A052847 -id:A052847 - cp's OEIS frontend

A264924 G.f.: 1 / Product_{n>=0} (1 - x^(n+4))^((n+1)(n+2)(n+3)/3!).

1, 0, 0, 0, 1, 4, 10, 20, 36, 60, 104, 180, 336, 620, 1174, 2160, 3961, 7100, 12690, 22424, 39651, 69820, 122970, 215904, 378470, 660872, 1150740, 1996200, 3452685, 5952916, 10237576, 17559460, 30049285, 51301020, 87390872, 148534232, 251916041, 426329040, 720003646, 1213481344, 2041155052, 3426721080

Offset: 0

Paul D. Hanna, Nov 28 2015

nonn

Number of partitions of n objects of 4 colors, where each part must contain at least one of each color. [Conjecture - see comment by Franklin T. Adams-Watters in A052847].

			G.f.: A(x) = 1 + x^4 + 4*x^5 + 10*x^6 + 20*x^7 + 36*x^8 + 60*x^9 + 104*x^10 + 180*x^11 +...
where
1/A(x) = (1-x^4) * (1-x^5)^4 * (1-x^6)^10 * (1-x^7)^20 * (1-x^8)^35 * (1-x^9)^56 * (1-x^10)^84 * (1-x^11)^120 * (1-x^12)^165 *...
Also,
log(A(x)) = (x/(1-x))^4 + (x^2/(1-x^2))^4/2 + (x^3/(1-x^3))^4/3 + (x^4/(1-x^4))^4/4 + (x^5/(1-x^5))^4/5 + (x^6/(1-x^6))^4/6 +...

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

Cf. A052847, A264923, A264925, A264926.

Cf. A000335, A255050, A258352.

nmax = 50; CoefficientList[Series[Product[1/(1-x^k)^((k-3)*(k-2)*(k-1)/6), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Dec 09 2015 *)

{a(n) = my(A=1); A = prod(k=0,n, 1/(1 - x^(k+4) +x*O(x^n) )^((k+1)*(k+2)*(k+3)/3!) ); polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

{a(n) = my(A=1); A = exp( sum(k=1,n+1, (x^k/(1 - x^k))^4 /k +x*O(x^n) ) ); polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

{L(n) = sumdiv(n,d, d*(d-1)*(d-2)*(d-3)/3! )}
{a(n) = my(A=1); A = exp( sum(k=1,n+1, L(k) * x^k/k +x*O(x^n) ) ); polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

G.f.: exp( Sum_{n>=1} ( x^n/(1-x^n) )^4 /n ).
G.f.: exp( Sum_{n>=1} L(n) * x^n/n ), where L(n) = Sum_{d|n} d*(d-1)*(d-2)*(d-3)/3!.
a(n) ~ Zeta(5)^(109/3600) / (2^(791/1800) * n^(1909/3600) * sqrt(5*Pi)) * exp(11*Zeta'(-1)/6 + log(2*Pi)/2 + Zeta(3)/(4*Pi^2) - Pi^16/(194400000 * Zeta(5)^3) + 11*Pi^8 * Zeta(3)/(108000 * Zeta(5)^2) - Pi^6/(1800*Zeta(5)) - 121*Zeta(3)^2/(360*Zeta(5)) + Zeta'(-3)/6 + (-Pi^12/(1350000 * 2^(2/5) * Zeta(5)^(11/5)) + 11*Pi^4 * Zeta(3)/(900 * 2^(2/5) * Zeta(5)^(6/5)) - Pi^2/(3*2^(7/5) * Zeta(5)^(1/5))) * n^(1/5) + (-Pi^8/(9000 * 2^(4/5) * Zeta(5)^(7/5)) + 11*Zeta(3)/(3*2^(9/5) * Zeta(5)^(2/5))) * n^(2/5) - Pi^4/(90 * 2^(1/5) * Zeta(5)^(3/5)) * n^(3/5) + 5*Zeta(5)^(1/5) / 2^(8/5) * n^(4/5)). - Vaclav Kotesovec, Dec 09 2015

A264925 G.f.: 1 / Product_{n>=0} (1 - x^(n+5))^((n+1)(n+2)(n+3)*(n+4)/4!).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 5, 15, 35, 70, 127, 215, 360, 605, 1080, 2003, 3890, 7570, 14715, 27960, 52255, 95705, 173295, 311060, 557400, 999032, 1795880, 3235130, 5835955, 10521060, 18931287, 33956485, 60692510, 108087835, 191883595, 339724144, 600203700, 1058605775, 1864535670, 3279862975, 5762287759, 10109925380

Offset: 0

Paul D. Hanna, Nov 28 2015

nonn

Number of partitions of n objects of 5 colors, where each part must contain at least one of each color. [Conjecture - see comment by Franklin T. Adams-Watters in A052847].

