OFFSET
0,5
COMMENTS
For a given partition cn(i,n) means the number of its parts equal to i modulo n.
Short: (2=3 := 0).
It appears that this sequence is related to the generalized heptagonal numbers A085787 in the same way as the partition numbers A000041 are related to the generalized pentagonal numbers A001318. (See the table in comments section of A195825.) Conjecture: Column 1 of triangle A195837. Also 1 together with the row sums of triangle A195837. Also column 3 of the square array A195825. - Omar E. Pol, Oct 08 2011
Note that this sequence contains two plateaus: [1, 1, 1, 1] and [4, 4]. For more information see A195825 and A210843. - Omar E. Pol, Jun 23 2012
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000
FORMULA
Euler transform of period 5 sequence [1, 0, 0, 1, 1, ...]. - Michael Somos, Feb 09 2012
Expansion of 1 / f(-x, -x^4) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Sep 08 2012
Convolution inverse of A113429. - Michael Somos, Feb 09 2012
G.f.: 1 / (Product_{k>0} (1 - x^(5*k)) * (1 - x^(5*k - 1)) * (1 - x^(5*k - 4))). - Michael Somos, Sep 08 2012
G.f.: 1 / (Sum_{k in Z} (-1)^k * x^(k * (5*k + 3) / 2)). - Michael Somos, Sep 08 2012
a(n) ~ sqrt(1+sqrt(5)) * exp(sqrt(2*n/5)*Pi) / (2^(5/2)*5^(1/4)*n). - Vaclav Kotesovec, Oct 06 2015
a(n) = (1/n)*Sum_{k=1..n} A284361(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
EXAMPLE
G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 4*x^7 + 5*x^8 + 7*x^9 + 10*x^10 + ...
G.f. = q^-9 + q^31 + q^71 + q^111 + 2*q^151 + 3*q^191 + 4*q^231 + 4*q^271 + 5*q^311 + ... - Michael Somos, Sep 08 2012
MAPLE
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*[1, 1, 0, 0, 1]
[1+irem(d, 5)], d=divisors(j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..60); # Alois P. Heinz, Apr 04 2014
MATHEMATICA
a[n_] := a[n] = If[n == 0, 1, Sum[ Sum[ d*{1, 1, 0, 0, 1}[[1 + Mod[d, 5]]], {d, Divisors[j]}] * a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Apr 15 2015, after Alois P. Heinz *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( 1 / prod( k=1, (n+4)\5, (1 - x^(5*k - 4)) * (1 - x^(5*k - 1)) * (1 - x^(5*k)), 1 + x * O(x^n)), n))}; /* Michael Somos, Feb 09 2012 */
(GW-BASIC)' A program with two A-numbers:
20 For n = 1 to 56: For j = 1 to n
40 Next j: Print a(n-1); : Next n ' Omar E. Pol, Jun 10 2012
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved