, 100}]
a[ n_] := SeriesCoefficient[ QPolyGamma[ 1, 1, q] / Log[q]^2, {q, 0, n}]; (* Michael Somos, Apr 25 2013 *)

makelist(divsum(n),n,1,1000); /* Emanuele Munarini, Mar 26 2011 */

numlib::sigma(n)$ n=1..81 // Zerinvary Lajos, May 13 2008

PARI
```
{a(n) = if( n<1, 0, sigma(n))};
```

{a(n) = if( n<1, 0, direuler( p=2, n, 1 / (1 - X) /(1 - p*X))[n])};

{a(n) = if( n<1, 0, polcoeff( sum( k=1, n, x^k / (1 - x^k)^2, x * O(x^n)), n))}; /* Michael Somos, Jan 29 2005 */

max_n = 30; ser = - sum(k=1,max_n,log(1-x^k)); a(n) = polcoeff(ser,n)*n \\ Gottfried Helms, Aug 10 2009

from sympy import divisor_sigma
def a(n): return divisor_sigma(n, 1)
print([a(n) for n in range(1, 71)]) # Michael S. Branicky, Jan 03 2021

from math import prod
from sympy import factorint
def a(n): return prod((p**(e+1)-1)//(p-1) for p, e in factorint(n).items())
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Feb 25 2024
(APL, Dyalog dialect) A000203 ← +/{ð←⍵{(0=⍵|⍺)/⍵}⍳⌊⍵*÷2 ⋄ 1=⍵:ð ⋄ ð,(⍵∘÷)¨(⍵=(⌊⍵*÷2)*2)↓⌽ð} ⍝ Antti Karttunen, Feb 20 2024

[sigma(n, 1) for n in range(1, 71)]  # Zerinvary Lajos, Jun 04 2009

(definec (A000203 n) (if (= 1 n) n (let ((p (A020639 n)) (e (A067029 n))) (* (/ (- (expt p (+ 1 e)) 1) (- p 1)) (A000203 (A028234 n)))))) ;; Uses macro definec from http://oeis.org/wiki/Memoization#Scheme - Antti Karttunen, Nov 25 2017

(define (A000203 n) (let ((r (sqrt n))) (let loop ((i (inexact->exact (floor r))) (s (if (integer? r) (- r) 0))) (cond ((zero? i) s) ((zero? (modulo n i)) (loop (- i 1) (+ s i (/ n i)))) (else (loop (- i 1) s)))))) ;; (Stand-alone program) - Antti Karttunen, Feb 20 2024

Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1). - David W. Wilson, Aug 01 2001
For the following bounds and many others, see Mitrinovic et al. - N. J. A. Sloane, Oct 02 2017
If n is composite, a(n) > n + sqrt(n).
a(n) < n*sqrt(n) for all n.
a(n) < (6/Pi^2)*n^(3/2) for n > 12.
G.f.: -x*deriv(eta(x))/eta(x) where eta(x) = Product_{n>=1} (1-x^n). - Joerg Arndt, Mar 14 2010
L.g.f.: -log(Product_{j>=1} (1-x^j)) = Sum_{n>=1} a(n)/n*x^n. - Joerg Arndt, Feb 04 2011
Dirichlet convolution of phi(n) and tau(n), i.e., a(n) = sum_{d|n} phi(n/d)*tau(d), cf. A000010, A000005.
a(n) is odd iff n is a square or twice a square. - Robert G. Wilson v, Oct 03 2001
a(n) = a(n*prime(n)) - prime(n)*a(n). - Labos Elemer, Aug 14 2003 (Clarified by Omar E. Pol, Apr 27 2016)
a(n) = n*A000041(n) - Sum_{i=1..n-1} a(i)*A000041(n-i). - Jon Perry, Sep 11 2003
a(n) = -A010815(n)*n - Sum_{k=1..n-1} A010815(k)*a(n-k). - Reinhard Zumkeller, Nov 30 2003
a(n) = f(n, 1, 1, 1), where f(n, i, x, s) = if n = 1 then s*x else if p(i)|n then f(n/p(i), i, 1+p(i)*x, s) else f(n, i+1, 1, s*x) with p(i) = i-th prime (A000040). - Reinhard Zumkeller, Nov 17 2004
Recurrence: n^2*(n-1)*a(n) = 12*Sum_{k=1..n-1} (5*k*(n-k) - n^2)*a(k)*a(n-k), if n>1. - Dominique Giard (dominique.giard(AT)gmail.com), Jan 11 2005
G.f.: Sum_{k>0} k * x^k / (1 - x^k) = Sum_{k>0} x^k / (1 - x^k)^2. Dirichlet g.f.: zeta(s)*zeta(s-1). - Michael Somos, Apr 05 2003. See the Hardy-Wright reference, p. 312. first equation, and p. 250, Theorem 290. - Wolfdieter Lang, Dec 09 2016
For odd n, a(n) = A000593(n). For even n, a(n) = A000593(n) + A074400(n/2). - Jonathan Vos Post, Mar 26 2006
Equals the inverse Moebius transform of the natural numbers. Equals row sums of A127093. - Gary W. Adamson, May 20 2007
A127093 * [1/1, 1/2, 1/3, ...] = [1/1, 3/2, 4/3, 7/4, 6/5, 12/6, 8/7, ...]. Row sums of triangle A135539. - Gary W. Adamson, Oct 31 2007
a(n) = A054785(2*n) - A000593(2*n). - Reinhard Zumkeller, Apr 23 2008
a(n) = n*Sum_{k=1..n} A060642(n,k)/k*(-1)^(k+1). - Vladimir Kruchinin, Aug 10 2010
Dirichlet convolution of A037213 and A034448. - R. J. Mathar, Apr 13 2011
G.f.: A(x) = x/(1-x)*(1 - 2*x*(1-x)/(G(0) - 2*x^2 + 2*x)); G(k) = -2*x - 1 - (1+x)*k + (2*k+3)*(x^(k+2)) - x*(k+1)*(k+3)*((-1 + (x^(k+2)))^2)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 06 2011
a(n) = A001065(n) + n. - Mats Granvik, May 20 2012
a(n) = A006128(n) - A220477(n). - Omar E. Pol, Jan 17 2013
a(n) = Sum_{k=1..A003056(n)} (-1)^(k-1)*A196020(n,k). - conjectured by Omar E. Pol, Feb 02 2013, and proved by Max Alekseyev, Nov 17 2013
a(n) = Sum_{k=1..A003056(n)} (-1)^(k-1)*A000330(k)*A000716(n-A000217(k)). - Mircea Merca, Mar 05 2014