			G.f.: A(x) = 1 + x^5 + 5*x^6 + 15*x^7 + 35*x^8 + 70*x^9 + 127*x^10 + 215*x^11 + 360*x^12 +...
where
1/A(x) = (1-x^5) * (1-x^6)^5 * (1-x^7)^15 * (1-x^8)^35 * (1-x^9)^70 * (1-x^10)^126 * (1-x^11)^210 * (1-x^12)^330 * (1-x^13)^495 *...
Also,
log(A(x)) = (x/(1-x))^5 + (x^2/(1-x^2))^5/2 + (x^3/(1-x^3))^5/3 + (x^4/(1-x^4))^5/4 + (x^5/(1-x^5))^5/5 + (x^6/(1-x^6))^5/6 +...

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

Cf. A052847, A264923, A264924, A264926.

Cf. A000391, A255052.

nmax = 50; CoefficientList[Series[Product[1/(1-x^k)^((k-4)*(k-3)*(k-2)*(k-1)/24), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Dec 09 2015 *)

{a(n) = my(A=1); A = prod(k=0,n, 1/(1 - x^(k+4) +x*O(x^n) )^((k+1)*(k+2)*(k+3)/3!) ); polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

{a(n) = my(A=1); A = exp( sum(k=1,n+1, (x^k/(1 - x^k))^4 /k +x*O(x^n) ) ); polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

{L(n) = sumdiv(n,d, d*(d-1)*(d-2)*(d-3)*(d-4)/4!)}
{a(n) = my(A=1); A = exp( sum(k=1,n+1, L(k) * x^k/k +x*O(x^n) ) ); polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

G.f.: exp( Sum_{n>=1} ( x^n/(1-x^n) )^5 /n ).
G.f.: exp( Sum_{n>=1} L(n) * x^n/n ), where L(n) = Sum_{d|n} d*(d-1)*(d-2)*(d-3)*(d-4)/4!.
a(n) ~ Pi^(95/288) / (2 * 3^(527/576) * 7^(239/1728) * n^(1103/1728)) * exp(-25*Zeta'(-1)/12 - log(2*Pi)/2 + 595*Zeta(3)/(48*Pi^2) - 29291*Zeta(5) / (128*Pi^4) - 2480625 * Zeta(3) * Zeta(5)^2 / (2*Pi^12) + 72930375 * Zeta(5)^3 / (2*Pi^14) - 1063324867500 * Zeta(5)^5/Pi^24 - 5*Zeta'(-3)/12 + (41 * 7^(1/6) * Pi/(768*sqrt(3)) - 2625 * sqrt(3) * 7^(1/6) * Zeta(3) * Zeta(5)/(2*Pi^7) + 540225 * sqrt(3) * 7^(1/6) * Zeta(5)^2/(16*Pi^9) - 4740474375 * sqrt(3) * 7^(1/6) * Zeta(5)^4/(4*Pi^19)) * n^(1/6) + (-25 * 7^(1/3) * Zeta(3)/(4*Pi^2) + 735 * 7^(1/3) * Zeta(5) /(8*Pi^4) - 3969000 * 7^(1/3) * Zeta(5)^3 / Pi^14) * n^(1/3) + (7*sqrt(7/3)*Pi/24 - 4725 * sqrt(21) * Zeta(5)^2 / Pi^9) * sqrt(n) - 45 * 7^(2/3) * Zeta(5)/(2*Pi^4) * n^(2/3) + 2*sqrt(3)*Pi / (5*7^(1/6)) * n^(5/6)). - Vaclav Kotesovec, Dec 09 2015

A264926 G.f.: 1 / Product_{n>=0} (1 - x^(n+6))^((n+1)(n+2)(n+3)(n+4)(n+5)/5!).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 6, 21, 56, 126, 252, 463, 798, 1329, 2184, 3696, 6552, 12405, 24486, 49524, 99722, 197967, 383796, 727609, 1350174, 2466534, 4457844, 8022819, 14448168, 26142810, 47603010, 87222576, 160522228, 295996791, 545445468, 1002392105, 1834644210, 3342375099, 6061611192, 10949981496, 19720143366, 35440268956

Offset: 0

Paul D. Hanna, Nov 28 2015

nonn

Number of partitions of n objects of 6 colors, where each part must contain at least one of each color. [Conjecture - see comment by Franklin T. Adams-Watters in A052847].

