cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A280456 Expansion of Product_{k>=0} (1 + x^(6*k+1)).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 3, 2, 0, 0, 0, 1, 3, 3, 1, 0, 0, 1, 4, 4, 1, 0, 0, 1, 4, 5, 2, 0, 0, 1, 5, 7, 3, 0, 0, 1, 5, 8, 5, 1, 0, 1, 6, 10, 6, 1, 0, 1, 6, 12, 9, 2, 0, 1, 7, 14, 11, 3, 0, 1, 7, 16, 15, 5, 0, 1, 8, 19, 18, 7, 1, 1, 8, 21, 23, 10, 1, 1, 9, 24
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 03 2017

Keywords

Comments

Number of partitions of n into distinct parts congruent to 1 mod 6.
Convolution of A281244 and A280456 is A098884. - Vaclav Kotesovec, Jan 18 2017

Examples

			a(32) = 3 because we have [31, 1], [25, 7] and [19, 13].

Links

Crossrefs

Cf. A000700, A016921, A109701, A169975, A261612, A280454, A280457.
Cf. A262928, A147599, A281243, A281244, A281245.

Programs

  • Mathematica
    nmax = 105; CoefficientList[Series[Product[(1 + x^(6 k + 1)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 6] == 1, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 18 2017 *)

Formula

G.f.: Product_{k>=0} (1 + x^(6*k+1)).
a(n) ~ exp(Pi*sqrt(n)/(3*sqrt(2)))/(2*2^(5/12)*sqrt(3)*n^(3/4)) * (1 + (Pi/(144*sqrt(2)) - 9/(4*sqrt(2)*Pi)) / sqrt(n)). - Ilya Gutkovskiy, Jan 03 2017, extended by Vaclav Kotesovec, Jan 18 2017

A262949 Expansion of Product_{k>=1} (1 + x^(3*k-2))^(3*k-2).

Original entry on oeis.org

1, 1, 0, 0, 4, 4, 0, 7, 13, 6, 10, 38, 32, 17, 74, 103, 59, 139, 266, 191, 247, 593, 581, 513, 1175, 1487, 1190, 2223, 3453, 2938, 4158, 7264, 7095, 8052, 14430, 16308, 16246, 27364, 35347, 34096, 50997, 72595, 72163, 94707, 142522, 151435, 178047, 270112
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 05 2015

Keywords

Links

Crossrefs

Cf. A261612, A262879, A262884, A262924, A262948.

Programs

  • Mathematica
    nmax=60; CoefficientList[Series[Product[(1 + x^(3*k-2))^(3*k-2),{k,1,nmax}],{x,0,nmax}],x]

Formula

a(n) ~ exp(3 * 2^(-4/3) * Zeta(3)^(1/3) * n^(2/3)) * Zeta(3)^(1/6) / (2^(7/12) * sqrt(3*Pi) * n^(2/3)).

A284312 Expansion of Product_{k>=0} (1 - x^(3*k+1)) in powers of x.

Original entry on oeis.org

1, -1, 0, 0, -1, 1, 0, -1, 1, 0, -1, 2, -1, -1, 2, -1, -1, 3, -2, -1, 3, -3, 0, 4, -4, 0, 4, -5, 1, 5, -7, 2, 5, -8, 4, 5, -10, 5, 5, -12, 8, 5, -14, 10, 4, -16, 14, 3, -19, 17, 1, -20, 22, -1, -23, 26, -4, -25, 33, -8, -27, 38, -13, -28, 46, -19, -30, 53, -26, -29
Offset: 0

Views

Author

Seiichi Manyama, Mar 24 2017

Keywords

Links

Crossrefs

Cf. Product_{k>=0} (1 - x^(m*k+1)): A081362 (m=2), this sequence (m=3), A284313 (m=4), A284314 (m=5).
Cf. A078181, A261612.

Programs

  • Mathematica
    CoefficientList[Series[Product[1 - x^(3k + 1), {k, 0, 100}], {x, 0, 100}], x] (* Indranil Ghosh, Mar 25 2017 *)
  • PARI
    Vec(prod(k=0, 100, 1 - x^(3*k + 1)) + O(x^101)) \\ Indranil Ghosh, Mar 25 2017

Formula

a(n) = -(1/n)*Sum_{k=1..n} A078181(k)*a(n-k), a(0) = 1.

A337547 Number of compositions (ordered partitions) of n into distinct parts congruent to 1 mod 3.

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 0, 1, 2, 0, 1, 4, 6, 1, 4, 6, 1, 6, 12, 1, 6, 18, 25, 8, 24, 25, 8, 30, 49, 10, 42, 73, 10, 48, 121, 132, 60, 145, 132, 72, 217, 254, 84, 265, 374, 96, 361, 616, 114, 433, 856, 846, 553, 1218, 864, 649, 1578, 1602, 817, 2180, 2340, 937, 2780, 3798, 1129, 3622
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 22 2020

Keywords

Examples

			a(12) = 6 because we have [7, 4, 1], [7, 1, 4], [4, 7, 1], [4, 1, 7], [1, 7, 4] and [1, 4, 7].

Links

Crossrefs

Cf. A000930, A016777, A032020, A032021, A035382, A261612, A337548, A339059, A339060.

Programs

  • Mathematica
    nmax = 65; CoefficientList[Series[Sum[k! x^(k (3 k - 1)/2)/Product[1 - x^(3 j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=0} k! * x^(k*(3*k - 1)/2) / Product_{j=1..k} (1 - x^(3*j)).

A261616 Expansion of Product_{k>=0} 1/(1 - x^(3*k+1))^2.

Original entry on oeis.org

1, 2, 3, 4, 7, 10, 13, 18, 26, 34, 44, 58, 76, 96, 123, 156, 196, 244, 304, 374, 461, 566, 690, 836, 1015, 1224, 1470, 1762, 2110, 2512, 2987, 3542, 4191, 4944, 5825, 6842, 8025, 9392, 10971, 12788, 14891, 17300, 20068, 23242, 26883, 31034, 35787, 41204
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 26 2015

Keywords

Comments

Self-convolution of A035382.
In general, if a > 0, b > 0, GCD(a,b) = 1 and g.f. = Product_{k>=0} 1/(1 - x^(a*k+b))^2, then a(n) ~ Gamma(b/a)^2 * a^(b/a - 3/4) * exp(2*Pi*sqrt(n/(3*a))) * Pi^(2*b/a - 2) / (4 * 3^(b/a - 1/4) * n^(b/a + 1/4)).

Crossrefs

Cf. A022567, A035382, A261610, A261612, A261615.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Product[1/(1 - x^(3*k+1))^2, {k, 0, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(2*Pi*sqrt(n)/3) * Gamma(1/3)^2 / (4 * sqrt(3) * Pi^(4/3) * n^(7/12)).

A261615 Expansion of Product_{k>=0} (1 + x^(3*k+1))^2.

Original entry on oeis.org

1, 2, 1, 0, 2, 4, 2, 2, 5, 4, 3, 8, 10, 6, 9, 14, 11, 14, 22, 18, 17, 30, 32, 28, 41, 46, 39, 54, 68, 60, 73, 94, 85, 96, 131, 128, 130, 170, 175, 176, 229, 246, 237, 294, 330, 320, 386, 446, 430, 492, 582, 578, 642, 762, 763, 818, 977, 1008, 1061, 1254, 1311
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 26 2015

Keywords

Comments

Self-convolution of A261612.
In general, if a > 0, b > 0, GCD(a,b) = 1 and g.f. = Product_{k>=0} (1 + x^(a*k+b))^2, then a(n) ~ exp(Pi*sqrt(2*n/(3*a))) / (2^(2*b/a + 1/4) * 3^(1/4) * a^(1/4) * n^(3/4)).

Crossrefs

Cf. A022567, A035382, A261610, A261612, A261616.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Product[(1 + x^(3*k+1))^2, {k, 0, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(Pi*sqrt(2*n)/3) / (2^(11/12) * sqrt(3) * n^(3/4)).

A374064 Expansion of Product_{k>=1} 1 / (1 + x^(3*k-1)).

Original entry on oeis.org

1, 0, -1, 0, 1, -1, -1, 1, 0, -1, 1, 0, -1, 1, 0, -2, 2, 1, -3, 1, 3, -3, 0, 3, -3, -1, 4, -3, -1, 5, -3, -3, 7, -3, -5, 7, -1, -7, 8, 0, -8, 8, 1, -11, 10, 3, -14, 9, 8, -17, 8, 10, -18, 6, 14, -22, 6, 19, -24, 1, 26, -26, -3, 30, -25, -9, 37, -27, -13, 42, -26, -23, 51, -25, -31, 56
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 27 2024

Keywords

Crossrefs

Cf. A035386, A081360, A081362, A109389, A132463, A261612, A262928, A284315, A374065.

Programs

  • Mathematica
    nmax = 75; CoefficientList[Series[Product[1/(1 + x^(3 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[DivisorSum[k, (-1)^(k/#) # &, Mod[#, 3] == 2 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

Formula

a(0) = 1; a(n) = -Sum_{k=1..n} A262928(k) * a(n-k).
a(n) = Sum_{k=0..n} A081362(k) * A132463(n-k).
a(n) = Sum_{k=0..n} A109389(k) * A261612(n-k).

A374065 Expansion of Product_{k>=1} 1 / (1 + x^(3*k-2)).

Original entry on oeis.org

1, -1, 1, -1, 0, 0, 0, -1, 2, -2, 1, 0, -1, 0, 2, -3, 3, -1, -1, 1, 1, -4, 5, -3, 0, 2, 0, -4, 7, -6, 1, 3, -2, -3, 9, -10, 4, 3, -5, -1, 11, -15, 10, 1, -8, 3, 10, -20, 17, -3, -10, 9, 7, -24, 26, -10, -10, 15, 2, -27, 37, -21, -8, 22, -6, -28, 49, -36, -2, 30, -19, -24, 61, -56, 10, 35
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 27 2024

Keywords

Crossrefs

Cf. A035382, A081360, A081362, A109389, A132462, A261612, A262928, A284312, A374064.

Programs

  • Mathematica
    nmax = 75; CoefficientList[Series[Product[1/(1 + x^(3 k - 2)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[DivisorSum[k, (-1)^(k/#) # &, Mod[#, 3] == 1 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

Formula

a(0) = 1; a(n) = -Sum_{k=1..n} A261612(k) * a(n-k).
a(n) = Sum_{k=0..n} A081362(k) * A132462(n-k).
a(n) = Sum_{k=0..n} A109389(k) * A262928(n-k).

A261634 Expansion of Product_{k>=0} (1+x^(4*k+1))^3.

Original entry on oeis.org

1, 3, 3, 1, 0, 3, 9, 9, 3, 3, 12, 18, 12, 6, 18, 37, 33, 15, 22, 54, 66, 42, 36, 81, 114, 84, 57, 112, 189, 171, 109, 156, 279, 294, 201, 222, 405, 486, 360, 328, 564, 747, 617, 504, 783, 1123, 1017, 783, 1065, 1602, 1605
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 27 2015

Keywords

Crossrefs

Cf. A261612, A261615, A261633, A169975, A261630.

Programs

  • Mathematica
    nmax=50; CoefficientList[Series[Product[(1+x^(4*k+1))^3, {k, 0, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(Pi*sqrt(n)/2) / (2^(5/4) * n^(3/4)).

A261638 Expansion of Product_{k>=0} (1+x^(4*k+1))^4.

Original entry on oeis.org

1, 4, 6, 4, 1, 4, 16, 24, 16, 8, 22, 48, 52, 32, 38, 92, 128, 96, 70, 140, 245, 244, 172, 228, 417, 488, 374, 380, 680, 924, 798, 676, 1044, 1560, 1542, 1256, 1625, 2524, 2778, 2304, 2537, 3892, 4716, 4156, 4076, 5908, 7650, 7196, 6592, 8796, 11938
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 27 2015

Keywords

Comments

In general, if j > 0, a > 0, b > 0, GCD(a,b) = 1 and g.f. = Product_{k>=0} (1 + x^(a*k+b))^j, then a(n) ~ 2^((j-3)/2 - j*b/a) * j^(1/4) * exp(Pi*sqrt(j*n/(3*a))) / ((3*a)^(1/4) * n^(3/4)).

Crossrefs

Cf. A261612, A261615, A261637, A169975, A261630, A261634.

Programs

  • Mathematica
    nmax=50; CoefficientList[Series[Product[(1+x^(4*k+1))^4, {k, 0, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(Pi*sqrt(n/3)) / (sqrt(2) * 3^(1/4) * n^(3/4)).
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