cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A264923 G.f.: 1 / Product_{n>=0} (1 - x^(n+3))^((n+1)*(n+2)/2!).

Original entry on oeis.org

1, 0, 0, 1, 3, 6, 11, 18, 33, 57, 105, 183, 330, 567, 990, 1693, 2904, 4917, 8343, 14010, 23511, 39171, 65100, 107592, 177352, 290931, 475905, 775381, 1259637, 2039094, 3291613, 5296467, 8499339, 13599292, 21702795, 34541724, 54839894, 86847255, 137212197, 216274466, 340129773, 533726442, 835732774, 1305877914, 2036369010
Offset: 0

Views

Author

Paul D. Hanna, Nov 28 2015

Keywords

Comments

Number of partitions of n objects of 3 colors, where each part must contain at least one of each color. [Conjecture - see comment by Franklin T. Adams-Watters in A052847].

Examples

			G.f.: A(x) = 1 + x^3 + 3*x^4 + 6*x^5 + 11*x^6 + 18*x^7 + 33*x^8 + 57*x^9 + 105*x^10 +...
where
1/A(x) = (1-x^3) * (1-x^4)^3 * (1-x^5)^6 * (1-x^6)^10 * (1-x^7)^15 * (1-x^8)^21 * (1-x^9)^28 * (1-x^10)^36 * (1-x^11)^45 *...
Also,
log(A(x)) = (x/(1-x))^3 + (x^2/(1-x^2))^3/2 + (x^3/(1-x^3))^3/3 + (x^4/(1-x^4))^3/4 + (x^5/(1-x^5))^3/5 + (x^6/(1-x^6))^3/6 +...

Links

Crossrefs

Cf. A052847, A264924, A264925, A264926.
Cf. A000294, A217093, A258349.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[1/(1-x^k)^((k-2)*(k-1)/2), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Dec 09 2015 *)
  • PARI
    {a(n) = my(A=1); A = prod(k=0,n, 1/(1 - x^(k+3) +x*O(x^n) )^((k+1)*(k+2)/2) ); polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))
  • PARI
    {a(n) = my(A=1); A = exp( sum(k=1,n+1, (x^k/(1 - x^k))^3 /k +x*O(x^n) ) ); polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))
  • PARI
    {L(n) = sumdiv(n,d, d*(d-1)*(d-2)/2! )}
{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),", "))

Formula

G.f.: exp( Sum_{n>=1} ( x^n/(1-x^n) )^3 /n ).
G.f.: exp( Sum_{n>=1} L(n) * x^n/n ), where L(n) = Sum_{d|n} d*(d-1)*(d-2)/2!.
a(n) ~ Pi^(3/8) / (2^(55/32) * 15^(7/32) * n^(23/32)) * exp(29*Zeta(3)/(8*Pi^2) - log(2*Pi)/2 - 3*Zeta'(-1)/2 - 2025*Zeta(3)^3/(2*Pi^8) + (5^(1/4)*Pi/6^(3/4) - 135*15^(1/4)*Zeta(3)^2/(2^(7/4)*Pi^5)) * n^(1/4) - 3*sqrt(15*n/2)*Zeta(3)/Pi^2 + 2^(7/4)*Pi/(3*15^(1/4)) * n^(3/4)). - Vaclav Kotesovec, Dec 09 2015

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

Original entry on oeis.org

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

Views

Author

Paul D. Hanna, Nov 28 2015

Keywords

Comments

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].

Examples

			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 +...

Links

Crossrefs

Cf. A052847, A264923, A264925, A264926.
Cf. A000335, A255050, A258352.

Programs

  • Mathematica
    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 *)
  • PARI
    {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),", "))
  • PARI
    {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),", "))
  • PARI
    {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),", "))

Formula

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

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

Views

Author

Paul D. Hanna, Nov 28 2015

Keywords

Comments

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].

Examples

			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 +...

Links

Crossrefs

Cf. A052847, A264923, A264924, A264925.
Cf. A000417.

Programs

  • Mathematica
    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 *)
  • PARI
    {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),", "))
  • PARI
    {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),", "))
  • PARI
    {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),", "))

Formula

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
Showing 1-3 of 3 results.

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