A027347 Number of partitions of n into distinct odd parts, the least being congruent to 1 mod 4.

1, 0, 0, 1, 1, 1, 0, 1, 2, 1, 1, 2, 3, 2, 2, 3, 4, 3, 3, 5, 6, 6, 5, 7, 9, 8, 8, 11, 12, 12, 12, 15, 18, 17, 18, 22, 25, 25, 26, 30, 34, 34, 36, 42, 47, 48, 50, 57, 64, 65, 69, 78, 85, 89, 93, 104, 114, 118, 125, 139, 151, 157, 166, 183, 198, 207, 219, 240

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

Crossrefs

Cf. A026832.

Programs

Maple

G := add( q^(4*n+1)*mul( 1 + q^(2*k+1), k = 2*n+1..50 ), n = 0..25 ):
S := series(G, q, 101):
seq(coeff(S, q, j), j = 1..100); # Peter Bala, Jan 31 2021

Formula

From Peter Bala, Jan 31 2021: (Start)

G.f.: A(q) = Sum_{n >= 0} q^(4*n+1) * Product_{k >= 2*n+1} 1 + q^(2*k+1).

A(q) = Limit_{N -> oo} Sum_{n = 0..2*N+1} (-1)^n * Product_{k = n..2*N+1} 1 + q^(2*k+1) = Limit_{N -> oo} Sum_{n = 0..2*N+1} (-1)^n * Product_{k >= n} 1 + q^(2*k+1). (End)