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.

Showing 1-6 of 6 results.

A281425 a(n) = [q^n] (1 - q)^n / Product_{j=1..n} (1 - q^j).

Original entry on oeis.org

1, 0, 1, -1, 2, -4, 9, -21, 49, -112, 249, -539, 1143, -2396, 5013, -10550, 22420, -48086, 103703, -223806, 481388, -1029507, 2187944, -4625058, 9742223, -20490753, 43111808, -90840465, 191773014, -405523635, 858378825, -1817304609, 3845492204, -8129023694, 17162802918, -36191083386
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 05 2017

Keywords

Comments

a(n) is n-th term of the Euler transform of -n + 1, 1, 1, 1, ...
Inverse zero-based binomial transform of A000041. The version for strict partitions is A380412, or A293467 up to sign. - Gus Wiseman, Feb 06 2025

Links

Crossrefs

Cf. A128566, A292463, A292541, A292613.
Cf. A000041, A002865, A053445, A275638, A275639, A275640, A275641, A275642, A275643, A275644.
Cf. A218481, A294466, A095051.
Cf. A266232, A294467, A293467, A294468.
Row n=0 of A175804 (row n=1 is A320590 except first term).
Cf. A000009, A008284, A087897, A116608, A129519, A225486, A320591, A377056, A378622.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(k=0,
      combinat[numbpart](n), b(n, k-1)-b(n-1, k-1))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 21 2024
  • Mathematica
    Table[SeriesCoefficient[(1 - q)^n / Product[(1 - q^j), {j, 1, n}], {q, 0, n}], {n, 0, 35}]
Table[SeriesCoefficient[(1 - q)^n QPochhammer[q^(1 + n), q]/QPochhammer[q, q], {q, 0, n}], {n, 0, 35}]
Table[SeriesCoefficient[1/QFactorial[n, q], {q, 0, n}], {n, 0, 35}]
Table[Differences[PartitionsP[Range[0, n]], n], {n, 0, 35}] // Flatten
Table[Sum[(-1)^j*Binomial[n, j]*PartitionsP[n-j], {j, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 06 2017 *)

Formula

a(n) = [q^n] 1/((1 + q)*(1 + q + q^2)*...*(1 + q + ... + q^(n-1))).
a(n) = Sum_{j=0..n} (-1)^j * binomial(n, j) * A000041(n-j). - Vaclav Kotesovec, Oct 06 2017
a(n) ~ (-1)^n * 2^(n - 3/2) * exp(Pi*sqrt(n/12) + Pi^2/96) / (sqrt(3)*n). - Vaclav Kotesovec, May 07 2018

A293467 a(n) = Sum_{k=0..n} (-1)^k * binomial(n, k) * q(k), where q(k) is A000009 (partitions into distinct parts).

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

Vaclav Kotesovec, Oct 09 2017

Keywords

Comments

Multiply by (-1)^n to get A380412, which is the first term of the n-th differences of the strict partition numbers, or column n=0 of A378622. - Gus Wiseman, Feb 04 2025

Links

Crossrefs

Cf. A025147, A266232, A294467, A294468.
Cf. A218481, A294466, A095051.
The non-strict version is the absolute value of A281425; see A175804, A320590.
Up to sign, same as A380412. See A320591, A377285, A378970, A378971.
A000009 counts strict integer partitions, differences A087897.
Cf. A000041, A002865, A007442, A008284, A116608.

Programs

  • Mathematica
    Table[Sum[(-1)^k * Binomial[n, k] * PartitionsQ[k], {k, 0, n}], {n, 0, 50}]

A320590 Expansion of Product_{k>=1} 1/(1 - x^k/(1 + x)^k).

Original entry on oeis.org

1, 1, 1, 0, 1, -2, 5, -12, 28, -63, 137, -290, 604, -1253, 2617, -5537, 11870, -25666, 55617, -120103, 257582, -548119, 1158437, -2437114, 5117165, -10748530, 22621055, -47728657, 100932549, -213750621, 452855190, -958925784, 2028187595, -4283531490, 9033779224
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 16 2018

Keywords

Comments

The zero-based binomial transform of this sequence is A000070, and if we remove first terms it becomes A000041.