			G.f.: A(x) = 1 + x^6 + 6*x^7 + 21*x^8 + 56*x^9 + 126*x^10 + 252*x^11 + 463*x^12 +...
where
1/A(x) = (1-x^6) * (1-x^7)^6 * (1-x^8)^21 * (1-x^9)^56 * (1-x^10)^126 * (1-x^11)^252 * (1-x^12)^462 * (1-x^13)^792 * (1-x^14)^1287 * (1-x^15)^2002 *...
Also,
log(A(x)) = (x/(1-x))^6 + (x^2/(1-x^2))^6/2 + (x^3/(1-x^3))^6/3 + (x^4/(1-x^4))^6/4 + (x^5/(1-x^5))^6/5 + (x^6/(1-x^6))^6/6 +...

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

Cf. A052847, A264923, A264924, A264925.

Cf. A000417.

nmax = 50; CoefficientList[Series[Product[1/(1-x^k)^((k-5)*(k-4)*(k-3)*(k-2)*(k-1)/120), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Dec 09 2015 *)

{a(n) = my(A=1); A = prod(k=0,n, 1/(1 - x^(k+6) +x*O(x^n) )^((k+1)*(k+2)*(k+3)*(k+4)*(k+5)/5!) ); polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

{a(n) = my(A=1); A = exp( sum(k=1,n+1, (x^k/(1 - x^k))^6 /k +x*O(x^n) ) ); polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

{L(n) = sumdiv(n,d, d*(d-1)*(d-2)*(d-3)*(d-4)*(d-5)/5! )}
{a(n) = my(A=1); A = exp( sum(k=1,n+1, L(k) * x^k/k +x*O(x^n) ) ); polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

G.f.: exp( Sum_{n>=1} ( x^n/(1-x^n) )^6 /n ).
G.f.: exp( Sum_{n>=1} L(n) * x^n/n ), where L(n) = Sum_{d|n} d*(d-1)*(d-2)*(d-3)*(d-4)*(d-5)/5!.
a(n) ~ (3*Zeta(7))^(11153/423360) / (2^(200527/423360) * n^(222833/423360) * sqrt(7*Pi)) * exp(137*Zeta'(-1)/60 + log(2*Pi)/2 + 15*Zeta(3) / (32*Pi^2) - 3*Zeta(5) / (32*Pi^4) - Pi^36 / (102098378167208640 * Zeta(7)^5) + 17 * Pi^24 * Zeta(5) / (571643448768 * Zeta(7)^4) - Pi^22 / (9073705536 * Zeta(7)^3) - 289 * Pi^12 * Zeta(5)^2 / (12002256 * Zeta(7)^3) + 137 * Pi^12 * Zeta(3) / (60011280 * Zeta(7)^2) + 17 * Pi^10 * Zeta(5) / (127008 * Zeta(7)^2) + 4913 * Zeta(5)^3 / (1512 * Zeta(7)^2) - 253 * Pi^8 / (1016064 * Zeta(7)) - 2329 * Zeta(3) * Zeta(5) / (1260 * Zeta(7)) + Zeta'(-5)/120 + 17 * Zeta'(-3)/24 + (-11*Pi^30 / (1544080410553464 * 6^(1/7) * Zeta(7)^(29/7)) + 85 * Pi^18 * Zeta(5) / (4631370534 * 6^(1/7) * Zeta(7)^(22/7)) - Pi^16 / (14002632 * 6^(1/7) * Zeta(7)^(15/7)) - 289 * Pi^6 * Zeta(5)^2 / (27783 * 6^(1/7) * Zeta(7)^(15/7)) + 137 * Pi^6 * Zeta(3) / (79380 * 6^(1/7) * Zeta(7)^(8/7)) + 17 * Pi^4 * Zeta(5) / (336 * 6^(1/7) * Zeta(7)^(8/7)) - Pi^2 / (6^(8/7) * Zeta(7)^(1/7))) * n^(1/7) + (-Pi^24 / (194517562428 * 6^(2/7) * Zeta(7)^(23/7)) + 17 * Pi^12 * Zeta(5) / (1555848 * 6^(2/7) * Zeta(7)^(16/7)) - Pi^10 / (21168 * 6^(2/7) * Zeta(7)^(9/7)) - 289 * Zeta(5)^2 / (84 * 6^(2/7) * Zeta(7)^(9/7)) + 137 * Zeta(3) / (60 * (6*Zeta(7))^(2/7))) * n^(2/7) + (-5*Pi^18 / (1323248724 * 6^(3/7) * Zeta(7)^(17/7)) + 17 * Pi^6 * Zeta(5) / (2646 * 6^(3/7) * Zeta(7)^(10/7)) - Pi^4 /(24 * (6*Zeta(7))^(3/7))) * n^(3/7) + (-Pi^12 / (333396 * 6^(4/7) * Zeta(7)^(11/7)) + 17 * Zeta(5) / (4 * (6*Zeta(7))^(4/7))) * n^(4/7) - Pi^6 / (315*(6*Zeta(7))^(5/7)) * n^(5/7) + 7 * Zeta(7)^(1/7) / 6^(6/7) * n^(6/7)). - Vaclav Kotesovec, Dec 09 2015

