A052812 -id:A052812 - cp's OEIS frontend

A052847 G.f.: 1 / Product_{k>=1} (1-x^k)^(k-1).

1, 0, 1, 2, 4, 6, 12, 18, 33, 52, 88, 138, 229, 354, 568, 880, 1378, 2110, 3260, 4942, 7527, 11320, 17031, 25394, 37842, 55956, 82630, 121300, 177677, 258980, 376626, 545352, 787784, 1133764, 1627657, 2329020, 3324559, 4731396, 6717774, 9512060

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Euler transform of sequence [0,1,2,3,...]. - Michael Somos, Jul 02 2004
Number of partitions of n objects of 2 colors, where each part must contain at least one of each color. - Franklin T. Adams-Watters, Jan 23 2006
Number of partitions of n without 1s, one kind of 2s, two kinds of 3s, etc. - Joerg Arndt, Jul 31 2011
From Vaclav Kotesovec, Oct 17 2015: (Start)
In general, if v>=0 and g.f. = Product_{k>=1} 1/(1-x^(k+v))^k, then a(n) ~ d1(v) * n^(v^2/6 - 25/36) * exp(-Pi^4 * v^2 / (432*Zeta(3)) + 3*Zeta(3)^(1/3) * n^(2/3)/2^(2/3) - v * Pi^2 * n^(1/3) / (3 * 2^(4/3) * Zeta(3)^(1/3))) / (sqrt(3*Pi) * 2^(v^2/6 + 11/36) * Zeta(3)^(v^2/6 - 7/36)), where Zeta(3) = A002117.
d1(v) = exp(Integral_{x=0..infinity} (1/(x*exp((v-1)*x) * (exp(x)-1)^2) - (6*v^2-1) / (12*x*exp(x)) + v/x^2 - 1/x^3) dx).
d1(v) = (exp(Zeta'(-1) - v*Zeta'(0))) / Product_{j=0..v-1} j!, where Zeta'(0) = -A075700, Zeta'(-1) = A084448 and Product_{j=0..v-1} j! = A000178(v-1).
d1(v) = exp(1/12) * (2*Pi)^(v/2) / (A * G(v+1)), where A = A074962 is the Glaisher-Kinkelin constant and G is the Barnes G-function.
(End)

			1 + x^2 + 2*x^3 + 4*x^4 + 6*x^5 + 12*x^6 + 18*x^7 + 33*x^8 + 52*x^9 + ...
From _Gus Wiseman_, Jan 22 2019: (Start)
The partitions described in Franklin T. Adams-Watters's comment are (n = 2 through 6):
  {{12}}  {{112}}  {{1112}}    {{11112}}    {{111112}}
          {{122}}  {{1122}}    {{11122}}    {{111122}}
                   {{1222}}    {{11222}}    {{111222}}
                   {{12}{12}}  {{12222}}    {{112222}}
                               {{12}{112}}  {{122222}}
                               {{12}{122}}  {{112}{112}}
                                            {{112}{122}}
                                            {{12}{1112}}
                                            {{12}{1122}}
                                            {{12}{1222}}
                                            {{122}{122}}
                                            {{12}{12}{12}}
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 815
Vaclav Kotesovec, Graph - The asymptotic ratio

Cf. A005380, A000203, A001157, A052812.
Cf. A000219 (v=0), A052847 (v=1), A263358 (v=2), A263359 (v=3), A263360 (v=4), A263361 (v=5), A263362 (v=6), A263363 (v=7), A263364 (v=8).
Cf. A054974, A321407, A321760, A323654, A323655, A323656.

spec := [S,{B=Sequence(Z,1 <= card),C=Prod(B,B),S= Set(C)},unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
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-> n-1): seq(a(n), n=0..50); # Vaclav Kotesovec, Mar 04 2015 after Alois P. Heinz

Clear[a]; a[n_]:= a[n] = 1/n*Sum[(DivisorSigma[2,k]-DivisorSigma[1,k])*a[n-k],{k,1,n}]; a[0]=1; Table[a[n],{n,0,100}] (* Vaclav Kotesovec, Mar 04 2015 *)
nmax = 40; CoefficientList[Series[Product[1/(1-x^(k+1))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 16 2015 *)