a(n) = A240698(n, A000005(n)). - Reinhard Zumkeller, Apr 10 2014
a(n) = Sum_{d^2|n} A001615(n/d^2) = Sum_{d^3|n} A254981(n/d^3). - Álvar Ibeas, Mar 06 2015
a(3*n) = A144613(n). a(3*n + 1) = A144614(n). a(3*n + 2) = A144615(n). - Michael Somos, Jul 19 2015
a(n) = Sum{i=1..n} Sum{j=1..i} cos((2*Pi*n*j)/i). - Michel Lagneau, Oct 14 2015
a(n) = A000593(n) + A146076(n). - Omar E. Pol, Apr 05 2016
a(n) = A065475(n) + A048050(n). - Omar E. Pol, Nov 28 2016
a(n) = (Pi^2*n/6)*Sum_{q>=1} c_q(n)/q^2, with the Ramanujan sums c_q(n) given in A054533 as a c_n(k) table. See the Hardy reference, p. 141, or Hardy-Wright, Theorem 293, p. 251. - Wolfdieter Lang, Jan 06 2017
G.f. also (1 - E_2(q))/24, with the g.f. E_2 of A006352. See e.g., Hardy, p. 166, eq. (10.5.5). - Wolfdieter Lang, Jan 31 2017
From Antti Karttunen, Nov 25 2017: (Start)
a(n) = A048250(n) + A162296(n).
a(n) = A092261(n) * A295294(n). [This can be further expanded, see comment in A291750.] (End)
a(n) = A000593(n) * A038712(n). - Ivan N. Ianakiev and Omar E. Pol, Nov 26 2017
a(n) = Sum_{q=1..n} c_q(n) * floor(n/q), where c_q(n) is the Ramanujan's sum function given in A054533. - Daniel Suteu, Jun 14 2018
a(n) = Sum_{k=1..n} gcd(n, k) / phi(n / gcd(n, k)), where phi(k) is the Euler totient function. - Daniel Suteu, Jun 21 2018
a(n) = (2^(1 + (A000005(n) - A001227(n))/(A000005(n) - A183063(n))) - 1)*A000593(n) = (2^(1 + (A183063(n)/A001227(n))) - 1)*A000593(n). - Omar E. Pol, Nov 03 2018
a(n) = Sum_{i=1..n} tau(gcd(n, i)). - Ridouane Oudra, Oct 15 2019
From Peter Bala, Jan 19 2021: (Start)
G.f.: A(x) = Sum_{n >= 1} x^(n^2)*(x^n + n*(1 - x^(2*n)))/(1 - x^n)^2 - differentiate equation 5 in Arndt w.r.t. x, and set x = 1.
A(x) = F(x) + G(x), where F(x) is the g.f. of A079667 and G(x) is the g.f. of A117004. (End)
a(n) = Sum_{k=1..n} tau(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021
With the convention that a(n) = 0 for n <= 0 we have the recurrence a(n) = t(n) + Sum_{k >= 1} (-1)^(k+1)*(2*k + 1)*a(n - k*(k + 1)/2), where t(n) = (-1)^(m+1)*(2*m+1)*n/3 if n = m*(m + 1)/2, with m positive, is a triangular number else t(n) = 0. For example, n = 10 = (4*5)/2 is a triangular number, t(10) = -30, and so a(10) = -30 + 3*a(9) - 5*a(7) + 7*a(4) = -30 + 39 - 40 + 49 = 18. - Peter Bala, Apr 06 2022
Recurrence: a(p^x) = p*a(p^(x-1)) + 1, if p is prime and for any integer x. E.g., a(5^3) = 5*a(5^2) + 1 = 5*31 + 1 = 156. - Jules Beauchamp, Nov 11 2022
Sum_{n>=1} a(n)/exp(2*Pi*n) = 1/24 - 1/(8*Pi) = A319462. - Vaclav Kotesovec, May 07 2023
a(n) < (7n*A001221(n) + 10*n)/6 [Duncan, 1961] (see Duncan and Tattersall). - Stefano Spezia, Jul 13 2025

