A166861 -id:A166861 - cp's OEIS frontend

A022567 Expansion of Product_{m>=1} (1+x^m)^2.

1, 2, 3, 6, 9, 14, 22, 32, 46, 66, 93, 128, 176, 238, 319, 426, 562, 736, 960, 1242, 1598, 2048, 2608, 3306, 4175, 5248, 6570, 8198, 10190, 12622, 15589, 19190, 23552, 28830, 35190, 42842, 52034, 63040, 76198, 91904, 110604, 132832, 159216, 190464, 227417

Offset: 0

N. J. A. Sloane, Jun 14 1998

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number of partitions of n into distinct parts, with 2 types of each part. E.g., for n=4, we consider k and k* to be different versions of k and so we have 4, 4*, 31, 31*, 3*1, 3*1*, 22*, 211*, 2*11*, thus a(4)=9. - Jon Perry, Apr 04 2004
Number of partitions of n into odd parts, each part being of two kinds. E.g., a(3)=6 because we have 3, 3', 1+1+1, 1+1+1', 1+1'+1', 1'+1'+1'. - Emeric Deutsch, Mar 22 2005
Euler transform of period 2 sequence [2,0,2,0,...]. - Emeric Deutsch, Mar 22 2005
Equals A000041 convolved with A010054. - Gary W. Adamson, Jun 11 2009
The sum of the least gaps in all partitions of n. The "least gap" of a partition is the least positive integer that is not a part of the partition. Example: a(4) = 9 because the least gaps in [4], [3,1], [2,2], [2,1,1], and [1,1,1,1] are 1, 2, 1, 3, and 2, respectively. - Emeric Deutsch, May 18 2015
Number of 2-regular bipartitions of n. - N. J. A. Sloane, Oct 20 2019
The least gap is also known as the minimal excludant or mex; see Andrews and Newman. - George Beck, Dec 10 2020

			G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 9*x^4 + 14*x^5 + 22*x^6 + 32*x^7 + 46*x^8 + ...
G.f. = q + 2*q^13 + 3*q^25 + 6*q^37 + 9*q^49 + 14*q^61 + 22*q^73 + 32*q^85 + ...

P. J. Grabner, A. Knopfmacher, Analysis of some new partition statistics, Ramanujan J., 12, 2006, 439-454.
Kathiravan, T., and S. N. Fathima. "On L-regular bipartitions modulo L." The Ramanujan Journal 44.3 (2017): 549-558.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
George E. Andrews, David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
Cristina Ballantine, Mircea Merca, Bisected theta series, least r-gaps in partitions, and polygonal numbers, arXiv:1710.05960 [math.CO], 2017.
Helena Bergold, Lukas Egeling, and Hung. P. Hoang, Signotopes with few plus signs, arXiv:2411.19208 [math.CO], 2024. See p. 14.
J. Currie, N. Rampersad, Binary words avoiding xx^Rx and strongly unimodal sequences, JIS 18 (2015) #15.10.3.
Alejandro Erickson, Frank Ruskey, Enumerating maximal tatami mat coverings of square grids with v vertical dominoes, arXiv:1304.0070 [math.CO], 2013.
Alejandro Erickson and Mark Schurch, Monomer-dimer tatami tilings of square regions, arXiv preprint arXiv:1110.5103 [math.CO], 2011.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 852
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.
Mircea Merca, A new look on the generating function for the number of divisors, Journal of Number Theory, Volume 149, April 2015, Pages 57-69.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75. See q(n)'.
Mbavhalelo Mulokwe and Konstantinos Zoubos, Free fermions, neutrality and modular transformations, arXiv:2403.08531 [hep-th], 2024.
Michael Somos, Introduction to Ramanujan theta functions
Jacob Sprittulla, On Colored Factorizations, arXiv:2008.09984 [math.CO], 2020.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000009, A022597, A285221, A304048.
Cf. A010054. - Gary W. Adamson, Jun 11 2009
Column k=2 of A286335.
Number of r-regular bipartitions of n for r = 2,3,4,5,6: A022567, A328547, A001936, A263002, A328548.

