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A281704 Expansion of Sum_{i>=1} x^(i^2) / (1 - Sum_{j>=1} x^(j^2))^2. 1
1, 2, 3, 5, 9, 15, 23, 35, 55, 87, 134, 202, 305, 463, 700, 1049, 1565, 2334, 3478, 5168, 7654, 11314, 16705, 24632, 36260, 53295, 78237, 114728, 168059, 245916, 359483, 525021, 766144, 1117107, 1627587, 2369609, 3447549, 5012588, 7283577, 10577198, 15351519, 22268890, 32286666, 46788056, 67770831 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Total number of parts in all compositions (ordered partitions) of n into squares (A000290).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..6575

Index entries for sequences related to sums of squares

Index entries for sequences related to compositions

FORMULA

G.f.: Sum_{i>=1} x^(i^2) / (1 - Sum_{j>=1} x^(j^2))^2.

EXAMPLE

a(6) = 15 because we have [4, 1, 1], [1, 4, 1], [1, 1, 4], [1, 1, 1, 1, 1, 1] and 3 + 3 + 3 + 6 = 15.

MAPLE

b:= proc(n) option remember; `if`(n=0, [1, 0], add(

      (p-> p+[0, p[1]])(b(n-j^2)), j=1..isqrt(n)))

    end:

a:= n-> b(n)[2]:

seq(a(n), n=1..45);  # Alois P. Heinz, Aug 07 2019

MATHEMATICA

nmax = 45; Rest[CoefficientList[Series[Sum[x^i^2, {i, 1, nmax}]/(1 - Sum[x^j^2, {j, 1, nmax}])^2, {x, 0, nmax}], x]]

nmax = 45; Rest[CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)/(2 (1 + (1 - EllipticTheta[3, 0, x])/2)^2), {x, 0, nmax}], x]]

CROSSREFS

Cf. A000290, A006456.

Sequence in context: A306829 A067798 A205536 * A074693 A147322 A143282

Adjacent sequences:  A281701 A281702 A281703 * A281705 A281706 A281707

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Jan 30 2017

STATUS

approved

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Last modified August 7 17:44 EDT 2020. Contains 336278 sequences. (Running on oeis4.)