{a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, (1 - x^k + x*O(x^n))^(k-1)), n))}

a(n) = 1/n*Sum_{k=1..n} (sigma[2](k)-sigma[1](k))*a(n-k).
G.f.: exp( Sum_{k>0} ( x^k / (1 - x^k) )^2 / k ).
G.f.: exp( sum(n>=0, (sigma[2](n)-sigma[1](n)) *x^n/n ) ). - Joerg Arndt, Jul 31 2011
a(n) ~ 2^(1/36) * Zeta(3)^(1/36) * exp(1/12 - Pi^4/(432*Zeta(3)) - Pi^2 * n^(1/3) / (3 * 2^(4/3) * Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (A * 3^(1/2) * n^(19/36)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 07 2015

Edited by Vladeta Jovovic, Sep 10 2002

A255835 G.f.: Product_{k>=1} (1+x^k)^(2*k-1).

Original entry on oeis.org

1, 1, 3, 8, 15, 34, 67, 133, 255, 486, 901, 1649, 2984, 5312, 9373, 16342, 28221, 48283, 81928, 137858, 230278, 381919, 629156, 1029933, 1675856, 2711288, 4362575, 6983196, 11122327, 17630798, 27820283, 43706461, 68375137, 106534093, 165340844, 255643289

Offset: 0

Vaclav Kotesovec, Mar 07 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A026007, A219555, A052812, A253289, A255834.

nmax=50; CoefficientList[Series[Product[(1+x^k)^(2*k-1),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ Zeta(3)^(1/6) * exp(-Pi^4 / (2592*Zeta(3)) - Pi^2 * n^(1/3) / (12*(3*Zeta(3))^(1/3)) + 3^(4/3)/2 * Zeta(3)^(1/3) * n^(2/3)) / (2^(1/6) * 3^(1/3) * sqrt(Pi) * n^(2/3)), where Zeta(3) = A002117.

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + x^k)/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, Jun 07 2018

A219555 Number of bipartite partitions of (i,j) with i+j = n into distinct pairs.

Original entry on oeis.org

1, 2, 4, 10, 19, 38, 73, 134, 242, 430, 749, 1282, 2171, 3622, 5979, 9770, 15802, 25334, 40288, 63560, 99554, 154884, 239397, 367800, 561846, 853584, 1290107, 1940304, 2904447, 4328184, 6422164, 9489940, 13967783, 20480534, 29920277, 43557272, 63194864

Offset: 0

Alois P. Heinz, Nov 22 2012

nonn

			a(2) = 4: [(2,0)], [(1,1)], [(1,0),(0,1)], [(0,2)].

Alois P. Heinz, Table of n, a(n) for n = 0..8000 (terms n=101..1000 from Vaclav Kotesovec)

Row sums of A054242.

Cf. A026007, A052812, A005380, A255834, A255836.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
       b(n-i*j, min(n-i*j, i-1))*binomial(i+1, j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..42);  # Alois P. Heinz, Sep 19 2019

nmax=50; CoefficientList[Series[Product[(1+x^k)^(k+1),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 07 2015 *)

a(n) = Sum_{i+j=n} [x^i*y^j] 1/2 * Product_{k,m>=0} (1+x^k*y^m).
G.f.: Product_{k>=1} (1+x^k)^(k+1). - Vaclav Kotesovec, Mar 07 2015
a(n) ~ Zeta(3)^(1/6) * exp(-Pi^4 / (1296*Zeta(3)) + Pi^2 * n^(1/3) / (2^(5/3) * 3^(4/3) * Zeta(3)^(1/3)) + (3/2)^(4/3) * Zeta(3)^(1/3) * n^(2/3)) / (2^(5/4) * 3^(1/3) * sqrt(Pi) * n^(2/3)), where Zeta(3) = A002117. - Vaclav Kotesovec, Mar 07 2015
G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k*(2 - x^k)/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, Aug 11 2018

A255836 G.f.: Product_{k>=1} (1+x^k)^(3*k+1).