Coefficients(&*[(1+x^m)^2:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 26 2018

A022567 := proc(n)
    local x,m;
    product((1+x^m)^2,m=1..n) ;
    expand(%) ;
    coeff(%,x,n) ;
end proc: # R. J. Mathar, Jun 18 2016

a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2]^-2, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ Product[ 1 + q^k, {k, n}]^2, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
(QPochhammer[-1, x]^2/4 + O[x]^30)[[3]] (* Vladimir Reshetnikov, Sep 22 2016 *)
nmax = 50; poly = ConstantArray[0, nmax+1]; poly[[1]] = 1; poly[[2]] = 2; poly[[3]] = 1; Do[Do[Do[poly[[j+1]] += poly[[j-k+1]], {j, nmax, k, -1}]; , {p, 1, 2}], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 14 2017 *)

{a(n) = if( n<0, 0, polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n))^2, n))}; /* Michael Somos, Mar 21 2004 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A))^2, n))}; /* Michael Somos, Jun 03 2005 */

# uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, 1, 0, 2)
a = EulerTransform(b)
print([a(n) for n in range(45)]) # Peter Luschny, Nov 11 2020

a(n) = p(n)+p(n-1)+p(n-3)+p(n-6)+...+p(n-k*(k+1)/2)+..., where p() is A000041(). E.g. a(8) = p(8)+p(7)+p(5)+p(2) = 22+15+7+2 = 46. - Vladeta Jovovic, Aug 09 2004
Expansion of q^(-1/12) * (eta(q^2) / eta(q))^2 in powers of q. - Michael Somos, Apr 27 2008
Expansion of chi(-q)^(-2) in powers of q where chi() is a Ramanujan theta function. - Michael Somos, Apr 27 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A022597. - Michael Somos, Apr 27 2008
G.f.: Product_{k>0} (1 + x^k)^2.
Convolution square of A000009. Convolution inverse of A022597. - Michael Somos, Apr 27 2008
Parity result: a(n) is even except when n is twice a generalized pentagonal number (i.e., of the form 2*A001318(m) for some m). - Peter Bala, Mar 19 2009
a(n) ~ exp(Pi * sqrt(2*n/3)) / (4 * 6^(1/4) * n^(3/4)) * (1 + (Pi/(12*sqrt(6)) - 3*sqrt(3/2)/(8*Pi)) / sqrt(n) + (Pi^2/1728 - 45/(256*Pi^2) - 5/64)/n). - Vaclav Kotesovec, Mar 05 2015, extended Jan 22 2017
a(0) = 1, a(n) = (2/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017
G.f.: exp(2*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

A073592 Euler transform of negative integers.

Original entry on oeis.org

1, -1, -2, -1, 0, 4, 4, 7, 3, -2, -9, -17, -25, -24, -13, -1, 32, 61, 97, 111, 112, 74, 8, -108, -243, -392, -512, -569, -542, -358, -33, 473, 1078, 1788, 2395, 2865, 2955, 2569, 1496, -245, -2751, -5783, -9121, -12299, -14739, -15806, -14719, -10930, -3813, 6593, 20284, 36139, 53081, 68620, 80539

Offset: 0

Vladeta Jovovic, Aug 28 2002

sign

1/A(x) is g.f. for A000219.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vaclav Kotesovec)
E. M. Wright, Coefficients of a reciprocal generating function, Quart. J. Math. 17 (1) (1966) 39-43, ADS Abstracts.
N. J. A. Sloane, Transforms

Column k=1 of A283272.

Cf. A000219, A001157, A001478, A026007, A156616, A255528.

a:= proc(n) option remember; `if`(n=0, 1, -add(
      numtheory[sigma][2](j)*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 12 2015

nmax=50; CoefficientList[Series[Exp[Sum[-x^k/(k*(1-x^k)^2),{k,1,nmax}]],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 02 2015 *)
a[n_]:= a[n] = -1/n*Sum[DivisorSigma[2,k]*a[n-k],{k,1,n}]; a[0]=1; Table[a[n],{n,0,100}] (* Vaclav Kotesovec, Mar 02 2015 *)

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: -n)
print([b(n) for n in range(55)]) # Peter Luschny, Nov 11 2020

G.f.: Product_{k>0} (1-x^k)^k.
a(n) = -1/n*Sum_{k=1..n} sigma[2](k)*a(n-k).
G.f.: exp( Sum_{n>=1} -sigma_2(n)*x^n/n ). - Seiichi Manyama, Mar 04 2017

A000335 Euler transform of A000292.

Original entry on oeis.org

1, 5, 15, 45, 120, 331, 855, 2214, 5545, 13741, 33362, 80091, 189339, 442799, 1023192, 2340904, 5302061, 11902618, 26488454, 58479965, 128120214, 278680698, 602009786, 1292027222, 2755684669, 5842618668, 12317175320, 25825429276, 53865355154, 111786084504, 230867856903, 474585792077, 971209629993

Offset: 1

N. J. A. Sloane

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..500
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100.
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy]
Srivatsan Balakrishnan, Suresh Govindarajan and Naveen S. Prabhakar, On the asymptotics of higher-dimensional partitions, arXiv:1105.6231 [cond-mat.stat-mech], 2011, p. 20.
N. J. A. Sloane, Transforms

Cf. A000041, A000219, A000294, A000391, A000417, A000428, A255965.