A055887 Number of ordered partitions of partitions.

Original entry on oeis.org

1, 1, 3, 8, 22, 59, 160, 431, 1164, 3140, 8474, 22864, 61697, 166476, 449210, 1212113, 3270684, 8825376, 23813776, 64257396, 173387612, 467856828, 1262431711, 3406456212, 9191739970, 24802339472, 66924874539, 180585336876, 487278670744, 1314838220172

Offset: 0

Christian G. Bower, Jun 09 2000

nonn

Jordan matrices are upper bidiagonal matrices such that (A) the diagonal entries are in sorted order, (B) there are only 1's and 0's on the superdiagonal, (C) for each superdiagonal 1, the two diagonal entries to the left and below it must be equal. Let J(N) be the number of N X N Jordan matrices where the diagonal values are, without loss of generality, taken to be a prefix of some fixed strictly increasing sequence x_1, x_2, x_3, ... If Jordan blocks sorted by eigenvalue with ties broken by block size during the sorting, then J(1, 2, 3, ...) is this sequence. - Warren D. Smith, Jan 28 2002
Number of compositions of n into parts k >= 1 where there are A000041(k) sorts of part k. - Joerg Arndt, Sep 30 2012
Also number of chains of multisets that partition a normal multiset of weight n, where a multiset is normal if it spans an initial interval of positive integers. - Gus Wiseman, Oct 28 2015
From Gus Wiseman, Jul 31 2022: (Start)
Also the number of ways to choose a multiset partition into constant multisets of a multiset of length n covering an initial interval of positive integers. This interpretation involves only multisets, not sequences. For example, the a(1) = 1 through a(3) = 8 multiset partitions are:
  {{1}}  {{1,1}}    {{1,1,1}}
         {{1},{1}}  {{1},{1,1}}
         {{1},{2}}  {{1},{2,2}}
                    {{2},{1,1}}
                    {{1},{1},{1}}
                    {{1},{1},{2}}
                    {{1},{2},{2}}
                    {{1},{2},{3}}
Factorizations into prime powers, are counted by A000688.
The strongly normal case is A063834.
The strongly normal strict case is A270995.
Twice-partitions of type PPR are counted by A279784, factorizations A295935.
The strict case is A304969.
(End)

			The a(4) = 22 chains of multisets, where notation x-y means "y is a submultiset of x", are: (o-o-o-o) (oo-o-o) (oo-oo) (ooo-o) (oooo) (oe-o-o) (ooe-o) (oooe) (oe-oe) (ooe-e) (oee-o) (ooee) (oei-o) (ooei) (oe-e-e) (oee-e) (oeee) (oei-e) (oeei) (oei-i) (oeii) (oeis).
From _Gus Wiseman_, Jul 31 2022: (Start)
a(n) is the number of ways to choose an integer partition of each part of an integer composition of n. The a(0) = 1 through a(3) = 8 choices are:
  ()  ((1))  ((2))     ((3))
             ((11))    ((21))
             ((1)(1))  ((111))
                       ((1)(2))
                       ((2)(1))
                       ((1)(11))
                       ((11)(1))
                       ((1)(1)(1))
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..2320
N. J. A. Sloane, Transforms
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89, DOI:10.1007/s11083-004-9460-9.
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.

Cf. A055888, A083355, A095975, A246828.
Row sums of A060642.
Cf. A326346.
The unordered version is A001970, row-sums of A061260.
A000041 counts integer partitions, strict A000009.
A011782 counts integer compositions.
A072233 counts partitions by sum and length.
Cf. A001055, A050361, A063834, A279784, A355743.

with(combstruct); SeqSetSetU := [T, {T=Sequence(S), S=Set(U,card >= 1), U=Set(Z,card >=1)},unlabeled];
P := (x) -> product( 1/(1-x^k), k=1..20 ) - 1; F := (x) -> series( 1/(1-P(x)) - 1, x, 21 ); # F(x) is g.f. for this sequence # Warren D. Smith, Jan 28 2002
A055887rec:= proc(n::integer) local k; option remember; with(combinat): if n = 0 then 1 else add(numbpart(k) *procname(n - k), k=1..n); end if; end proc: seq (A055887rec(n), n=0..10); # Thomas Wieder, Nov 26 2007

a = 1/Product[(1 - x^k), {k, 1, \[Infinity]}] - 1; CoefficientList[Series[1/(1 - a), {x, 0, 20}], x] (* Geoffrey Critzer, Dec 23 2010 *)
(1/(2 - 1/QPochhammer[x]) + O[x]^30)[[3]] (* Vladimir Reshetnikov, Sep 22 2016 *)
Table[Sum[Times@@PartitionsP/@c,{c,Join@@Permutations/@IntegerPartitions[n]}],{n,0,10}] (* Gus Wiseman, Jul 31 2022 *)

