|
| |
|
|
A264923
|
|
G.f.: 1 / Product_{n>=0} (1 - x^(n+3))^((n+1)*(n+2)/2!).
|
|
4
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
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].
|
|
|
LINKS
|
|
|
|
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
|
|
|
EXAMPLE
|
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 +...
|
|
|
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 *)
|
|
|
PROG
|
(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), ", "))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