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> binomial(n+2,3)): seq(a(n), n=1..26); # Alois P. Heinz, Sep 08 2008

max = 33; f[x_] := Exp[ Sum[ x^k/(1-x^k)^4/k, {k, 1, max}]]; Drop[ CoefficientList[ Series[ f[x], {x, 0, max}], x], 1](* Jean-François Alcover, Nov 21 2011, after Joerg Arndt *)
nmax=50; Rest[CoefficientList[Series[Product[1/(1-x^k)^(k*(k+1)*(k+2)/6),{k,1,nmax}],{x,0,nmax}],x]] (* Vaclav Kotesovec, Mar 11 2015 *)
etr[p_] := Module[{b}, b[n_] := b[n] = If[n==0, 1, Sum[DivisorSum[j, #*p[#] &]*b[n-j], {j, 1, n}]/n]; b]; a = etr[Binomial[#+2, 3]&]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Nov 24 2015, after Alois P. Heinz *)

a(n)=if(n<1, 0, polcoeff(exp(sum(k=1, n, x^k/(1-x^k)^4/k, x*O(x^n))), n)) /* Joerg Arndt, Apr 16 2010 */

N=66; x='x+O('x^66); gf=-1 + exp(sum(k=1, N, x^k/(1-x^k)^4/k)); Vec(gf) /* Joerg Arndt, Jul 06 2011 */

# uses[EulerTransform from A166861] and prepends a(0) = 1.
a = EulerTransform(lambda n: n*(n+1)*(n+2)//6)
print([a(n) for n in range(33)]) # Peter Luschny, Nov 17 2022

a(n) ~ Zeta(5)^(379/3600) / (2^(521/1800) * sqrt(5*Pi) * n^(2179/3600)) * exp(Zeta'(-1)/3 - Zeta(3) / (8*Pi^2) - Pi^16 / (3110400000 * Zeta(5)^3) + Pi^8*Zeta(3) / (216000 * Zeta(5)^2) - Zeta(3)^2 / (90*Zeta(5)) + Zeta'(-3)/6 + (Pi^12 / (10800000 * 2^(2/5) * Zeta(5)^(11/5)) - Pi^4 * Zeta(3) / (900 * 2^(2/5) * Zeta(5)^(6/5))) * n^(1/5) + (Zeta(3) / (3 * 2^(4/5) * Zeta(5)^(2/5)) - Pi^8 / (36000 * 2^(4/5) * Zeta(5)^(7/5))) * n^(2/5) + Pi^4 / (180 * 2^(1/5) * Zeta(5)^(3/5)) * n^(3/5) + 5*Zeta(5)^(1/5) / 2^(8/5) * n^(4/5)). - Vaclav Kotesovec, Mar 12 2015

A038348 Expansion of (1/(1-x^2))*Product_{m>=0} 1/(1-x^(2m+1)).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 11, 14, 19, 24, 31, 39, 49, 61, 76, 93, 114, 139, 168, 203, 244, 292, 348, 414, 490, 579, 682, 801, 938, 1097, 1278, 1487, 1726, 1999, 2311, 2667, 3071, 3531, 4053, 4644, 5313, 6070, 6923, 7886, 8971, 10190, 11561

Offset: 0

N. J. A. Sloane

nonn

Number of partitions of n+2 with exactly one even part. - Vladeta Jovovic, Sep 10 2003
Also, number of partitions of n with at most one even part. - Vladeta Jovovic, Sep 10 2003
Also total number of parts, counted without multiplicity, in all partitions of n into odd parts, offset 1. - Vladeta Jovovic, Mar 27 2005
a(n) = Sum_{k>=1} k*A116674(n+1,k). - Emeric Deutsch, Feb 22 2006
Equals row sums of triangle A173305. - Gary W. Adamson, Feb 15 2010
Equals partial sums of A025147 (observed by Jonathan Vos Post, proved by several correspondents).
Conjecture: The n-th derivative of Gamma(x+1) at x = 0 has a(n+1) terms. For example, d^4/dx^4_(x = 0) Gamma(x+1) = 8*eulergamma*zeta(3) + eulergamma^4 + eulergamma^2*Pi^2 + 3*Pi^4/20 which has a(5) = 4 terms. - David Ulgenes, Dec 05 2023