Vec(1/(2-1/eta(x+O(x^66)))) \\ Joerg Arndt, Sep 30 2012

Invert transform of partitions numbers A000041.
Let p(k) be the number of integer partitions of k. Furthermore, set a(0)=1. Then a(n) = Sum_{k=1..n} p(k)*a(n-k). - Thomas Wieder, Nov 26 2007
G.f.: 1/( 1 - Sum_{k>=1} p(k)*x^k ) where p(k) = A000041(k) is the number of integer partitions of k. - Joerg Arndt, Sep 30 2012
a(n) ~ c * d^n, where d = 2.698329106474211231263998666188376330713465125913986356769... (see A246828) and c = 0.414113793172792357745578049739573823627306487211379286647... - Vaclav Kotesovec, Mar 29 2014
G.f.: 1/(2 - 1/(x)inf), where (x)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Sep 22 2016

A144064 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->k).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 3, 0, 1, 4, 9, 10, 5, 0, 1, 5, 14, 22, 20, 7, 0, 1, 6, 20, 40, 51, 36, 11, 0, 1, 7, 27, 65, 105, 108, 65, 15, 0, 1, 8, 35, 98, 190, 252, 221, 110, 22, 0, 1, 9, 44, 140, 315, 506, 574, 429, 185, 30, 0, 1, 10, 54, 192, 490, 918, 1265, 1240, 810, 300, 42, 0

Offset: 0

Alois P. Heinz, Sep 09 2008

A(n,k) is also the number of partitions of n into parts of k kinds.
In general, column k > 0 is asymptotic to k^((k+1)/4) * exp(Pi*sqrt(2*k*n/3)) / (2^((3*k+5)/4) * 3^((k+1)/4) * n^((k+3)/4)) * (1 - (Pi*k^(3/2)/(24*sqrt(6)) + sqrt(3)*(k+1)*(k+3)/(8*Pi*sqrt(2*k))) / sqrt(n)). - Vaclav Kotesovec, Feb 28 2015, extended Jan 16 2017
When k is a prime power greater than 1, A(n,k) is the number of conjugacy classes of n X n matrices over a field with k elements that contain an upper-triangular matrix. - Geoffrey Critzer, Nov 11 2022

			Square array begins:
  1,   1,   1,   1,   1,   1, ...
  0,   1,   2,   3,   4,   5, ...
  0,   2,   5,   9,  14,  20, ...
  0,   3,  10,  22,  40,  65, ...
  0,   5,  20,  51, 105, 190, ...
  0,   7,  36, 108, 252, 506, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
G. E. Bergum and V. E. Hoggatt, Jr., Numerator polynomial coefficient array for the convolved Fibonacci sequence, Fib. Quart., 14 (1976), 43-44. (Annotated scanned copy)
G. E. Bergum and V. E. Hoggatt, Jr., Numerator polynomial coefficient array for the convolved Fibonacci sequence, Fib. Quart., 14 (1976), 43-48. See Table 1.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 8.
N. J. A. Sloane, Transforms
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Columns k=0-24 give: A000007, A000041, A000712, A000716, A023003, A023004, A023005, A023006, A023007, A023008, A023009, A023010, A005758, A023011, A023012, A023013, A023014, A023015, A023016, A023017, A023018, A023019, A023020, A023021, A006922.
Cf. A082556 (k=30), A082557 (k=32), A082558 (k=48), A082559 (k=64).
Rows n=0-4 give: A000012, A001477, A000096, A006503, A006504.
Main diagonal gives A008485.
Antidiagonal sums give A067687.
Cf. A060642, A122768, A246935, A255961, A261718.