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

A086666 a(n) = sigma_2(n) - sigma_1(n).

Original entry on oeis.org

0, 2, 6, 14, 20, 38, 42, 70, 78, 112, 110, 182, 156, 226, 236, 310, 272, 416, 342, 504, 468, 574, 506, 790, 620, 808, 780, 994, 812, 1228, 930, 1302, 1172, 1396, 1252, 1820, 1332, 1750, 1644, 2120, 1640, 2404, 1806, 2478, 2288, 2578, 2162, 3286, 2394, 3162

Offset: 1

Jon Perry, Jul 27 2003

Total area of all distinct L X W rectangles such that s + t = n, 1 <= s <= t, s | n, L = n/s and W = t/s. - Wesley Ivan Hurt, Aug 01 2025

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).

Cf. A000203, A001157, A002117, A052847, A069153, A092348.

Table[DivisorSigma[2,n]-DivisorSigma[1,n],{n,50}] (* Harvey P. Dale, Aug 01 2020 *)

PARI
```
a(n) = sigma(n,2)-sigma(n,1);
```

a(n) = my(f = factor(n)); sigma(f, 2) - sigma(f); \\ Amiram Eldar, Jan 01 2025

G.f.: Sum_{n>=1} n*(n-1) * x^n/(1-x^n). - Joerg Arndt, Jan 30 2011
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k-1)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 21 2018
From Peter Bala, Jan 21 2021: (Start)
a(n) = 2*A069153(n).
G.f.: A(x) = Sum_{n >= 1} 2*x^(2*n)/(1 - x^n)^3.
A faster converging series: A(x) = Sum_{n >= 1} x^(n^2)*( n*(n-1)*x^(3*n) - (n^2 + n - 2)*x^(2*n) + n*(3 - n)*x^n + n*(n - 1) )/(1 - x^n)^3 - differentiate equation 5 in Arndt twice w.r.t x and set x = 1. (End)
From Amiram Eldar, Jan 01 2025: (Start)
Dirichlet g.f.: zeta(s) * (zeta(s-2) - zeta(s-1)).
Sum_{k=1..n} a(k) ~ (zeta(3)/3) * n^3. (End)
a(n) = Sum_{d|n} d*(d-1). - Wesley Ivan Hurt, Aug 01 2025

A081285 Table of coefficients of polynomials f_n(q) defined by 1/Product_{i>=1} (1-a q^i)^i = Sum_{n>=0} a^n q^n f_n(q) / ((q)n)^2, where (x)_n is the q-Pochhammer symbol, defined to be Product{i=0..n-1} (1-x q^i).

