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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A358910 Number of integer partitions of n whose parts do not have weakly decreasing numbers of prime factors (A001222).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 4, 8, 11, 19, 25, 41, 56, 84, 113, 164, 218, 306, 401, 547, 711, 949, 1218, 1599, 2034, 2625, 3310, 4224, 5283, 6664, 8271, 10336, 12747, 15791, 19343, 23791, 28979, 35398, 42887, 52073, 62779, 75804, 90967, 109291, 130605
Offset: 0

Views

Author

Gus Wiseman, Dec 09 2022

Keywords

Examples

			The a(9) = 1 through a(14) = 11 partitions:
  (54)  (541)  (74)    (543)    (76)      (554)
               (542)   (741)    (544)     (743)
               (5411)  (5421)   (742)     (761)
                       (54111)  (5422)    (5432)
                                (5431)    (5441)
                                (7411)    (7421)
                                (54211)   (54221)
                                (541111)  (54311)
                                          (74111)
                                          (542111)
                                          (5411111)

Crossrefs

For sequences of partitions see A141199, compositions A218482.
The case of equality is A319169, for compositions A358911.
The complement is counted by A358909.
A001222 counts prime factors, distinct A001221.
A063834 counts twice-partitions.
Cf. A056239, A300335, A320324, A358831, A358902, A358903, A358908.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],!GreaterEqual@@PrimeOmega/@#&]],{n,0,30}]

A307500 Expansion of Product_{k>=1} 1/(1 - (x*(1 - x))^k).

Original entry on oeis.org

1, 1, 1, -1, -2, -4, 3, -1, 17, -16, 21, -57, 67, -130, 305, -536, 995, -1726, 2652, -4286, 7320, -13043, 24458, -45405, 81415, -141724, 239755, -400603, 676872, -1171076, 2072334, -3695550, 6519951, -11279015, 19188230, -32462795, 55334284, -95718737, 167673672, -294894076
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 11 2019

Keywords

Links

Crossrefs

Cf. A000041, A030528, A218482, A238441, A286955, A307501.

Programs

  • Mathematica
    nmax = 39; CoefficientList[Series[Product[1/(1 - (x (1 - x))^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 39; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k] (x (1 - x))^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
Table[Sum[(-1)^(n - k) Binomial[k, n - k] PartitionsP[k], {k, 0, n}], {n, 0, 39}]

Formula

G.f.: exp(Sum_{k>=1} sigma(k)*(x*(1 - x))^k/k).
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(k,n-k)*A000041(k).

A318570 Expansion of Product_{k>=1} ((1 - x)^k + x^k)/((1 - x)^k - x^k).

Original entry on oeis.org

1, 2, 6, 18, 52, 146, 402, 1090, 2916, 7708, 20160, 52236, 134222, 342304, 867024, 2182384, 5461696, 13595918, 33677550, 83036878, 203859820, 498470998, 1214230586, 2947204870, 7129403128, 17191258642, 41328057106, 99067295658, 236822823336, 564650823162, 1342921372126
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 15 2018

Keywords

Comments

First differences of the binomial transform of A015128.
Convolution of A129519 and A218482.

Links

Crossrefs

Cf. A000203, A015128, A129519, A218482, A266497.

Programs

  • Maple
    a:=series(mul(((1-x)^k+x^k)/((1-x)^k-x^k),k=1..100),x=0,31): seq(coeff(a,x,n),n=0..30); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 30; CoefficientList[Series[Product[((1 - x)^k + x^k)/((1 - x)^k - x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[1/EllipticTheta[4, 0, x/(1 - x)], {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, 2 k] - DivisorSigma[1, k]) x^k/(k (1 - x)^k), {k, 1, nmax}]], {x, 0, nmax}], x]

Formula

G.f.: 1/theta_4(x/(1 - x)), where theta_4() is the Jacobi theta function.
G.f.: exp(Sum_{k>=1} (sigma(2*k) - sigma(k))*x^k/(k*(1 - x)^k)).
a(n) ~ 2^(n-3) * exp(Pi*sqrt(n/2) + Pi^2/16) / n. - Vaclav Kotesovec, Oct 15 2018

A358904 Number of finite sets of compositions with all equal sums and total sum n.

