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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A347915 Expansion of e.g.f. Product_{k>=1} (1 + x^k)^exp(x).

Original entry on oeis.org

1, 1, 4, 24, 150, 1235, 11725, 126987, 1512084, 20313897, 296921623, 4700713787, 80221988726, 1468879687145, 28661345212981, 594457831566757, 13027193829914920, 301079987772726257, 7318797530268562203, 186496088631167771143, 4971371842655844396298, 138384071439982000722737
Offset: 0

Views

Author

Seiichi Manyama, Sep 18 2021

Keywords

Crossrefs

Cf. A000593, A088311, A346547, A347916, A354507.
Cf. A354503, A354504.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[(1 + x^k)^Exp[x], {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Aug 17 2022 *)
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, (1+x^k)^exp(x))))
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(exp(exp(x)*sum(k=1, N, sigma(k>>valuation(k, 2))*x^k/k))))
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(exp(exp(x)*sum(k=1, N, x^k/(k*(1-x^(2*k)))))))
  • PARI
    a354507(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d)/(k*(n-k)!));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354507(j)*binomial(i-1, j-1)*v[i-j+1])); v; \\ Seiichi Manyama, Aug 16 2022

Formula

E.g.f.: exp( exp(x) * Sum_{k>=1} A000593(k)*x^k/k ).
E.g.f.: exp( exp(x) * Sum_{k>=1} x^k/(k*(1 - x^(2*k))) ).
a(0) = 1; a(n) = Sum_{k=1..n} A354507(k) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Aug 16 2022

A239837 Number of ordered pairs of permutation functions on n elements satisfying f(f(x)) = g(f(g(x))).

Original entry on oeis.org

1, 1, 2, 12, 96, 600, 6480, 85680, 1048320, 16692480, 315705600, 5468601600, 117834393600, 2951607859200, 68958028339200, 1856897602560000, 58228124258304000, 1721171464556544000, 56379302852640768000, 2111880588197732352000, 76033053559536353280000
Offset: 0

Views

Author

Chad Brewbaker, Mar 27 2014

Keywords

Links

Crossrefs

Cf. A000012, A000085, A000142, A088311, A053529, A001044.

Extensions

a(8)-a(9) from Giovanni Resta, Mar 27 2014
a(10)-a(13) from Paul Boddington, Feb 23 2015
a(14)-a(20) from Hiroaki Yamanouchi, Mar 12 2015

A239838 Number of ordered pairs of permutation functions on n elements where f(f(f(x))) = g(g(g(x))).

Original entry on oeis.org

1, 1, 2, 12, 96, 600, 9360, 146160, 1935360, 41368320, 1092268800, 23111827200, 700300339200, 26141033318400, 810322216704000, 31624796915712000, 1537594906079232000, 63980698878480384000, 3128302230597992448000, 187447679378885173248000
Offset: 0

Views

Author

Chad Brewbaker, Mar 27 2014

Keywords

Links

Crossrefs

Cf. A000012, A000085, A000142, A088311, A053529, A001044.
a(n) = n! * A232207(n) for n>0. - Alois P. Heinz, Jul 23 2014

Extensions

a(8)-a(12) from Giovanni Resta, Mar 27 2014
a(13) from Alois P. Heinz, Jul 23 2014
a(14)-a(19) from Hiroaki Yamanouchi, Mar 12 2015

A239839 Number of ordered pairs of permutation functions on n elements satisfying f(f(f(x))) = g(f(g(x))).

Original entry on oeis.org

1, 1, 4, 18, 168, 1560, 20880, 267120, 5080320, 93623040, 2184537600, 49896000000, 1451853849600, 41739720422400, 1426847092070400, 47989033956864000, 1919268439216128000, 76229151152394240000, 3471527082588364800000, 156226856133456396288000
Offset: 0

Views

Author

Chad Brewbaker, Mar 27 2014

Keywords

Links

Crossrefs

Cf. A000012, A000085, A000142, A088311, A053529, A001044.

Extensions

a(8)-a(9) from Giovanni Resta, Mar 27 2014
a(10)-a(13) from Paul Boddington, Feb 23 2015
a(14)-a(19) from Hiroaki Yamanouchi, Mar 12 2015

A239840 Number of ordered pairs of permutation functions (f,g) on n elements satisfying f(x) = f(g(g(x))).

Original entry on oeis.org

1, 1, 4, 24, 240, 3120, 54720, 1169280, 30804480, 950745600, 34459084800, 1424870092800, 67133032243200, 3540086232883200, 208397961547776000, 13533822947893248000, 966773828738285568000, 75334352557782269952000, 6385175803136642383872000
Offset: 0

Views

Author

Chad Brewbaker, Mar 27 2014

Keywords

Links

Crossrefs

Cf. A000012, A000085, A000142, A088311, A053529, A001044.

