A321139 -id:A321139 - cp's OEIS frontend

A300446 Expansion of Product_{k>0} (Sum_{m>=0} x^(k*m^2)).

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

Offset: 0

Seiichi Manyama, May 11 2018

nonn

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

			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;

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 [math.CO], 2018, p. 30.

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

m:=60; R:=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

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

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

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

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

A321179 a(n) = [x^(n^2)] Product_{k=1..n} theta_3(q^k), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 2, 14, 44, 174, 988, 4314, 20780, 126320, 692328, 3836166, 23160914, 135752866, 803203484, 4902966108, 29745996950, 181712320506, 1124481497694, 6965802854354, 43360326335154, 271658784580760, 1706393926177980, 10757142052998054, 68081390206251952, 432001821971576352

Offset: 0

Seiichi Manyama, Oct 29 2018

nonn

Also the number of integer solutions (a_1, a_2, ... , a_n) to the equation a_1^2 + 2*a_2^2 + ... + n*a_n^2 = n^2.

			Solutions (a_1, a_2, a_3) to the equation a_1^2 + 2*a_2^2 + 3*a_3^2 = 9.
------------------------------------------------------------------------
( 1,  2,  0), ( 1, -2,  0),
(-1,  2,  0), (-1, -2,  0),
( 2,  1,  1), ( 2,  1, -1),
( 2, -1,  1), ( 2, -1, -1),
(-2,  1,  1), (-2,  1, -1),
(-2, -1,  1), (-2, -1, -1),
( 3,  0,  0), (-3,  0,  0).

Vaclav Kotesovec, Table of n, a(n) for n = 0..400 (first 91 terms from Seiichi Manyama)

Cf. A000122, A000290, A320067, A320931, A321139.

nmax = 20; Table[SeriesCoefficient[Product[EllipticTheta[3, 0, x^k], {k, 1, n}], {x, 0, n^2}], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 29 2018 *)

{a(n) = polcoeff(prod(i=1, n, 1+2*sum(j=1, sqrtint(n^2\i), x^(i*j^2)+x*O(x^(n^2)))), n^2)}

a(n) ~ c * d^n / n^(7/4), where d = 6.8137220913147... and c = 0.178176349247... - Vaclav Kotesovec, Oct 30 2018

A321238 a(n) = [x^(n^3)] Product_{k=1..n} Sum_{m>=0} x^(k*m^2).

Original entry on oeis.org

1, 1, 1, 4, 16, 87, 911, 8081, 82494, 1108584, 14559487, 206462480, 3300362073, 54235076625, 939612600043, 17366394088532, 332129019947772, 6615538793829307, 137564490944940832, 2954281836759475893, 65572183746807351880, 1503752010271535590284, 35476544827929325305961

Offset: 0

Seiichi Manyama, Nov 01 2018

nonn

Also the number of nonnegative integer solutions (a_1, a_2, ... , a_n) to the equation a_1^2 + 2*a_2^2 + ... + n*a_n^2 = n^3.

			1*0^2 + 2*0^2 + 3*0^2 + 4*4^2 = 64.
1*0^2 + 2*0^2 + 3*4^2 + 4*2^2 = 64.
1*1^2 + 2*0^2 + 3*3^2 + 4*3^2 = 64.
1*1^2 + 2*4^2 + 3*3^2 + 4*1^2 = 64.
1*2^2 + 2*2^2 + 3*4^2 + 4*1^2 = 64.
1*2^2 + 2*4^2 + 3*2^2 + 4*2^2 = 64.
1*4^2 + 2*0^2 + 3*2^2 + 4*3^2 = 64.
1*4^2 + 2*0^2 + 3*4^2 + 4*0^2 = 64.
1*4^2 + 2*4^2 + 3*0^2 + 4*2^2 = 64.
1*4^2 + 2*4^2 + 3*2^2 + 4*1^2 = 64.
1*5^2 + 2*0^2 + 3*1^2 + 4*3^2 = 64.
1*5^2 + 2*2^2 + 3*3^2 + 4*1^2 = 64.
1*5^2 + 2*4^2 + 3*1^2 + 4*1^2 = 64.
1*6^2 + 2*0^2 + 3*2^2 + 4*2^2 = 64.
1*7^2 + 2*2^2 + 3*1^2 + 4*1^2 = 64.
1*8^2 + 2*0^2 + 3*0^2 + 4*0^2 = 64.
So a(4) = 16.

Cf. A300446, A321139, A321239.

{a(n) = polcoeff(prod(i=1, n, sum(j=0, sqrtint(n^3\i), x^(i*j^2)+x*O(x^(n^3)))), n^3)}

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A300446 Expansion of Product_{k>0} (Sum_{m>=0} x^(k*m^2)).

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A321179 a(n) = [x^(n^2)] Product_{k=1..n} theta_3(q^k), where theta_3() is the Jacobi theta function.

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A321238 a(n) = [x^(n^3)] Product_{k=1..n} Sum_{m>=0} x^(k*m^2).

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