			From _Gus Wiseman_, Sep 23 2019: (Start)
Also the number of integer partitions of n that are strict except possibly for any number of 1's. For example, the a(1) = 1 through a(7) = 11 partitions are:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (31)    (32)     (42)      (43)
             (111)  (211)   (41)     (51)      (52)
                    (1111)  (311)    (321)     (61)
                            (2111)   (411)     (421)
                            (11111)  (3111)    (511)
                                     (21111)   (3211)
                                     (111111)  (4111)
                                               (31111)
                                               (211111)
                                               (1111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Cristina Ballantine and Mircea Merca, On identities of Watson type, Ars Mathematica Contemporanea (2019) Vol. 17, 277-290.
Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.
P. Flajolet and B. Salvy, Euler sums and contour integral representations, Experimental Mathematics, Vol. 7 Issue 1 (1998).
J. Fulman, 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. 2, with k=1. - _N. J. A. Sloane_, Aug 31 2014
Rebekah Ann Gilbert, A Fine Rediscovery, 2014.
Amrik Singh Nimbran and Paul Levrie, Series of the form Sum {a_n*binomial(2n, n)}, Math. Student (2023) Vol. 92, Nos. 3-4, 155-173. See pp. 10, 16.

Cf. A067588, A090867, A116674, A173305.

Cf. A000009, A007360, A051424, A259936, A266591, A302569, A306200.

f:=1/(1-x^2)/product(1-x^(2*j-1),j=1..32): fser:=series(f,x=0,62): seq(coeff(fser,x,n),n=0..58); # Emeric Deutsch, Feb 22 2006

mmax = 47; CoefficientList[ Series[ (1/(1-x^2))*Product[1/(1-x^(2m+1)), {m, 0, mmax}], {x, 0, mmax}], x] (* Jean-François Alcover, Jun 21 2011 *)

# uses[EulerTransform from A166861]
def g(n): return n % 2 if n > 2 else 1
a = EulerTransform(g)
print([a(n) for n in range(48)]) # Peter Luschny, Dec 04 2020

a(n) = A036469(n) - a(n-1) = Sum_{k=0..n} (-1)^k*A036469(n-k). - Vladeta Jovovic, Sep 10 2003
a(n) = A000009(n) + a(n-2). - Vladeta Jovovic, Feb 10 2004
G.f.: 1/((1-x^2)*Product_{j>=1} (1 - x^(2*j-1))). - Emeric Deutsch, Feb 22 2006
From Vaclav Kotesovec, Aug 16 2015: (Start)
a(n) ~ (1/2) * A036469(n).
a(n) ~ 3^(1/4) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)). (End)
Euler transform of the sequence [1, 1, period(1, 0)] (A266591). - Georg Fischer, Dec 04 2020

A023871 Expansion of Product_{k>=1} (1 - x^k)^(-k^2).

Original entry on oeis.org

1, 1, 5, 14, 40, 101, 266, 649, 1593, 3765, 8813, 20168, 45649, 101591, 223654, 486046, 1045541, 2225167, 4692421, 9804734, 20318249, 41766843, 85218989, 172628766, 347338117, 694330731, 1379437080, 2724353422, 5350185097, 10449901555, 20304465729, 39254599832

Offset: 0

Olivier Gérard

In general, if g.f. = Product_{k>=1} 1/(1 - x^k)^(c2*k^2 + c1*k + c0) and c2 > 0, then a(n) ~ exp(4*Pi * c2^(1/4) * n^(3/4) / (3*15^(1/4)) + c1*Zeta(3) / Pi^2 * sqrt(15*n/c2) + (Pi * 5^(1/4) * c0 / (2*3^(3/4) * c2^(1/4)) - 15^(5/4) * c1^2 * Zeta(3)^2 / (2*c2^(5/4) * Pi^5)) * n^(1/4) + c1/12 + 75 * c1^3 * Zeta(3)^3 / (c2^2 * Pi^8) - 5*c0 * c1 * Zeta(3) / (4*c2 * Pi^2) - c2*Zeta(3) / (4*Pi^2)) * Pi^(c1/12) * (c2/15)^(1/8 + c0/8 + c1/48) / (A^c1 * 2^((c0 + 3)/2) * n^(5/8 + c0/8 + c1/48)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 08 2017
Let A(x) = Product_{k >= 1} (1 - x^k)^(-k^2). The sequence defined by u(n) := [x^n] A(x)^n is conjectured to satisfy the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 7 and all positive integers n and r. See A380290. - Peter Bala, Feb 02 2025
a(n) is the number of partitions of n where there are k^2 sorts of part k. - Joerg Arndt, Feb 02 2025

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 1001 terms from Alois P. Heinz)
G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
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. 21.