# DedekindEta is defined in A000594.
A144064Column(k, len) = DedekindEta(len, -k)
for n in 0:8 A144064Column(n, 6) |> println end # Peter Luschny, Mar 10 2018

with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: A:= (n,k)-> etr(j->k)(n): seq(seq(A(n, d-n), n=0..d), d=0..14);

a[0, ] = 1; a[, 0] = 0; a[n_, k_] := SeriesCoefficient[ Product[1/(1 - x^j)^k, {j, 1, n}], {x, 0, n}]; Table[a[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 06 2013 *)
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[n_, k_] := etr[k&][n]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Mar 30 2015, after Alois P. Heinz *)

Mat(apply( {A144064_col(k,nMax=9)=Col(1/eta('x+O('x^nMax))^k,nMax)}, [0..9])) \\ M. F. Hasler, Aug 04 2024

G.f. of column k: Product_{j>=1} 1/(1-x^j)^k.

A(n,k) = Sum_{i=0..k} binomial(k,i) * A060642(n,k-i):

A048574 Self-convolution of 1 2 3 5 7 11 15 22 30 42 56 77 ... (A000041).

Original entry on oeis.org

1, 4, 10, 22, 43, 80, 141, 240, 397, 640, 1011, 1568, 2395, 3604, 5360, 7876, 11460, 16510, 23588, 33418, 47006, 65640, 91085, 125596, 172215, 234820, 318579, 430060, 577920, 773130, 1030007, 1366644, 1806445, 2378892, 3121835, 4082796

Offset: 2

Alford Arnold

Number of proper partitions of n into parts of two kinds (i.e. both kinds must be present). - Franklin T. Adams-Watters, Feb 08 2006

			a(4) = 22 because (1,2,3,5)*(5,3,2,1) = 5 + 6 + 6 + 5 = 22

Reinhard Zumkeller, Table of n, a(n) for n = 2..5000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 804

Cf. A000041, A000712, A023626.
Essentially the same as A052837.
Cf. A122768.
Column k=2 of A060642.

a048574 n = a048574_list !! (n-2)
a048574_list = f (drop 2 a000041_list) [1] where
f (p:ps) rs = (sum $ zipWith (*) rs $ tail a000041_list) : f ps (p : rs)
-- Reinhard Zumkeller, Nov 09 2015

spec := [S,{C=Sequence(Z,1 <= card),B=Set(C,1 <= card),S=Prod(B,B)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20); # Franklin T. Adams-Watters, Feb 08 2006
# second Maple program:
a:= n-> (p-> add(p(j)*p(n-j), j=1..n-1))(combinat[numbpart]):
seq(a(n), n=2..40);  # Alois P. Heinz, May 26 2018

a[n_] := First[ ListConvolve[ pp = Array[ PartitionsP, n], pp]]; Table[ a[n], {n, 1, 36}] (* Jean-François Alcover, Oct 21 2011 *)
Table[ListConvolve[PartitionsP[Range[n]],PartitionsP[Range[n]]],{n,40}]// Flatten (* Harvey P. Dale, Oct 29 2020 *)

a(n) = sum(k=1, n-1, numbpart(k)*numbpart(n-k)); \\ Michel Marcus, Dec 11 2016

From Franklin T. Adams-Watters, Feb 08 2006: (Start)
a(0) = 0, a(n) = A000712(n)-2*A000041(n) for n>0.
a(n) = Sum_{k=1..n-1} A000041(k)*A000041(n-k).
G.f.: ((Product_{k>0} 1/(1-x^k))-1)^2 = (exp(Sum_{k>0} (x^k/(1-x^k)/k))-1)^2. (End)
a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*3^(3/4)*n^(5/4)). - Vaclav Kotesovec, Mar 10 2018

More terms from Larry Reeves (larryr(AT)acm.org), Sep 29 2000

A261719 Number T(n,k) of partitions of n where each part i is marked with a word of length i over a k-ary alphabet whose letters appear in alphabetical order and all k letters occur at least once in the partition; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 3, 12, 10, 0, 5, 40, 81, 47, 0, 7, 104, 396, 544, 246, 0, 11, 279, 1751, 4232, 4350, 1602, 0, 15, 654, 6528, 25100, 44475, 36744, 11481, 0, 22, 1577, 23892, 136516, 369675, 512787, 352793, 95503, 0, 30, 3560, 80979, 666800, 2603670, 5413842, 6170486, 3641992, 871030

Offset: 0

Alois P. Heinz, Aug 29 2015

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k<=n. T(n,k) = 0 for k>n.