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 1, 0, 1, 2, 4, 2, 4, 2, 4, 2, 1, 0, 1, 1, 0, 1, 2, 4, 6, 7, 8, 12, 12, 14, 12, 12, 8, 7, 6, 4, 2, 1, 0, 1, 1, 0, 1, 2, 4, 6, 12, 12, 21, 26, 37, 40, 55, 52, 61, 60, 61, 52, 55, 40, 37, 26, 21, 12, 12, 6, 4, 2, 1, 0, 1, 1, 0, 1, 2, 4, 6, 12, 18, 26, 38, 57, 76

Offset: 0

Dean Hickerson, using information supplied by Moshe Shmuel Newman and Richard Stanley, Mar 15 2003

f_n has degree n(n-1), so n-th row of table has n(n-1)+1 entries. Each row is palindromic. The sum of the terms in the n-th row is n!. The first n+1 terms of the n-th row are the same as the first n terms of A052847.

The 'major index' maj(p) of a permutation p = a_1 a_2 ... a_n is the sum of all i such that a_i > a_(i+1). f_n(q) = Sum_p q^(maj(p)+maj(p^(-1))), where the sum is over all permutations of {1,2,...,n}.

			f_0 = f_1 = 1, f_2 = 1+q^2, f_3 = 1+q^2+2q^3+q^4+q^6, so sequence begins 1; 1; 1,0,1; 1,0,1,2,1,0,1; ...

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; Exercise 4.20.

Alois P. Heinz, Rows n = 0..40, flattened
Zhipeng Lu, Symmetric permutation invariants in some tensor products, arXiv:2103.02168 [math.CO], 2021.
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.

Row sums give A000142.

Cf. A052847.

b:= proc(u, o) option remember; expand(`if`(u+o=0, 1,
      add(b(u-j, o+j-1)*x^(-u), j=1..u)+
      add(b(u+j-1, o-j)*x^( o), j=1..o)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=ldegree(p)..degree(p)))(b(n, 0)):
seq(T(n), n=0..10);  # Alois P. Heinz, Apr 28 2018

qpoch[x_, n_] := Product[1-x*q^i, {i, 0, n-1}]; f[0]=1; f[n_] := f[n]=Together[Sum[ -(-1)^r q^Binomial[r, 2] qpoch[q^(n-r+1), r-1]*qpoch[q^(r+1), n-r]/qpoch[q, n-r] f[n-r], {r, 1, n}]]; Join@@Table[CoefficientList[f[n], q], {n, 0, 7}]

f_n(q) = Sum_{r=1..n} (-1)^(r+1) q^(r(r-1)/2) (q)(n-1) (q)_n / ((q)(r) ((q)(n-r))^2) f(n-r)(q) for n>=1.

A319110 Expansion of Product_{k>=1} 1/(1 - (k - 1)*x^k).

Original entry on oeis.org

1, 0, 1, 2, 4, 6, 13, 18, 37, 56, 101, 152, 285, 410, 713, 1118, 1830, 2780, 4618, 6934, 11278, 17092, 26894, 40822, 64435, 96372, 149299, 225104, 345131, 515394, 788176, 1169962, 1772957, 2632458, 3950365, 5849260, 8748993, 12867848, 19135894, 28126614, 41598695

Offset: 0

Ilya Gutkovskiy, Sep 10 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..8190
Vaclav Kotesovec, Closed form of the asymptotics

Cf. A006906, A034008, A052847, A074141, A267007.

b:= proc(n, i) option remember; `if`(n=0 or i=1,
       0^n, b(n, i-1)+(i-1)*b(n-i, min(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..42);  # Alois P. Heinz, Aug 19 2019

nmax = 40; CoefficientList[Series[Product[1/(1 - (k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[Sum[(j - 1)^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (d - 1)^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 40}]

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (j - 1)^k*x^(j*k)/k).
From Vaclav Kotesovec, Sep 11 2018: (Start)
a(n) ~ c * 2^(2*n/5), where
c = 28108804.248904780960402246466460350520790117596512766842168... if mod(n,5) = 0
c = 28108804.010850549080284030388905319123062152339902207992657... if mod(n,5) = 1
c = 28108804.067769166625741650205643600577757560110636366636106... if mod(n,5) = 2
c = 28108804.083581827971851596540314974909801290757084687583764... if mod(n,5) = 3
c = 28108804.058853893104368046896759214442695016905368229405793... if mod(n,5) = 4
(End)

A319108 Expansion of Product_{k>=1} (1 - x^k)^(k-1).