Euler transform of squares (A000290).
Cf. A000219, A023872-A023878, A294530, A380290.
Column k=2 of A144048. - Alois P. Heinz, Nov 02 2012

m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^2: k in [1..m]]) )); // G. C. Greubel, Oct 29 2018

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1,
      add(add(d*d^2, d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35); # Alois P. Heinz, Nov 02 2012

max = 31; Series[ Product[ 1/(1-x^k)^k^2, {k, 1, max}], {x, 0, max}] // CoefficientList[#, x]& (* Jean-François Alcover, Mar 05 2013 *)

m=40; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^k^2)) \\ G. C. Greubel, Oct 29 2018

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: n^2)
print([b(n) for n in range(32)]) # Peter Luschny, Nov 11 2020

a(n) = 1/n * Sum_{k=1..n} a(n-k)*sigma_3(k), n > 0, a(0)=1, where sigma_3(n) = A001158(n) = sum of cubes of divisors of n. - Vladeta Jovovic, Jan 20 2002
G.f.: Prod_{n>=1} exp(sigma_3(n)*x^n/n), where sigma_3(n) is the sum of cubes of divisors of n (=A001158(n)). - N-E. Fahssi, Mar 28 2010
G.f. (conjectured): 1/Product_{n>=1} E(x^n)^J2(n) where E(x) = Product_{n>=1} 1-x^n and J2(n) = A007434(n) [follows from the identity Sum_{d|n} J2(d) = n^2 - Peter Bala, Feb 02 2025]. - Joerg Arndt, Jan 25 2011
a(n) ~ exp(4 * Pi * n^(3/4) / (3^(5/4) * 5^(1/4)) - Zeta(3) / (4*Pi^2)) / (2^(3/2) * 15^(1/8) * n^(5/8)), where Zeta(3) = A002117 = 1.2020569031595942853997... . - Vaclav Kotesovec, Feb 27 2015

Definition corrected by Franklin T. Adams-Watters and R. J. Mathar, Dec 04 2006

A023872 Expansion of Product_{k>=1} (1 - x^k)^(-k^3).

Original entry on oeis.org

1, 1, 9, 36, 136, 477, 1703, 5746, 19099, 61622, 195366, 607069, 1856516, 5586870, 16579850, 48549116, 140438966, 401592524, 1136121837, 3181700219, 8825733603, 24261363403, 66124058839, 178757752892, 479513547399, 1276792213203, 3375707760306, 8864712158225

Offset: 0

Olivier Gérard

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..5422 (first 1001 terms from Alois P. Heinz)
G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
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. 21.

Column k=3 of A144048.

Cf. A248882.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^3: k in [1..m]]) )); // G. C. Greubel, Oct 30 2018

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1,
      add(add(d*d^3, d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 02 2012

max = 27; Series[ Product[ 1/(1-x^k)^k^3, {k, 1, max}], {x, 0, max}] // CoefficientList[#, x]& (* Jean-François Alcover, Mar 05 2013 *)

m=30; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^k^3)) \\ G. C. Greubel, Oct 30 2018

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: n^3)
print([b(n) for n in range(30)]) # Peter Luschny, Nov 11 2020

a(n) ~ (3*Zeta(5))^(59/600) * exp(5 * n^(4/5) * (3*Zeta(5))^(1/5) / 2^(7/5) + Zeta'(-3)) / (2^(41/200) * n^(359/600) * sqrt(5*Pi)), where Zeta(5) = A013663 = 1.036927755143369926..., Zeta'(-3) = ((gamma + log(2*Pi) - 11/6)/30 - 3*Zeta'(4)/Pi^4)/4 = 0.00537857635777430114441697421... . - Vaclav Kotesovec, Feb 27 2015
G.f.: exp( Sum_{n>=1} sigma_4(n)*x^n/n ). - Seiichi Manyama, Mar 04 2017
a(n) = (1/n)*Sum_{k=1..n} sigma_4(k)*a(n-k). - Seiichi Manyama, Mar 04 2017

Definition corrected by Franklin T. Adams-Watters and R. J. Mathar, Dec 04 2006

A088311 Number of sets of lists with distinct list sizes, cf. A000262.

Original entry on oeis.org

1, 1, 2, 12, 48, 360, 2880, 25200, 241920, 2903040, 36288000, 479001600, 7185024000, 112086374400, 1917922406400, 35307207936000, 669529276416000, 13516122267648000, 294509190463488000, 6568835422076928000, 155705728523304960000, 3882911605049917440000

Offset: 0

Vladeta Jovovic, Nov 05 2003

nonn

a(n) also enumerates ordered pairs of permutation functions on n elements where f(g(x)) = g(g(f(x))). - Chad Brewbaker, Mar 27 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000009, A007837, A007838.