			A(3,2) = 12: 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 1a1a1b, 1a1b1a, 1a1b1b, 1b1a1a, 1b1a1b, 1b1b1a.
Triangle T(n,k) begins:
  1
  0,  1;
  0,  2,    3;
  0,  3,   12,    10;
  0,  5,   40,    81,     47;
  0,  7,  104,   396,    544,    246;
  0, 11,  279,  1751,   4232,   4350,   1602;
  0, 15,  654,  6528,  25100,  44475,  36744,  11481;
  0, 22, 1577, 23892, 136516, 369675, 512787, 352793, 95503;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000007, A000041 (for n>0), A293366, A293367, A293368, A293369, A293370, A293371, A293372, A293373, A293374.
Row sums give A035341.
Main diagonal gives A005651.
T(2n,n) gives A261732.
Cf. A060642, A261718, A261781 (same for compositions).

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k)+`if`(i>n, 0, b(n-i, i, k)*binomial(i+k-1, k-1))))
    end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, b[n - i, i, k]*Binomial[i + k - 1, k - 1]]]]; T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 21 2017, translated from Maple *)

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A261718(n,k-i).

A308680 Number T(n,k) of colored integer partitions of n such that all colors from a k-set are used and parts differ by size or by color; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 5, 3, 1, 0, 3, 8, 9, 4, 1, 0, 4, 14, 19, 14, 5, 1, 0, 5, 22, 39, 36, 20, 6, 1, 0, 6, 34, 72, 85, 60, 27, 7, 1, 0, 8, 50, 128, 180, 160, 92, 35, 8, 1, 0, 10, 73, 216, 360, 381, 273, 133, 44, 9, 1, 0, 12, 104, 354, 680, 845, 720, 434, 184, 54, 10, 1

Offset: 0

Alois P. Heinz, Aug 29 2019

For fixed k > 0, T(n,k) ~ exp(Pi*sqrt(k*n/3)) * k^(1/4) / (3^(1/4) * 2^((k+3)/2) * n^(3/4)). - Vaclav Kotesovec, Sep 16 2019

T is the convolution triangle of A000009 (see A357368). - Peter Luschny, Oct 19 2022

			T(4,1) = 2: 3a1a, 4a.
T(4,2) = 5: 2a1a1b, 2b1a1b, 2a2b, 3a1b, 3b1a.
T(4,3) = 3: 2a1b1c, 2b1a1c, 2c1a1b.
T(4,4) = 1: 1a1b1c1d.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,  1;
  0,  2,  2,   1;
  0,  2,  5,   3,   1;
  0,  3,  8,   9,   4,   1;
  0,  4, 14,  19,  14,   5,   1;
  0,  5, 22,  39,  36,  20,   6,   1;
  0,  6, 34,  72,  85,  60,  27,   7,  1;
  0,  8, 50, 128, 180, 160,  92,  35,  8, 1;
  0, 10, 73, 216, 360, 381, 273, 133, 44, 9, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
Wikipedia, Partition (number theory)

Columns k=0-10 give: A000007, A000009 (for n>0), A327380, A327381, A327382, A327383, A327384, A327385, A327386, A327387, A327388.
Main diagonal and lower diagonals give: A000012, A001477, A000096.
Row sums give A304969.
T(2n,n) gives A324595.
Cf. A060642, A286335, A325915.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
      b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..min(k, n/i))))
    end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
T:= proc(n, k) option remember;
      `if`(k=0, `if`(n=0, 1, 0), `if`(k=1, `if`(n=0, 0, b(n)),
          (q-> add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Jan 31 2021
# Uses function PMatrix from A357368.
PMatrix(10, A000009); # Peter Luschny, Oct 19 2022

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t,    b[t, Min[t, i - 1], k]*Binomial[k, j]][n - i*j], {j, 0, Min[k, n/i]}]]];
T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 06 2019, from Maple *)

T(n,k) = Sum_{i=0..k} (-1)^i * binomial(k,i) * A286335(n,k-i).
Sum_{k=1..n} k * T(n,k) = A325915(n).
G.f. of column k: (-1 + Product_{j>=1} (1 + x^j))^k. - Alois P. Heinz, Jan 29 2021

A341221 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^3.

