A300974 -id:A300974 - cp's OEIS frontend

A300975 a(n) = [x^n] Product_{k>=1} 1/(1 - x^(k^3))^n.

1, 1, 3, 10, 35, 126, 462, 1716, 6443, 24391, 92928, 355862, 1368458, 5280744, 20438148, 79302960, 308385355, 1201536286, 4689450021, 18330233110, 71747534460, 281177705490, 1103163479190, 4332522733560, 17031238725410, 67007449610751, 263841039245280, 1039628691988795

Offset: 0

Ilya Gutkovskiy, Mar 17 2018

nonn

Number of partitions of n into cubes of n kinds.

Cf. A000578, A003108, A008485, A023872, A298434, A300974.

a:= proc(m) option remember; local b; b:= proc(n, i)
      option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(m+j-1, j)*b(n-i^3*j, i-1), j=0..n/i^3)))
      end: b(n, iroot(n, 3))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 17 2018

Table[SeriesCoefficient[Product[1/(1 - x^k^3)^n, {k, 1, n}], {x, 0, n}], {n, 0, 27}]

a(n) ~ c * d^n / sqrt(n), where d = 4.0147940395164236614815683662796167488... and c = 0.2726202310726337579308600184572222... - Vaclav Kotesovec, Mar 23 2018

A301335 a(n) = [x^n] 1/(1 + (1/2)n(1 - theta_3(x))), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 1, 4, 27, 260, 3175, 47304, 833147, 16941120, 390611331, 10070060200, 287028156162, 8962583345856, 304255011200647, 11156593415089808, 439452231820920000, 18505340390664634384, 829599437871129843839, 39447684087807950938908, 1983038000428208822539998, 105080571577382659860160800

Offset: 0

Ilya Gutkovskiy, Mar 18 2018

nonn

Number of compositions (ordered partitions) of n into squares of n kinds.

Cf. A000290, A006456, A240944, A300974, A301334.

Table[SeriesCoefficient[1/(1 + (1/2) n (1 - EllipticTheta[3, 0, x])), {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[1/(1 - n Sum[x^k^2, {k, 1, n}]), {x, 0, n}], {n, 0, 20}]

a(n) = [x^n] 1/(1 - n*Sum_{k>=1} x^(k^2)).

a(n) ~ n^n * (1 + 1/n^2 - 3/n^3 + 1/(2*n^4) - 13/(2*n^5) + 127/(6*n^6) - 4/n^7 + 335/(8*n^8) - 665/(4*n^9) + 337/(15*n^10) + ...). - Vaclav Kotesovec, Mar 19 2018

A301518 a(n) = [x^n] Product_{k>=1} (1 + x^(k^2))^n.

Original entry on oeis.org

1, 1, 1, 1, 5, 26, 91, 246, 589, 1468, 4226, 13311, 41471, 122864, 351184, 1001876, 2920957, 8698612, 26070130, 77707056, 229959130, 679050870, 2011457295, 5986185690, 17866178695, 53343031301, 159149943668, 474683353849, 1416730630936, 4233405443596

Offset: 0

Vaclav Kotesovec, Mar 23 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A033461, A281790, A291649, A300974.

Table[SeriesCoefficient[Product[(1 + x^(k^2))^n, {k, 1, n}], {x, 0, n}], {n, 0, 30}]

a(n) ~ c * d^n / sqrt(n), where d = 3.04074590736461391963643911... and c = 0.2268848664201836146769277...

A301458 a(n) = [x^n] Product_{k>=1} 1/(1 - x^(k*(k+1)/2))^n.

Original entry on oeis.org

1, 1, 3, 13, 51, 201, 825, 3431, 14355, 60493, 256463, 1092268, 4669665, 20029036, 86148373, 371434173, 1604845715, 6946936628, 30121158813, 130795358333, 568709929191, 2475778867547, 10789659781640, 47069225185789, 205524447217185, 898163031782576, 3928112419640126

Offset: 0

Ilya Gutkovskiy, Mar 21 2018

nonn

Number of partitions of n into triangular numbers of n kinds.

Cf. A000217, A000294, A007294, A008485, A298435, A298730, A300974, A301334.

Table[SeriesCoefficient[Product[1/(1 - x^(k (k + 1)/2))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]

A302836 a(n) = [x^n] Product_{k>=1} 1/(1 - n*x^(k^2)).

Original entry on oeis.org

1, 1, 4, 27, 260, 3150, 46872, 825944, 16810048, 387952668, 10010010100, 285526191874, 8921263237056, 303013028232642, 11116057874586840, 438023675344410000, 18451248777413066768, 827408674110381669305, 39353155876513869320412, 1978708139249503877752798

Offset: 0

Ilya Gutkovskiy, Apr 13 2018

nonn

Cf. A001156, A124577, A285245, A291583, A292417, A300974.

Table[SeriesCoefficient[Product[1/(1 - n x^k^2), {k, 1, n}], {x, 0, n}], {n, 0, 19}]

a(n) ~ n^n * (1 + 1/n^3 + 1/n^6 + 1/n^8 + 1/n^9 + 1/n^11 + 1/n^12 + 1/n^14 + 2/n^15 + 1/n^16 + 1/n^17 + 2/n^18 + 1/n^19 + 1/n^20 + 2/n^21 + 1/n^22 + 2/n^23 + 4/n^24 + 1/n^25 + 2/n^26 + 4/n^27 + 1/n^28 + 2/n^29 + 5/n^30 + ...), for coefficients see A111178. - Vaclav Kotesovec, Apr 13 2018

cp's OEIS Frontend

A300975 a(n) = [x^n] Product_{k>=1} 1/(1 - x^(k^3))^n.

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A301335 a(n) = [x^n] 1/(1 + (1/2)*n*(1 - theta_3(x))), where theta_3() is the Jacobi theta function.

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A301518 a(n) = [x^n] Product_{k>=1} (1 + x^(k^2))^n.

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Programs

Mathematica

Formula

A301458 a(n) = [x^n] Product_{k>=1} 1/(1 - x^(k*(k+1)/2))^n.

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Author

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Programs

Mathematica

A302836 a(n) = [x^n] Product_{k>=1} 1/(1 - n*x^(k^2)).

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Author

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Programs

Mathematica

Formula

A301335 a(n) = [x^n] 1/(1 + (1/2)n(1 - theta_3(x))), where theta_3() is the Jacobi theta function.