|
| |
|
|
A035386
|
|
Number of partitions of n into parts congruent to 2 mod 3.
|
|
14
|
|
|
|
1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 3, 2, 3, 3, 4, 4, 6, 5, 7, 7, 9, 9, 12, 11, 15, 15, 18, 19, 23, 23, 29, 29, 35, 37, 43, 45, 53, 55, 64, 68, 78, 82, 95, 99, 114, 121, 136, 145, 164, 173, 196, 208, 232, 248, 276, 294, 328, 349, 386, 413, 456, 486, 537, 572, 629, 673, 737, 787
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,9
|
|
|
COMMENTS
|
|
|
|
LINKS
|
|
|
|
FORMULA
|
Euler transform of period 3 sequence [ 0, 1, 0, ...]. - Michael Somos, Jul 24 2007
a(n) ~ Gamma(2/3) * exp(sqrt(2*n)*Pi/3) / (2^(11/6) * sqrt(3) * Pi^(1/3) * n^(5/6)) * (1 + (Pi/72 - 5/(3*Pi)) / sqrt(2*n)). - Vaclav Kotesovec, Feb 26 2015, extended Jan 24 2017
G.f.: A(x) = Sum_{n >= 0} x^(n*(3*n-1))/Product_{k = 1..n} ((1 - x^(3*k)) *(1 - x^(3*k-1))). (Set z = x^2 and q = x^3 in McLaughlin et al., Section 1.3, Entry 7.) - Peter Bala, Feb 02 2021
|
|
|
MAPLE
|
g:=add(x^(n*(3*n-1))/mul((1-x^(3*k))*(1-x^(3*k-1)), k = 1..n), n = 0..6): gser:=series(g, x, 101): seq(coeff(gser, x, n), n = 0..100); # Peter Bala, Feb 02 2021
|
|
|
MATHEMATICA
|
nmax=100; CoefficientList[Series[Product[1/(1-x^(3*k+2)), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 26 2015 *)
nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 3] == 2, Do[poly[[j + 1]] -= poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly2 = Take[poly, {2, nmax + 1}]; poly3 = 1 + Sum[poly2[[n]]*x^n, {n, 1, Length[poly2]}]; CoefficientList[Series[1/poly3, {x, 0, Length[poly2]}], x] (* Vaclav Kotesovec, Jan 13 2017 *)
nmax = 50; s = Range[0, nmax/3]*3 + 2;
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 05 2020 *)
|
|
|
PROG
|
(PARI) {a(n)= if( n<0, 0, polcoeff( 1 / prod( k=1, n, 1 - (k%3==2) * x^k, 1 + x * O(x^n)), n))} /* Michael Somos, Jul 24 2007 */
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