Other ordered permutation function pair relations are A000012, A000085, A000142, A001044, A053529.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Laplace( (&*[1+x^j: j in [1..m+2]]) ))); // G. C. Greubel, Dec 14 2022

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:
a:= n-> n!*b(n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 15 2018

nn = 19; Drop[ Range[0, nn]! CoefficientList[ Series[ Product[1 + x^i, {i,nn}], {x,0,nn}], x], 0] (* Geoffrey Critzer, Aug 05 2013; adapted to new offset by Vincenzo Librandi, Mar 28 2014 *)
nmax = 20; CoefficientList[Series[Product[1/(1-x^(2*k-1)), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Aug 19 2015 *)

my(x='x+O('x^66)); Vec(serlaplace(eta(x^2)/eta(x))) \\ Joerg Arndt, Aug 06 2013

# uses[EulerTransform from A166861]
a = BinaryRecurrenceSequence(0, 1) #  Peter Luschny's code of A000009 and A166861
b = EulerTransform(a)
[factorial(n)*b(n) for n in range(41)] # G. C. Greubel, Dec 14 2022

E.g.f: Product_{m>0} (1+x^m).

a(n) = n! * A000009(n).

Prepended a(0) = 1, Joerg Arndt, Aug 06 2013

A002513 Number of "cubic partitions" of n: expansion of Product_{k>0} 1/((1-x^(2k))^2*(1-x^(2k-1))) in powers of x.

Original entry on oeis.org

1, 1, 3, 4, 9, 12, 23, 31, 54, 73, 118, 159, 246, 329, 489, 651, 940, 1242, 1751, 2298, 3177, 4142, 5630, 7293, 9776, 12584, 16659, 21320, 27922, 35532, 46092, 58342, 75039, 94503, 120615, 151173, 191611, 239060, 301086, 374026, 468342, 579408, 721638, 889287

Offset: 0

N. J. A. Sloane, Simon Plouffe

For a real polynomial equation of degree n, a(n) is the number of possibilities for the roots to be real and unequal, real and equal (in various combinations), or simple or multiple complex conjugates. For example, a(3)=4 because we can have: three equal roots, two equal roots, three distinct real roots and two complex roots (see the Monthly Problem reference). - Emeric Deutsch, Mar 22 2005
Number of partitions of n, the even parts being of two kinds. E.g. a(4)=9 because we have 4, 4', 3+1, 2+2, 2+2', 2'+2', 2+1+1, 2'+1+1, 1+1+1+1. - Emeric Deutsch, Mar 22 2005
For the name "cubic partition" see Xiong; Chen & Lin; Chern & Dastidar. - Michel Marcus, Jan 28 2016

			G.f. = 1 + x + 3*x^2 + 4*x^3 + 9*x^4 + 12*x^5 + 23*x^6 + 31*x^7 + 54*x^8 + 73*x^9 + ...
G.f. = 1/q + q^7 + 3*q^15 + 4*q^23 + 9*q^31 + 12*q^39 + 23*q^47 + 31*q^55 + 54*q^63 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence in two entries, N0930 and N0931).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
Zakir Ahmed, Nayandeep Deka Baruah, and Manosij Ghosh Dastidar, New congruences modulo 5 for the number of 2-color partitions, Journal of Number Theory, Volume 157, December 2015, Pages 184-198.
Koustav Banerjee, New Asymptotics and Inequalities Related to the Partition Function, Doctoral Thesis, Johannes Kepler Univ. (Linz, Austria 2022).
M. F. Capobianco and C. F. Pinzka, Problem 2055, Amer. Math. Monthly, 75 (1968), 188; 76 (1969), 194.
William Y. C. Chen and Bernard L. S. Lin, Congruences for the Number of Cubic Partitions Derived from Modular Forms, arXiv:0910.1263 [math.NT], 2016.
Shane Chern and Manosij Ghosh Dastidar, Congruences and recursions for the cubic partitions, arXiv:1601.06480 [math.NT], 2016.
Marston Conder, Tomaš Pisanski, and Arjana Žitnik, Vertex-transitive graphs and their arc-types, arXiv preprint arXiv:1505.02029 [math.CO], 2015.
R. K. Guy, Letter to Morris Newman, Aug 21 1986, concerning A2513 (annotated scanned copy, with permission).
David J. Hemmer, Generating functions for fixed points of the Mullineux map, arXiv:2402.03643 [math.CO], 2024. Table 1 p. 5 mentions this sequence.
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. 16.
Vaclav Kotesovec, Asymptotics of sequence A002513, 2019.
Lukas Mauth, Exact formula for cubic partitions, arXiv:2305.03396 [math.NT], 2023.
Morris Newman, Construction and application of a class of modular functions (II). Proc. London Math. Soc. (3) 9 1959 373-387.
Morris Newman, Construction and application of a class of modular functions, II, Proc. London Math. Soc. (3) 9 1959 373-387. [Annotated scanned copy, barely legible]
James A. Sellers, Elementary proofs of congruences for the cubic and overcubic partition functions, Australasian Journal of Combinatorics, Volume 60(2) (2014), Pages 191-197.
Xinhua Xiong, The number of cubic partitions modulo powers of 5, arXiv:1004.4737 [math.NT], 2010.