Original entry on oeis.org

1, 4, 13, 42, 117, 310, 785, 1896, 4433, 10062, 22248, 48080, 101821, 211682, 432795, 871520, 1730491, 3391894, 6568996, 12580316, 23841774, 44742634, 83193865, 153347110, 280336704, 508499474, 915540681, 1636805438, 2906642396, 5128530946, 8993376689

Offset: 0

Vaclav Kotesovec, Mar 07 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A026007, A219555, A052812, A255271, A255834, A255835, A255837.

nmax=50; CoefficientList[Series[Product[(1+x^k)^(3*k+1),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ Zeta(3)^(1/6) * exp(-Pi^4 / (3888*Zeta(3)) + Pi^2 * n^(1/3) / (6^(5/3) * Zeta(3)^(1/3)) + 3^(5/3)/2^(4/3) * Zeta(3)^(1/3) * n^(2/3)) / (2^(17/12) * 3^(1/6) * sqrt(Pi) * n^(2/3)), where Zeta(3) = A002117.

A005309 Fermionic string states.

Original entry on oeis.org

1, 0, 2, 4, 8, 16, 32, 60, 114, 212, 384, 692, 1232, 2160, 3760, 6480, 11056, 18728, 31474, 52492, 86976, 143176, 234224, 380988, 616288, 991624, 1587600, 2529560, 4011808, 6334656, 9960080, 15596532, 24327122, 37801568, 58525152, 90291232, 138825416

Offset: 0

N. J. A. Sloane

nonn

See the reference for precise definition.

The g.f. -(1-2*z+2*z**2)/(-1+2*z) conjectured by Simon Plouffe in his 1992 dissertation is not correct.

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

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
T. Curtright, Counting symmetry patterns in the spectra of strings, in H. J. de Vega and N. Sánchez, editors, String Theory, Quantum Cosmology and Quantum Gravity. Integrable and Conformal Invariant Theories. World Scientific, Singapore, 1987, pp. 304-333, eq. (3.39) and Table 3.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Cf. A156616, A261451, A261386, A261452, A261389.

G.f. Product_{k>=1} ((1+x^k)/(1-x^k))^(k-1). - Vaclav Kotesovec, Aug 19 2015
Convolution of A052847 and A052812. - Vaclav Kotesovec, Aug 19 2015
a(n) ~ 2^(7/18) * (7*Zeta(3))^(1/36) * exp(1/12 - Pi^4/(336*Zeta(3)) - Pi^2 * n^(1/3) / (2^(5/3)*(7*Zeta(3))^(1/3)) + 3/2 * (7*Zeta(3)/2)^(1/3) * n^(2/3)) / (A * sqrt(3) * n^(19/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 19 2015

A255837 G.f.: Product_{k>=1} (1+x^k)^(3*k+2).

Original entry on oeis.org

1, 5, 18, 61, 182, 506, 1338, 3369, 8172, 19197, 43833, 97636, 212748, 454461, 953505, 1968095, 4001627, 8024295, 15885484, 31074351, 60111277, 115071431, 218126868, 409662895, 762679151, 1408172844, 2579599582, 4690277001, 8467363674, 15182486586

Offset: 0

Vaclav Kotesovec, Mar 07 2015

nonn

In general, if g.f. = Product_{k>=1} (1+x^k)^(m*k+c), m > 0, then a(n) ~ (m*Zeta(3))^(1/6) * exp(-c^2 * Pi^4 / (1296*m*Zeta(3)) + (c * Pi^2 * n^(1/3)) / (2^(5/3) * 3^(4/3) * (m*Zeta(3))^(1/3)) + 3^(4/3) * (m*Zeta(3))^(1/3) * n^(2/3) / 2^(4/3)) / (2^(m/12 + c/2 + 2/3) * 3^(1/3) * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Mar 08 2015

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A026007 (k), A219555 (k+1), A052812 (k-1), A255834 (2*k+1), A255835 (2*k-1), A255836 (3*k+1).