Original entry on oeis.org

1, 6, 21, 59, 144, 321, 669, 1323, 2511, 4604, 8202, 14253, 24241, 40449, 66363, 107234, 170910, 269004, 418566, 644436, 982536, 1484482, 2223942, 3305484, 4876620, 7144455, 10398123, 15039564, 21624678, 30919323, 43973708, 62222844, 87619212, 122810585

Offset: 3

Ilya Gutkovskiy, Feb 07 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..10000

Column k=3 of A060642.

Cf. A000041, A000716, A048574, A327381, A341222, A341223, A341225, A341226, A341227, A341228.

b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, combinat[
      numbpart](n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 3):
seq(a(n), n=3..36);  # Alois P. Heinz, Feb 07 2021

nmax = 36; CoefficientList[Series[(-1 + Product[1/(1 - x^k), {k, 1, nmax}])^3, {x, 0, nmax}], x] // Drop[#, 3] &

a(n) ~ A000716(n). - Vaclav Kotesovec, Feb 20 2021

A341222 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^4.

Original entry on oeis.org

1, 8, 36, 124, 362, 944, 2266, 5100, 10903, 22340, 44168, 84692, 158137, 288452, 515344, 903740, 1558465, 2646820, 4432964, 7329916, 11977507, 19358524, 30970444, 49077936, 77081679, 120054268, 185514428, 284540060, 433360308, 655622392, 985604644, 1472751228

Offset: 4

Ilya Gutkovskiy, Feb 07 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..10000

Column k=4 of A060642.

Cf. A000041, A023003, A048574, A327382, A341221, A341223, A341225, A341226, A341227, A341228.

b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, combinat[
      numbpart](n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 4):
seq(a(n), n=4..35);  # Alois P. Heinz, Feb 07 2021

nmax = 35; CoefficientList[Series[(-1 + Product[1/(1 - x^k), {k, 1, nmax}])^4, {x, 0, nmax}], x] // Drop[#, 4] &

a(n) ~ A023003(n). - Vaclav Kotesovec, Feb 20 2021

A341223 Expansion of (-1 + Product_{k>=1} 1 / (1 - x^k))^5.

Original entry on oeis.org

1, 10, 55, 225, 765, 2287, 6215, 15680, 37265, 84300, 182933, 383070, 777705, 1536490, 2963120, 5592060, 10349465, 18817760, 33665870, 59341785, 103176877, 177131330, 300530125, 504318530, 837632700, 1377874861, 2246061540, 3630059510, 5819556060, 9258393655, 14622472250

Offset: 5

Ilya Gutkovskiy, Feb 07 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..10000

Column k=5 of A060642.

Cf. A000041, A023004, A048574, A327383, A341221, A341222, A341225, A341226, A341227, A341228.

b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, combinat[
      numbpart](n)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 5):
seq(a(n), n=5..35);  # Alois P. Heinz, Feb 07 2021

nmax = 35; CoefficientList[Series[(-1 + Product[1/(1 - x^k), {k, 1, nmax}])^5, {x, 0, nmax}], x] // Drop[#, 5] &

cp's OEIS Frontend

A119443 Distribution of A060642 in Abramowitz and Stegun order.

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A000203 a(n) = sigma(n), the sum of the divisors of n. Also called sigma_1(n).

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A055887 Number of ordered partitions of partitions.

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A144064 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->k).

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A048574 Self-convolution of 1 2 3 5 7 11 15 22 30 42 56 77 ... (A000041).

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