A282209 Expansion of Product_{k>=1} 1/(1 - k^2*x^(k^2)).

1, 1, 1, 1, 5, 5, 5, 5, 21, 30, 30, 30, 94, 130, 130, 130, 402, 546, 627, 627, 1715, 2291, 2615, 2615, 6967, 9440, 10736, 11465, 28873, 38765, 43949, 46865, 116753, 156321, 178578, 190242, 476391, 634663, 723691, 770347, 1914943, 2550735, 2906847, 3107160, 7685544

Offset

0,5

Comments

Sum of products of terms in all partitions of n into squares (A000290).

Links

Table of n, a(n) for n=0..44.

Index entries for related partition-counting sequences

Formula

G.f.: Product_{k>=1} 1/(1 - k^2*x^(k^2)).

From Vaclav Kotesovec, Feb 09 2017: (Start)

a(n) ~ c * 2^(n/2), where:

c = 1.84902025727376837058629436557644856279088... if n == 0 (mod 4),

c = 1.74739571210218418633067606853005648684028... if n == 1 (mod 4),

c = 1.41060067910504703778072732362810764186990... if n == 2 (mod 4),

c = 1.06705333199321743850009229910087278853310... if n == 3 (mod 4).

(End)

Example

a(8) = 21 because we have [4, 4], [4, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1], 4*4 = 16, 4*1*1*1*1 = 4, 1*1*1*1*1*1*1*1 = 1 and 16 + 4 + 1 = 21.

Mathematica

nmax = 44; CoefficientList[Series[Product[1/(1 - k^2 x^k^2), {k, 1, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A000290, A001156, A006906.

Sequence in context: A078097 A285243 A339704 * A082476 A365492 A024729

Adjacent sequences: A282206 A282207 A282208 * A282210 A282211 A282212

Keyword

nonn

Author

Ilya Gutkovskiy, Feb 09 2017

Status

approved