Cf. A000122, A015128, A001934, A089799, A215947, A273225, A319455.

N:= 50: # to get a(0) to a(N)
P:= mul((1-x^(2*k))^(-2)*(1-x^(2*k-1))^(-1),k=1..ceil(N/2)):
S:= series(P, x, N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Jan 26 2016
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      `if`(d::odd, d, 2*d), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Nov 05 2020

max = 50; f[x_] := Product[ 1/((1-x^(2 k))^2*(1-x^(2k-1))), {k, 1, Ceiling[max/2]} ]; CoefficientList[ Series[ f[x], {x, 0, max}], x] (* Jean-François Alcover, Nov 04 2011 *)
a[ n_] := SeriesCoefficient[ 1 / QPochhammer[ q] / QPochhammer[ q^2], {q, 0, n}];(* Michael Somos, Jul 17 2013 *)
Table[Sum[PartitionsP[k]*PartitionsP[n-2k],{k,0,n/2}],{n,0,50}] (* Vaclav Kotesovec, Jun 22 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( 1 / eta(x + A) / eta(x^2 + A), n))}; /* Michael Somos, Nov 10 2005 */

# uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, 1, 2)
a = EulerTransform(b)
print([a(n) for n in range(44)]) # Peter Luschny, Nov 17 2022

From Michael Somos, Mar 23 2003: (Start)
Expansion of q^(1/8) / (eta(q) * eta(q^2)) in powers of q.
Euler transform of period 2 sequence [1, 2, ...].
G.f.: Product_{k>0} 1/((1 - x^(2*k))^2 * (1 - x^(2*k-1))).
(End)
Given g.f. A(x), then B(q) = A(q)^8 / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = 16*v^4 + v^3*w + 256*u*v^3 + 16*u*v^2*w - u^2*w^2. - Michael Somos, Apr 03 2005
a(n) ~ exp(Pi*sqrt(n)) / (8*n^(5/4)) * (1 - (Pi/16 + 15/(8*Pi))/sqrt(n)). - Vaclav Kotesovec, Jun 22 2015, extended Jan 17 2017
From Michel Marcus, Jan 28 2016: (Start)
G.f.: Product_{k>0} 1/((1 - x^k) * (1 - x^(2*k))).
a(3n+2) = 0 (mod 3).
a(25n+22) = 0 (mod 5) (see Xiong).
a(49n+15) = a(49n+29) = a(49n+36) = a(49n+43) = 0 (mod 7) (see Chen & Lin).
a(297n+62) = a(297n+161) = 0 (mod 11) (see Chern & Dastidar).
(End)
G.f. is a period 1 Fourier series which satisfies f(-1 / (128 t)) = 2^(-7/2) (t/i)^-1 f(t) where q = exp(2 Pi i t). - Michael Somos, Oct 17 2017
G.f.: exp(Sum_{k>=1} x^k*(1 + 2*x^k)/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Aug 13 2018
From Peter Bala, Sep 25 2023: (Start)
The g.f. A(x) satisfies log(A(x)) = x + 5*x^2/2 + 4*x^3/3 + 13*x^4/4 + ... = Sum_{n >= 1} A215947(n)*x^n/n.
A(x^2) = 4/(F(x)*F(-x)) = 2/(F(x)*G(-x)), where F(x) = Sum_{n = -oo..oo} x^(n*(n+1)/2) is the g.f. of A089799 and G(x) = Sum_{n = -oo..oo} x^(n^2) is the g.f. of A000122. Cf. A001934. Note that 4/(F(-x)*F(-x)) is the g.f. of A273225.
The self-convolution A(x)^2 is the g.f. of A319455. (End)

More terms and information from Michael Somos, Mar 23 2003

A111317 Let f(a,q) = Product_{j>=0} (1 - a*q^j); g.f. is f(q^2,q^3) / f(q,q^3).

Original entry on oeis.org

1, 1, 0, 0, 1, 0, -1, 1, 1, -1, 0, 1, -1, 0, 2, -1, -1, 2, -1, -2, 3, 1, -3, 2, 1, -4, 2, 3, -4, 1, 4, -5, 0, 6, -5, -2, 7, -5, -4, 10, -3, -7, 10, -2, -10, 11, 1, -13, 11, 4, -16, 11, 9, -19, 8, 12, -22, 7, 19, -24, 2, 24, -26, -3, 32, -25, -10, 37, -25, -18, 45, -21, -29, 49, -17, -39, 56, -8, -51, 58, 0, -65, 61, 14, -78, 59, 27, -92

Offset: 0

N. J. A. Sloane, Nov 09 2005

Convolution inverse of A111165.

