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A303668 Expansion of 1/((1 - x)*(2 - theta_2(sqrt(x))/(2*x^(1/8)))), where theta_2() is the Jacobi theta function. 4
1, 2, 3, 5, 8, 12, 19, 30, 46, 71, 111, 172, 266, 413, 640, 991, 1537, 2383, 3692, 5722, 8869, 13745, 21303, 33018, 51172, 79308, 122917, 190503, 295251, 457597, 709207, 1099165, 1703546, 2640245, 4091988, 6341979, 9829132, 15233702, 23609994, 36592010, 56712212, 87895562 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Partial sums of A023361.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..5254

Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Index entries for sequences related to compositions

FORMULA

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

MAPLE

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

      add(`if`(issqr(8*j+1), b(n-j), 0), j=1..n))

    end:

a:= proc(n) option remember;

      `if`(n<0, 0, b(n)+a(n-1))

    end:

seq(a(n), n=0..50);  # Alois P. Heinz, Apr 28 2018

MATHEMATICA

nmax = 41; CoefficientList[Series[1/((1 - x) (2 - EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)))), {x, 0, nmax}], x]

nmax = 41; CoefficientList[Series[1/((1 - x) (1 - Sum[x^(k (k + 1)/2), {k, 1, nmax}])), {x, 0, nmax}], x]

a[0] = 1; a[n_] := a[n] = Sum[SquaresR[1, 8 k + 1] a[n - k], {k, 1, n}]/2; Accumulate[Table[a[n], {n, 0, 41}]]

CROSSREFS

Cf. A000217, A010054, A023361, A302835, A303667.

Sequence in context: A240523 A023436 A024567 * A060961 A225393 A243850

Adjacent sequences:  A303665 A303666 A303667 * A303669 A303670 A303671

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Apr 28 2018

STATUS

approved

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Last modified December 6 16:24 EST 2019. Contains 329808 sequences. (Running on oeis4.)