Links

Crossrefs

Cf. A000203, A103446, A218482, A320568.
Row n=1 of A175804 (except first term). Row n=0 is A281425.
The version for strict partitions is A320591, row n=1 of A378622, first column A293467.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865.
Cf. A047966, A008284, A377056, A378621.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1 - x^k/(1 + x)^k): k in [1..(m+2)]]) )); // G. C. Greubel, Oct 29 2018
  • Maple
    seq(coeff(series(mul(1/(1-x^k/(1+x)^k),k=1..n),x,n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Oct 16 2018
  • Mathematica
    nmax = 34; CoefficientList[Series[Product[1/(1 - x^k/(1 + x)^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 34; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k] x^k/(k (1 + x)^k), {k, 1, nmax}]], {x, 0, nmax}], x]
  • PARI
    m=50; x='x+O('x^m); Vec(prod(k=1, m+2, 1/(1 - x^k/(1 + x)^k))) \\ G. C. Greubel, Oct 29 2018

Formula

G.f.: exp(Sum_{k>=1} x^k/(k*((1 + x)^k - x^k))).
G.f.: exp(Sum_{k>=1} sigma(k)*x^k/(k*(1 + x)^k)).

A320589 Expansion of (1/(1 + x)) * Sum_{k>=1} k*x^k/(x^k + (1 + x)^k).

Original entry on oeis.org

1, -1, 4, -13, 36, -88, 197, -421, 895, -1946, 4346, -9832, 22140, -49043, 106389, -226213, 473366, -980413, 2022418, -4179198, 8687753, -18201140, 38398455, -81343408, 172383461, -364158198, 764854519, -1595107695, 3302884966, -6796646603, 13921482698, -28437025029, 58034908034
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 16 2018

Keywords

Comments

Inverse binomial transform of A000593.

Links

Crossrefs

Cf. A000593, A320568, A320586, A320591.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1/(1 + x))*(&+[k*x^k/(x^k + (1 + x)^k): k in [1..(m+2)]]) )); // G. C. Greubel, Oct 29 2018
  • Maple
    seq(coeff(series((1/(1+x))*add(k*x^k/(x^k+(1+x)^k),k=1..n),x,n+1), x, n), n = 1 .. 35); # Muniru A Asiru, Oct 16 2018
  • Mathematica
    nmax = 33; Rest[CoefficientList[Series[1/(1 + x) Sum[k x^k/(x^k + (1 + x)^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 33; Rest[CoefficientList[Series[(EllipticTheta[3, 0, x/(1 + x)]^4 + EllipticTheta[2, 0, x/(1 + x)]^4 - 1)/(24 (1 + x)), {x, 0, nmax}], x]]
Table[Sum[(-1)^(n - k) Binomial[n, k] Sum[(-1)^(k/d + 1) d, {d, Divisors[k]}], {k, n}], {n, 33}]
  • PARI
    m=50; x='x+O('x^m); Vec((1/(1 + x))*sum(k=1, m+2, k*x^k/(x^k + (1 + x)^k))) \\ G. C. Greubel, Oct 29 2018

Formula

G.f.: (theta_3(x/(1 + x))^4 + theta_2(x/(1 + x))^4 - 1)/(24*(1 + x)), where theta_() is the Jacobi theta function.
L.g.f.: Sum_{k>=1} A000593(k)*x^k/(k*(1 + x)^k) = Sum_{n>=1} a(n)*x^n/n.
a(n) = Sum_{k=1..n} (-1)^(n-k)*binomial(n,k)*A000593(k).
Conjecture: a(n) ~ -(-1)^n * c * 2^n * n, where c = Pi^2/48 = 0.205616758356... - Vaclav Kotesovec, Jun 26 2019

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).

A307548 Expansion of Product_{k>=1} (1 - (x/(1+x))^k).

Original entry on oeis.org

1, -1, 0, 1, -2, 4, -9, 21, -48, 105, -218, 429, -803, 1442, -2521, 4380, -7734, 14091, -26468, 50405, -94980, 172824, -296704, 467589, -644459, 678109, -177123, -1752141, 7003180, -19432494, 46778567, -104623822, 224830880, -473859273, 992825436, -2084921584
Offset: 0

Views

Author

Seiichi Manyama, Apr 14 2019

Keywords

Crossrefs

Convolution inverse of A320590.
Cf. A129519, A307310, A320591.

Programs

  • Mathematica
    m = 35; CoefficientList[Series[Product[1 - (x/(1+x))^k, {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 14 2021 *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(k=1, N, 1-(x/(1+x))^k))

Formula

O.g.f.: Sum_{n >= 0} (-1)^n * x^(n*(n+1)/2)/Product_{k = 1..n} ((1 + x)^k - x^k). Cf. A320591. - Peter Bala, Dec 22 2020
Showing 1-6 of 6 results.

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