OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
G. E. Andrews, R. P. Lewis, J. Lovejoy, Partitions with designated summands, Acta Arith. 105 (2002), no. 1, 51-66.
Nayandeep Deka Baruah and Kanan Kumari Ojah, Partitions with designated summands in which all parts are odd, INTEGERS 15 (2015), #A9.
FORMULA
Euler transform of period 12 sequence [1, 1, 2, 0, 1, 0, 1, 0, 2, 1, 1, 0, ...].
a(n) ~ 5^(1/4) * exp(sqrt(5*n)*Pi/3) / (2^(5/2)*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Nov 28 2015
G.f.: Product_{k>=1} (1 + Sum_{j>=1} j * x^(j*(2*k - 1))). - Ilya Gutkovskiy, Nov 06 2019
EXAMPLE
a(8)=22 because in the six partitions of 8 into odd parts, namely, 71,53,5111,3311,311111,11111111, the multiplicities of the parts are (1,1),(1,1),(1,3),(2,2),(1,5),(8) with products 1,1,3,4,5,8, having sum 22.
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-2) +add(b(n-i*j, i-2)*j, j=1..n/i)))
end:
a:= n-> b(n, iquo(1+n, 2)*2-1):
seq(a(n), n=0..50); # Alois P. Heinz, Feb 26 2013
MATHEMATICA
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 2] + Sum[b[n - i*j, i - 2]*j, {j, 1, n/i}]]]; a[n_] := b[n, Quotient[1 + n, 2]*2 - 1]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)
nmax=60; CoefficientList[Series[Product[(1-x^(4*k)) * (1+x^(3*k)) / ((1-x^k) * (1+x^(6*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2015 *)
Table[Total[l = Tally /@ Select[IntegerPartitions@n, VectorQ[#, OddQ] &];
Table[x = l[[i]]; Product[x[[j, 2]], {j, Length[x]}], {i, Length[l]}]], {n, 0, 47}] (* Robert Price, Jun 08 2020 *)
PROG
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^4+A)*eta(x^6+A)^2/ eta(x+A)/eta(x^3+A)/eta(x^12+A), n))} /* Michael Somos, Jul 30 2006 */
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Feb 16 2005
EXTENSIONS
More terms from Emeric Deutsch, Mar 28 2005
Name expanded by N. J. A. Sloane, Nov 21 2015
STATUS
approved