Original entry on oeis.org

1, 1, 2, 4, 9, 16, 38, 64, 156, 260, 632, 1024, 2601, 4096, 10208, 16944, 40966, 65536, 168672, 262144, 656980, 1090240, 2620928, 4194304, 10862100, 16781584, 41940992, 69872384, 168403448, 268435456, 693528552, 1073741824, 2695006177, 4473400320, 10737385472
Offset: 0

Views

Author

Gus Wiseman, Dec 13 2022

Keywords

Examples

			The a(1) = 1 through a(4) = 9 sets:
  {(1)}  {(2)}   {(3)}    {(4)}
         {(11)}  {(12)}   {(13)}
                 {(21)}   {(22)}
                 {(111)}  {(31)}
                          {(112)}
                          {(121)}
                          {(211)}
                          {(1111)}
                          {(2),(11)}

Crossrefs

This is the constant-sum case of A098407, ordered A358907.
The version for distinct sums is A304961, ordered A336127.
Allowing repetition gives A305552, ordered A074854.
The case of sets of partitions is A359041.
A001970 counts multisets of partitions.
A034691 counts multisets of compositions, ordered A133494.
A261049 counts sets of partitions, ordered A358906.
Cf. A000009, A063834, A075900, A218482, A296122.

Programs

  • Mathematica
    Table[If[n==0,1,Sum[Binomial[2^(d-1),n/d],{d,Divisors[n]}]],{n,0,30}]
  • PARI
    a(n) = if (n, sumdiv(n, d, binomial(2^(d-1), n/d)), 1); \\ Michel Marcus, Dec 14 2022

Formula

a(n>0) = Sum_{d|n} binomial(2^(d-1),n/d).

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).

A307311 Expansion of Product_{k>=1} 1/(1 + x^k/(1 - x)^k).

Original entry on oeis.org

1, -1, -1, -2, -3, -4, -5, -6, -6, -1, 19, 74, 200, 461, 977, 1987, 3976, 7902, 15559, 30105, 56778, 103833, 183765, 314882, 523007, 841752, 1305431, 1916607, 2540433, 2609983, 381628, -8814988, -36463325, -109113400, -285322360, -689608522, -1579574566, -3477967848
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 02 2019

Keywords

Comments

First differences of the binomial transform of A081362.
Convolution inverse of A129519.

Links

Crossrefs

Cf. A081362, A129519, A218482, A307310.

Programs

  • Magma
    m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( 1/(&*[(1+x^k/(1-x)^k): k in [1..m+2]]) )); // G. C. Greubel, Apr 03 2019
  • Maple
    a:=series(mul(1/(1+x^k/(1-x)^k),k=1..100), x=0,38): seq(coeff(a, x, n), n=0..37); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 40; CoefficientList[Series[Product[1/(1 + x^k/(1 - x)^k), {k, 1, nmax}], {x, 0, nmax}], x]
  • PARI
    m=40; my(x='x+O('x^m)); Vec( 1/prod(k=1,m+2, (1+x^k/(1-x)^k)) ) \\ G. C. Greubel, Apr 03 2019
  • Sage
    m=40; (1/product(1+x^k/(1-x)^k for k in (1..m+2))).series(x, m).coefficients(x, sparse=False) # G. C. Greubel, Apr 03 2019

A307261 Expansion of Product_{k>=1} 1/(1 - k*x^k/(1 - x)^k).

Original entry on oeis.org

1, 1, 4, 13, 42, 130, 397, 1197, 3566, 10517, 30760, 89293, 257397, 737220, 2099215, 5945594, 16756258, 47004829, 131286914, 365203797, 1012031772, 2794446326, 7690009600, 21094325177, 57687762889, 157306741287, 427777384499, 1160250104637, 3139067594584, 8472525405830, 22815639395641
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 01 2019

Keywords

Comments

First differences of the binomial transform of A006906.

Crossrefs

Cf. A006906, A218482, A307262, A318127, A320563.

Programs

  • Maple
    a:=series(mul(1/(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[Product[1/(1 - k x^k/(1 - x)^k), {k, 1, nmax}], {x, 0, nmax}], x]

A307679 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k/(1 - x)^k)^(1/k).

Original entry on oeis.org

1, 1, 5, 35, 323, 3679, 49819, 781465, 13923545, 277563617, 6118251461, 147715469131, 3875706370315, 109781717161375, 3338229675519803, 108443658227589329, 3747688533281296049, 137273241169036231105, 5311844045472206624005, 216505267421266611639667, 9270689769095765333645651
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 21 2019

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 35*x^3/3! + 323*x^4/4! + 3679*x^5/5! + 49819*x^6/6! + 781465*x^7/7! + 13923545*x^8/8! + ...
log(A(x)) = x + 4*x^2/2 + 11*x^3/3 + 27*x^4/4 + 62*x^5/5 + 137*x^6/6 + 296*x^7/7 + 630*x^8/8 + 1326*x^9/9 + ... + A160399(k)*x^k/k + ...

Crossrefs

Cf. A000005, A028342, A103446, A160399, A218482, A307680, A320563.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[1/(1 - x^k/(1 - x)^k)^(1/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[0, k] x^k/(k (1 - x)^k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: exp(Sum_{k>=1} d(k)*x^k/(k*(1 - x)^k)), where d(k) is the number of divisors of k (A000005).
a(n) = Sum_{k=0..n} binomial(n-1,k-1)*A028342(k)*n!/k!.

A330649 E.g.f.: Product_{k>=1} 1 / (1 - x^k/(k!*(1 - x)^k)).

Original entry on oeis.org

1, 1, 5, 34, 299, 3226, 41202, 607545, 10153831, 189628750, 3913009178, 88406043991, 2170372901534, 57531498837515, 1637713270797411, 49830222530823615, 1613950394999111903, 55444724259894089718, 2013760368429942861810, 77105255895256112519259
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 13 2020

Keywords

Crossrefs

Cf. A005651, A084358, A218482, A320566, A332024.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[Product[1/(1 - x^k/(k! (1 - x)^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n - 1, k - 1] Total[Apply[Multinomial, IntegerPartitions[k], {1}]] n!/k!, {k, 0, n}], {n, 0, 19}]
  • PARI
    seq(n)={Vec(serlaplace(prod(k=1, n, 1 / (1 - x^k/(k!*(1 - x)^k)) + O(x*x^n))))} \\ Andrew Howroyd, Feb 13 2020

Formula

a(n) = Sum_{k=0..n} binomial(n-1,k-1) * A005651(k) * n! / k!.
a(n) ~ c * 2^(n-1) * n!, where c = A247551 = 2.52947747207915264818... - Vaclav Kotesovec, Feb 16 2020

A307265 Expansion of Product_{k>=1} 1/(1 + (-x)^k/(1 - x)^k).

Original entry on oeis.org

1, 1, 1, 2, 5, 12, 27, 58, 122, 257, 549, 1190, 2600, 5683, 12367, 26749, 57530, 123202, 263115, 561131, 1196248, 2550975, 5443115, 11620526, 24814735, 52979512, 113038103, 240936717, 512916683, 1090501249, 2315608462, 4911611864, 10408318627, 22040127864
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 01 2019

Keywords

Comments

First differences of the binomial transform of A000700.

Crossrefs

Cf. A000700, A129519, A218482, A307264.

Programs

  • Maple
    a:=series(mul(1/(1+(-x)^k/(1-x)^k),k=1..50),x=0,34): seq(coeff(a,x,n),n=0..33); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 33; CoefficientList[Series[Product[1/(1 + (-x)^k/(1 - x)^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + x^(2*k-1)/(1 - x)^(2*k-1)).
a(n) ~ 2^(n-2) * exp(Pi*sqrt(n/3)/2 + Pi^2/96) / (3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Apr 01 2019
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