			From _Peter Bala_, Dec 2012: (Start)
F(1/10) = Sum_{n>=0} a(n)/10^n has the simple continued fraction expansion 1 + 1/(9 + 1/(1 + 1/(99 + 1/(1 + 1/(999 + 1/(1 + ...)))))).
F(-1/10) = Sum_{n>=0} (-1)^n*a(n)/10^n has the simple continued fraction expansion 1/(1 + 1/(9 + 1/(101 + 1/(999 + 1/(1001 + ...))))).
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000
G. E. Andrews and B. C. Berndt, Your Hit Parade: The Top Ten Most Fascinating Formulas in Ramanujan's Lost Notebook, Notices Amer. Math. Soc., 55 (No. 1, 2008), 18-30. See p. 25, Equation (39).

Cf. A005169, A143951.

a:= proc(n) option remember; `if`(n=0, 1,
      add(add(d*[0, 1, -1][irem(d, 3)+1],
      d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 01 2014

a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*{0, 1, -1}[[Mod[d, 3]+1]], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Apr 09 2014, after Alois P. Heinz *)

{a(n) = if( n<0, 0, polcoeff( prod(k=0, n\3, (1 - x^(3*k+2)) / (1 - x^(3*k+1)), 1 + x * O(x^n)), n))} /* Michael Somos, Dec 23 2007 */

{a(n)=polcoeff(exp(sum(m=1,n+1,1/(1+x^m+x^(2*m)+x*O(x^n))*x^m/m)),n)} \\ Paul D. Hanna, Jan 23 2010

# uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(-1, -1)
a = EulerTransform(b)
print([a(n) for n in range(88)]) # Peter Luschny, Nov 17 2022

Euler transform of period 3 sequence [ 1, -1, 0, ...]. - Michael Somos, Dec 23 2007
G.f.: Product_{k>=0} (1 - x^(3*k+2)) / (1 - x^(3*k+1)).
G.f.: exp( Sum_{n>=1} 1/(1 + x^n + x^(2n)) * x^n/n ). - Paul D. Hanna, Jan 23 2010
From Peter Bala, Dec 2012: (Start)
Let F(x) denote the o.g.f. of this sequence. For positive integer n >= 2, the real number F(1/n) has the simple continued fraction expansion 1 + 1/(n-1 + 1/(1 + 1/(n^2-1 + 1/(1 + 1/(n^3-1 + 1/(1 + ...)))))).
For n >= 2, F(-1/n) has the simple continued fraction expansion
1/(1 + 1/(n-1 + 1/(n^2+1 + 1/(n^3-1 + ...)))). Examples are given below. Cf. A005169 and A143951.
(End)

A283263 Expansion of exp( Sum_{n>=1} -sigma_3(n)*x^n/n ) in powers of x.

Original entry on oeis.org

1, -1, -4, -5, -1, 21, 49, 81, 45, -121, -484, -997, -1344, -840, 1624, 6931, 15149, 23155, 23469, 2240, -57596, -168929, -322587, -461165, -450668, -64135, 985621, 2935044, 5718865, 8597971, 9683008, 5596899, -8414092, -37295629, -83336988, -141108721

Offset: 0

Seiichi Manyama, Mar 04 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Column k=2 of A283272.
Cf. A023871 (exp( Sum_{n>=1} sigma_3(n)*x^n/n )).
Cf. exp( Sum_{n>=1} -sigma_k(n)*x^n/n ): A010815 (k=1), A073592 (k=2), this sequence (k=3), A283264 (k=4), A283271 (k=5).

a[n_] := If[n<1, 1,-(1/n) * Sum[DivisorSigma[3, k] a[n - k], {k, n}]]; Table[a[n], {n, 0, 35}] (* Indranil Ghosh, Mar 16 2017 *)

a(n) = if(n<1, 1, -(1/n) * sum(k=1, n, sigma(k, 3) * a(n - k)));
for(n=0, 35, print1(a(n), ", ")) \\ Indranil Ghosh, Mar 16 2017

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: -n^2)
print([b(n) for n in range(36)]) # Peter Luschny, Nov 11 2020

G.f.: Product_{n>=1} (1 - x^n)^(n^2).

a(n) = -(1/n)*Sum_{k=1..n} sigma_3(k)*a(n-k).

cp's OEIS Frontend

A022567 Expansion of Product_{m>=1} (1+x^m)^2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

SageMath

Formula

A073592 Euler transform of negative integers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

SageMath

Formula

A000335 Euler transform of A000292.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Sage

Formula

A038348 Expansion of (1/(1-x^2))*Product_{m>=0} 1/(1-x^(2m+1)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

SageMath

Formula

A023871 Expansion of Product_{k>=1} (1 - x^k)^(-k^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

Extensions

A023872 Expansion of Product_{k>=1} (1 - x^k)^(-k^3).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma