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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A320591 Expansion of Product_{k>=1} (1 + x^k/(1 + x)^k).

Original entry on oeis.org

1, 1, 0, 1, -2, 4, -7, 11, -16, 23, -36, 65, -129, 256, -473, 772, -1028, 835, 776, -5755, 17562, -41750, 86678, -165145, 299949, -541837, 1020029, -2068203, 4509512, -10252952, 23465297, -52762788, 115160832, -243018459, 496094524, -982431070, 1894710043, -3574095362
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 16 2018

Keywords

Comments

After the first term, this is the second term of the n-th differences of A000009, or column n=1 of A378622. - Gus Wiseman, Feb 03 2025

Links

Crossrefs

Cf. A000593, A320589, A307548.
The version for non-strict partitions is A320590, row n=1 of A175804.
Column n=1 (except first term) of A378622. See also A293467, A377285, A378970, A378971, A380412 (column n=0).
A000009 counts strict integer partitions, differences A087897, A378972.
A266232 gives zero-based binomial transform of strict partitions, differences A129519.
Cf. A000041, A007442, A002865, A008284, A095195, A103446, A116608, A218482, A281425, A320568.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1 + x^k/(1 + x)^k): k in [1..(m+2)]]) )); // G. C. Greubel, Oct 29 2018
  • Maple
    seq(coeff(series(mul((1+x^k/(1+x)^k),k=1..n),x,n+1), x, n), n = 0 .. 37); # Muniru A Asiru, Oct 16 2018
  • Mathematica
    nmax = 37; CoefficientList[Series[Product[(1 + x^k/(1 + x)^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 37; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k/d + 1) d, {d, Divisors[k]}] x^k/(k (1 + x)^k), {k, 1, nmax}]], {x, 0, nmax}], x]
Prepend[Table[Differences[PartitionsQ/@Range[0,k+1],k][[2]],{k,0,30}],1] (* Gus Wiseman, Jan 29 2025 *)
  • PARI
    m=50; x='x+O('x^m); Vec(prod(k=1, m+2, (1 + x^k/(1 + x)^k))) \\ G. C. Greubel, Oct 29 2018

Formula

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*((1 + x)^k - x^k))).
G.f.: exp(Sum_{k>=1} A000593(k)*x^k/(k*(1 + x)^k)).
From Peter Bala, Dec 22 2020: (Start)
O.g.f.: Sum_{n >= 0} x^(n*(n+1)/2)/Product_{k = 1..n} ((1 + x)^k - x^k). Cf. A307548.
Conjectural o.g.f.: (1/2) * Sum_{n >= 0} x^(n*(n-1)/2)*(1 + x)^n/( Product_{k = 1..n} ( (1 + x)^k - x^k ) ). (End)
a(n+1) = Sum_{k=0..n} (-1)^(n-k) binomial(n,k) A000009(k+1). - Gus Wiseman, Feb 03 2025

A294529 Binomial transform of A001156.

Original entry on oeis.org

1, 2, 4, 8, 17, 38, 86, 192, 420, 905, 1939, 4163, 8987, 19494, 42368, 91990, 199127, 429345, 921982, 1972553, 4206909, 8949412, 19001874, 40293048, 85373962, 180826115, 382957231, 811027414, 1717497958, 3636335170, 7695599294, 16275268520, 34389570596
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 02 2017

Keywords

Links

Crossrefs

Cf. A001156, A218481, A266232, A294500, A294530.

Programs

  • Mathematica
    nmax = 40; s = CoefficientList[Series[Product[1/(1 - x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x]; Table[Sum[Binomial[n, k] * s[[k+1]], {k, 0, n}], {n, 0, nmax}]

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * A001156(k).
a(n) ~ exp(3 * 2^(-5/3) * Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3)) * Zeta(3/2)^(2/3) * 2^(n - 7/6) / (sqrt(3) * Pi^(7/6) * n^(7/6)).
G.f.: (1/(1 - x))*Product_{k>=1} 1/(1 - x^(k^2)/(1 - x)^(k^2)). - Ilya Gutkovskiy, Aug 20 2018

A356268 a(n) = Sum_{k=0..n} binomial(2*n, k) * q(k), where q(k) is the number of partitions into distinct parts (A000009).

Original entry on oeis.org

1, 3, 11, 62, 289, 1472, 7581, 38014, 184453, 918512, 4548393, 22077762, 107423503, 516720332, 2483445404, 11959145079, 57022343425, 270173627092, 1282971321633, 6047971597490, 28446033085527, 133714464665108, 625893086713686, 2919093380089383, 13596052503945537
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 01 2022

Keywords

Crossrefs

Cf. A000009, A032443, A266232, A307496, A356267.

Programs

  • Mathematica
    Table[Sum[Binomial[2*n, k] * PartitionsQ[k], {k, 0, n}], {n, 0, 30}]

Formula

a(n) ~ erfc(Pi/(4*sqrt(3))) * 2^(2*n - 3) * exp(Pi*sqrt(n/3) + Pi^2/48) / (3^(1/4) * n^(3/4)).

A356281 a(n) = Sum_{k=0..n} binomial(2*n, n-k) * q(k), where q(k) is the number of partitions into distinct parts (A000009).

Original entry on oeis.org

1, 3, 11, 43, 172, 695, 2823, 11501, 46940, 191791, 784148, 3207196, 13119733, 53670793, 219545353, 897957702, 3672093558, 15013596535, 61370565546, 250803861369, 1024716136043, 4185683293934, 17093143284723, 69786349712519, 284847779542644, 1162385753008079
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 01 2022

Keywords

Crossrefs

Cf. A000009, A032443, A266232, A307496, A356268, A356280.

Programs

  • Mathematica
    Table[Sum[PartitionsQ[k]*Binomial[2*n, n-k], {k, 0, n}], {n, 0, 30}]
nmax = 30; CoefficientList[Series[Sum[PartitionsQ[k]*((1-2*x-Sqrt[1-4*x])/(2*x))^k / Sqrt[1-4*x], {k, 0, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ 2^(2*n - 1/2) * exp(3^(1/3) * Pi^(4/3) * n^(1/3) / 2^(8/3)) / sqrt(3*Pi*n).

A294530 Binomial transform of A023871.

Original entry on oeis.org

1, 2, 8, 33, 131, 497, 1834, 6635, 23622, 82942, 287656, 986552, 3349165, 11263951, 37558235, 124240204, 407951848, 1330340478, 4310385956, 13881618570, 44451643311, 141578435571, 448634389388, 1414774796929, 4441038400458, 13879652908322, 43197263002063
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 02 2017

Keywords

Links

Crossrefs

Cf. A023871, A218481, A266232, A294500, A294529.

Programs

  • Mathematica
    nmax = 40; s = CoefficientList[Series[Product[1/(1 - x^k)^(k^2), {k, 1, nmax}], {x, 0, nmax}], x]; Table[Sum[Binomial[n, k] * s[[k+1]], {k, 0, n}], {n, 0, nmax}]

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * A023871(k).
a(n) ~ exp(2^(5/4) * 3^(-5/4) * 5^(-1/4) * Pi * n^(3/4) + Pi^2 * sqrt(n) / (4*sqrt(30)) - Pi^3 * n^(1/4) / (32 * 2^(1/4) * 15^(3/4)) + Pi^4/3840 - Zeta(3)/(4*Pi^2)) * 2^(n - 7/8) / (15^(1/8) * n^(5/8)).
G.f.: (1/(1 - x))*exp(Sum_{k>=1} sigma_3(k)*x^k/(k*(1 - x)^k)). - Ilya Gutkovskiy, Aug 20 2018

A307756 Exponential convolution of number of partitions into distinct parts (A000009) with themselves.

Original entry on oeis.org

1, 2, 4, 10, 26, 66, 184, 472, 1268, 3340, 8748, 22772, 59102, 151590, 386830, 983914, 2489384, 6263284, 15703204, 39221884, 97498736, 241538472, 596115898, 1465958522, 3595196600, 8788765304, 21421616934, 52080152238, 126268822824, 305365334180, 736770528064
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 26 2019

Keywords

Links

Crossrefs

Cf. A000009, A022567, A266232, A307755.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= n-> add(binomial(n, j)*b(j)*b(n-j), j=0..n):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 26 2019
  • Mathematica
    nmax = 30; CoefficientList[Series[Sum[PartitionsQ[k] x^k/k!, {k, 0, nmax}]^2, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] PartitionsQ[k] PartitionsQ[n - k], {k, 0, n}], {n, 0, 30}]

Formula

E.g.f.: (Sum_{k>=0} A000009(k)*x^k/k!)^2.
a(n) = Sum_{k=0..n} binomial(n,k)*A000009(k)*A000009(n-k).
a(n) ~ exp(Pi*sqrt(2*n/3)) * 2^(n - 5/2) / (sqrt(3) * n^(3/2)). - Vaclav Kotesovec, May 06 2019

A356270 a(n) = Sum_{k=0..n} binomial(2*k, k) * q(k), where q(k) is the number of partitions into distinct parts (A000009).

Original entry on oeis.org

1, 3, 9, 49, 189, 945, 4641, 21801, 99021, 487981, 2335541, 10800725, 51363065, 238573865, 1121139065, 5309312105, 24543884585, 113220920945, 530677144745, 2439321389945, 11261499234425, 52169097691865, 239433905462945, 1095710701133345, 5029918350471545
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 01 2022

Keywords

Crossrefs

Cf. A000009, A006134, A032443, A266232, A307496, A356268, A356269.

Programs

  • Mathematica
    Table[Sum[Binomial[2*k, k] * PartitionsQ[k], {k, 0, n}], {n, 0, 30}]

Formula

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

A380412 First term of the n-th differences of the strict partition numbers. Inverse zero-based binomial transform of A000009.

Original entry on oeis.org

1, 0, 0, 1, -3, 7, -14, 25, -41, 64, -100, 165, -294, 550, -1023, 1795, -2823, 3658, -2882, -2873, 20435, -62185, 148863, -314008, 613957, -1155794, 2175823, -4244026, 8753538, -19006490, 42471787, -95234575, 210395407, -453413866, 949508390, -1931939460
Offset: 0

Views

Author

Gus Wiseman, Feb 03 2025

Keywords

Comments

Up to sign, same as A293467.

Crossrefs

The version for non-strict partitions is A281425, row n=0 of A175804.
Column n=0 of A378622.
A000009 counts strict integer partitions, differences A087897, A378972.
A266232 gives zero-based binomial transform of A000009, differences A129519.
Cf. A000041, A007442, A008284, A218482, A293467, A320590, A320591, A377285, A378970.

Programs

  • Mathematica
    nn=10;Table[First[Differences[PartitionsQ/@Range[0,nn],n]],{n,0,nn}]

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) binomial(n,k) A000041(k).

A307259 Expansion of (1/(1 - x)) * Product_{k>=1} (1 + k*x^k/(1 - x)^k).

Original entry on oeis.org

1, 2, 5, 15, 44, 126, 357, 1003, 2783, 7618, 20627, 55421, 148021, 393140, 1038123, 2724992, 7112022, 18465708, 47726767, 122861732, 315123476, 805428727, 2051556778, 5207982062, 13177117709, 33235023381, 83574705456, 209576713721, 524181331710, 1307849984089, 3255539133109
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 01 2019

Keywords

Comments

Binomial transform of A022629.

Crossrefs

Cf. A022629, A266232, A294502, A307260, A318127.

Programs

  • Maple
    a:=series((1/(1-x))*mul(1+k*x^k/(1-x)^k,k=1..100),x=0,31): seq(coeff(a,x,n),n=0..30); # Paolo P. Lava, Apr 03 2019
  • Mathematica
    nmax = 30; CoefficientList[Series[1/(1 - x) Product[(1 + k x^k/(1 - x)^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) = Sum_{k=0..n} binomial(n,k)*A022629(k).

A324237 Expansion of e.g.f. 1 / (1 - Sum_{k>=1} q(k)*x^k/k!), where q(k) = number of partitions of k into distinct parts (A000009).

Original entry on oeis.org

1, 1, 3, 14, 84, 633, 5730, 60485, 729710, 9904064, 149358998, 2477662364, 44837516675, 879028693860, 18558771941586, 419815668642109, 10129704474860688, 259695154154923814, 7049438079064414206, 201988316828399901634, 6092203404985463075656
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 02 2019

Keywords

Links

Crossrefs

Cf. A000009, A266232, A300514, A304969, A324236.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-k)*binomial(n, k)*b(k), k=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 02 2019
  • Mathematica
    nmax = 21; CoefficientList[Series[1/(1 - Sum[PartitionsQ[k] x^k/k!, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] PartitionsQ[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * q(k) * a(n-k).
a(n) ~ c * d^n * n!, where d = 1.5080583621492799630678624980320180394686208919872154400104169910221003637... and c = 0.67652958824662835367141799671720225317465169475061770258661897351... - Vaclav Kotesovec, Sep 03 2019
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