Original entry on oeis.org

1, 0, -1, -2, -3, -2, -1, 4, 8, 14, 17, 18, 9, -4, -27, -58, -88, -114, -122, -106, -48, 48, 200, 376, 577, 746, 862, 840, 646, 208, -486, -1450, -2622, -3888, -5086, -5950, -6204, -5492, -3547, -44, 5036, 11732, 19582, 28034, 35932, 42042, 44519, 41660, 31450, 12382, -16721

Offset: 0

Ilya Gutkovskiy, Sep 10 2018

sign

Convolution of A000041 and A073592.

Convolution inverse of A052847.

Cf. A000041, A000203, A001157, A052847, A073592, A299019.

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

G.f.: exp(Sum_{k>=1} (sigma_1(k) - sigma_2(k))*x^k/k), where sigma_1(k) = sum of divisors of k (A000203) and sigma_2(k) = sum of squares of divisors of k (A001157).

A385053 Number of non-isomorphic simple games with n players and one minimal winning vector.

Original entry on oeis.org

1, 3, 7, 15, 29, 55, 99, 176, 305, 522, 877, 1461, 2399, 3905, 6291, 10055, 15929, 25063, 39139, 60742, 93665, 143619, 218967, 332157, 501303, 753079, 1126155, 1676908, 2486641, 3673000, 5404711, 7924206, 11577465, 16858381, 24468317, 35402812, 51068703

Offset: 1

Dani Samaniego Vidal, Jun 16 2025

nonn

a(n) is also the number of non-isomorphic monotonic boolean functions with one minimal model.

Sascha Kurz and Dani Samaniego, Simple games with minimum, arXiv:2501.18966 [math.CO], 2025.

Second differences appear to be A052847.

sgnvnn[0,0] = 1; sgnvnn[_,0] = 0;
sgnvnn[n_,t_] := sgnvnn[n,t] = (1/t) Sum[If[Mod[k,l]==0, (k/l-1)sgnvnn[n-k,t-l], 0], {l,t},{k,n}];
sg[n_,1] := sgnvnn[n,1] + 1;
sg[n_,2] := sgnvnn[n,2] + 2 Sum[sgnvnn[n-i,1],{i,n-2}] + n - 1;
sg[n_,t_] := sgnvnn[n,t] + 2 Sum[sgnvnn[n-i,t-1],{i,n-2}] + Sum[(i-1) sgnvnn[n-i,t-2],{i,2,n-2}];
a[n_] := Sum[sg[n,t], {t,Quotient[n,2]+1}];
Table[a[n], {n,50}] (* Andrei Zabolotskii, Jul 24 2025 *)

Edited and extended by Andrei Zabolotskii, Jul 24 2025

cp's OEIS Frontend

A264924 G.f.: 1 / Product_{n>=0} (1 - x^(n+4))^((n+1)*(n+2)*(n+3)/3!).

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A264925 G.f.: 1 / Product_{n>=0} (1 - x^(n+5))^((n+1)*(n+2)*(n+3)*(n+4)/4!).

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A264926 G.f.: 1 / Product_{n>=0} (1 - x^(n+6))^((n+1)*(n+2)*(n+3)*(n+4)*(n+5)/5!).

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A005309 Fermionic string states.

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A086666 a(n) = sigma_2(n) - sigma_1(n).

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A081285 Table of coefficients of polynomials f_n(q) defined by 1/Product_{i>=1} (1-a q^i)^i = Sum_{n>=0} a^n q^n f_n(q) / ((q)n)^2, where (x)_n is the q-Pochhammer symbol, defined to be Product{i=0..n-1} (1-x q^i).

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A319110 Expansion of Product_{k>=1} 1/(1 - (k - 1)*x^k).

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A264924 G.f.: 1 / Product_{n>=0} (1 - x^(n+4))^((n+1)(n+2)(n+3)/3!).

A264925 G.f.: 1 / Product_{n>=0} (1 - x^(n+5))^((n+1)(n+2)(n+3)*(n+4)/4!).

A264926 G.f.: 1 / Product_{n>=0} (1 - x^(n+6))^((n+1)(n+2)(n+3)(n+4)(n+5)/5!).