Cf. A255803.

nmax=50; CoefficientList[Series[Product[(1+x^k)^(3*k+2),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ Zeta(3)^(1/6) * exp(-Pi^4/(972*Zeta(3)) + Pi^2 * n^(1/3) / (2^(2/3) * 3^(5/3) * Zeta(3)^(1/3)) + 3^(5/3)/2^(4/3) * Zeta(3)^(1/3) * n^(2/3)) / (2^(23/12) * 3^(1/6) * sqrt(Pi) * n^(2/3)), where Zeta(3) = A002117.

A363599 Number of partitions of n into distinct parts where there are k^2-1 kinds of part k.

Original entry on oeis.org

1, 0, 3, 8, 18, 48, 109, 264, 594, 1360, 2988, 6552, 14115, 30048, 63288, 131800, 271953, 555792, 1126583, 2264472, 4518051, 8948544, 17603781, 34405272, 66828247, 129040704, 247765665, 473160696, 898924929, 1699331808, 3197083220, 5987288352, 11162934948

Offset: 0

Seiichi Manyama, Jun 10 2023

Cf. A027998, A052812, A255835, A363600.

Cf. A363601.

my(N=40, x='x+O('x^N)); Vec(prod(k=1, N, (1+x^k)^(k^2-1)))

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

a(0) = 1; a(n) = (-1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d) * d * (d^2-1) ) * a(n-k).

Name suggested by Joerg Arndt, Jun 11 2023

A319109 Expansion of Product_{k>=1} 1/(1 + x^k)^(k-1).

Original entry on oeis.org

1, 0, -1, -2, -2, -2, 0, 2, 7, 8, 12, 10, 9, -2, -10, -32, -40, -62, -62, -70, -37, -20, 57, 106, 224, 272, 388, 376, 431, 272, 192, -184, -414, -1012, -1321, -2020, -2157, -2700, -2318, -2352, -1014, -272, 2280, 3798, 7464, 9200, 13257, 13958, 17098, 14846, 15266

Offset: 0

Ilya Gutkovskiy, Sep 10 2018

sign

Convolution of A000009 and A255528.

Convolution inverse of A052812.

Cf. A000009, A052812, A255528, A305628.

a:=series(mul(1/(1+x^k)^(k-1),k=1..100),x=0,51): seq(coeff(a,x,n),n=0..50); # Paolo P. Lava, Apr 02 2019

nmax = 50; CoefficientList[Series[Product[1/(1 + x^k)^(k - 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[Exp[Sum[(-1)^k x^(2 k)/(k (1 - x^k)^2), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d) d (d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 50}]

G.f.: exp(Sum_{k>=1} (-1)^k*x^(2*k)/(k*(1 - x^k)^2)).

A371480 Expansion of e.g.f. Product_{k>=1} (1 + x^k/k!)^(k-1).

Original entry on oeis.org

1, 0, 1, 2, 3, 24, 70, 300, 1365, 10136, 48636, 253560, 2069430, 13716780, 107972579, 695118606, 6155201325, 53001340112, 483401450344, 4129645998960, 36957700537146, 398523920633940, 3954961713704673, 42541112692428746, 430044528651402148

Offset: 0

Ilya Gutkovskiy, Mar 25 2024

nonn

"EGJ" (unordered, element, labeled) transform of 0,1,2,3,4,...

C. G. Bower, Transforms (2)

Cf. A007837, A032315, A032316, A052812.

nmax = 24; CoefficientList[Series[Product[(1 + x^k/k!)^(k - 1), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

cp's OEIS Frontend

A052847 G.f.: 1 / Product_{k>=1} (1-x^k)^(k-1).

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A255835 G.f.: Product_{k>=1} (1+x^k)^(2*k-1).

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A219555 Number of bipartite partitions of (i,j) with i+j = n into distinct pairs.

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A255836 G.f.: Product_{k>=1} (1+x^k)^(3*k+1).

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A005309 Fermionic string states.

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A255837 G.f.: Product_{k>=1} (1+x^k)^(3*k+2).

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A363599 Number of partitions of n into distinct parts where there are k^2-1 kinds of part k.

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A319109 Expansion of Product_{k>=1} 1/(1 + x^k)^(k-1).

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