login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A300446 Expansion of Product_{k>0} (Sum_{m>=0} x^(k*m^2)). 11
1, 1, 1, 2, 3, 3, 5, 6, 8, 12, 12, 17, 23, 27, 32, 41, 52, 61, 77, 91, 110, 134, 159, 188, 228, 271, 314, 380, 444, 518, 612, 713, 832, 976, 1128, 1308, 1529, 1756, 2023, 2343, 2698, 3091, 3555, 4072, 4657, 5343, 6074, 6922, 7912, 8986, 10194, 11590, 13135, 14855 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Also the number of partitions of n in which each part occurs a square number (>=0) of times.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Seiichi Manyama)

Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449, 2018, p. 30.

FORMULA

G.f.: Product_{k>=1} (theta_3(x^k) + 1)/2, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Oct 25 2018

EXAMPLE

n | Partitions of n in which each part occurs a square number (>=0) of times

--+-------------------------------------------------------------------------

1 | 1;

2 | 2;

3 | 3 = 2+1;

4 | 4 = 3+1 = 1+1+1+1;

5 | 5 = 4+1 = 3+2;

6 | 6 = 5+1 = 4+2 = 3+2+1 = 2+1+1+1+1;

7 | 7 = 6+1 = 5+2 = 4+3 = 4+2+1 = 3+1+1+1+1;

8 | 8 = 7+1 = 6+2 = 5+3 = 5+2+1 = 4+3+1 = 4+1+1+1+1 = 2+2+2+2;

MAPLE

b:= proc(n, i) option remember; local j; if n=0 then 1

      elif i<1 then 0 else b(n, i-1); for j while

        i*j^2<=n do %+b(n-i*j^2, i-1) od; % fi

    end:

a:= n-> b(n$2):

seq(a(n), n=0..60);  # Alois P. Heinz, May 11 2018

MATHEMATICA

nmax = 60; CoefficientList[Series[Product[(EllipticTheta[3, 0, x^k] + 1)/2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 25 2018 *)

PROG

(PARI) N=99; x='x+O('x^N); Vec(prod(i=1, N, sum(j=0, sqrtint(N\i), x^(i*j^2)))) \\ Seiichi Manyama, Oct 28 2018

(MAGMA) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(&+[x^(k*j^2):j in [0..2*m]]): k in [1..2*m]]) ));  // G. C. Greubel, Oct 29 2018

CROSSREFS

Cf. A000041, A000122, A010052, A158441, A232173, A298329, A304329, A320932, A321139.

Sequence in context: A027586 A039860 A084338 * A039876 A239312 A317167

Adjacent sequences:  A300443 A300444 A300445 * A300447 A300448 A300449

KEYWORD

nonn

AUTHOR

Seiichi Manyama, May 11 2018

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 2 17:02 EDT 2020. Contains 334787 sequences. (Running on oeis4.)