Programs

  • Maple
    a:= proc(n) a(n):= `if`(n<2, 1, n*a(n-1) +n*(n-1)^2*a(n-2)) end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 23 2014
  • Mathematica
    a[n_] := a[n] = n a[n-1] + n(n-1)^2 a[n-2]; a[0] = a[1] = 1;
a /@ Range[0, 20] (* Jean-François Alcover, Oct 04 2019 *)

Formula

From Alois P. Heinz, Jul 23 2014: (Start)
a(n) = n! * A000085(n) = A000142(n) * A000085(n).
a(n) = n*a(n-1) + n*(n-1)^2*a(n-2) for n>=2, a(0) = a(1) = 1. (End)
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x + x^2 / 2). - Ilya Gutkovskiy, Jul 15 2021

Extensions

a(8)-a(9) from Giovanni Resta, Mar 27 2014
a(10)-a(18) from Alois P. Heinz, Jul 23 2014

A293140 E.g.f.: Product_{m>0} (1-x^m).

Original entry on oeis.org

1, -1, -2, 0, 0, 120, 0, 5040, 0, 0, 0, 0, -479001600, 0, 0, -1307674368000, 0, 0, 0, 0, 0, 0, 1124000727777607680000, 0, 0, 0, 403291461126605635584000000, 0, 0, 0, 0, 0, 0, 0, 0, -10333147966386144929666651337523200000000, 0, 0, 0, 0
Offset: 0

Views

Author

Seiichi Manyama, Oct 01 2017

Keywords

Crossrefs

Column k=1 of A293139.
Cf. A003056, A010815, A088311, A351884.

Formula

a(n) = n! * A010815(n).
a(n) = Sum_{k=0..A003056(n)} (-1)^k * A351884(n,k). - Alois P. Heinz, Feb 23 2022

A347817 E.g.f.: Product_{k>=1} (1 + x^k)^sin(x).

Original entry on oeis.org

1, 0, 2, 3, 40, 80, 1760, 8211, 139256, 763272, 19466578, 147696835, 3372858476, 33370016316, 872184749046, 10340382875655, 289042962136272, 3884706041971728, 118640349946950738, 1911641854423398435, 59577007012206421356, 1086774235381609797540, 37138839666110194130670
Offset: 0

Views

Author

Seiichi Manyama, Sep 18 2021

Keywords

Crossrefs

Cf. A000593, A088311, A265024, A335629, A346841, A347893, A347894, A347898.

Programs

  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, (1+x^k)^sin(x))))
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace(exp(sin(x)*sum(k=1, N, sigma(k>>valuation(k, 2))*x^k/k))))

Formula

E.g.f.: exp( sin(x) * Sum_{k>=1} x^k / (k*(1 - x^(2*k))) ). - Ilya Gutkovskiy, Sep 18 2021
E.g.f.: exp( sin(x) * Sum_{k>=1} A000593(k)*x^k/k ). - Seiichi Manyama, Sep 18 2021

A386474 Number of sets of lists of [n] such that no list is longer than than the total number of lists.

Original entry on oeis.org

1, 1, 1, 7, 25, 141, 1171, 9913, 85233, 907273, 11010691, 143824341, 1988010553, 29605763773, 475664908083, 8284952367721, 153508912353121, 2997209814190353, 61485486404453443, 1326994255131585373, 30144049509450774441, 718905298680190094341, 17940822818538396541843
Offset: 0

Views

Author

John Tyler Rascoe, Jul 23 2025

Keywords

Comments

Here sets of lists are set partitions of [n] such that the elements within each block are ordered but the blocks themselves are unordered.

Examples

			a(3) = 7 counts: {(1),(2),(3)}, {(1),(2,3)}, {(1),(3,2)}, {(1,2),(3)}, {(1,3),(2)}, {(2),(3,1)}, {(2,1),(3)}.

Links

Crossrefs

Cf. A000262, A077229, A088009, A088311, A088312, A088313, A113775, A351884.

Programs

  • Maple
    b:= proc(n, m, l) option remember; `if`(m>n+l, 0, `if`(n=0, 1,
      add(b(n-j, max(m, j), l+1)*(n-1)!*j/(n-j)!, j=1..n)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..22);  # Alois P. Heinz, Jul 23 2025
  • Mathematica
    With[{m = 22}, CoefficientList[1 + Series[Sum[((x - x^(i + 1))/(1 - x))^i/i!, {i, 1, m}], {x, 0, m}], x] * Range[0, m]!] (* Amiram Eldar, Jul 24 2025 *)
  • PARI
    R_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(sum(i=0,N,((x-x^(i+1))/(1-x))^i/i!)))}

Formula

E.g.f.: Sum_{i>=0} ((x - x^(i+1))/(1 - x))^i / i!.

A330387 Expansion of e.g.f. Sum_{k>=1} (-1)^(k + 1) * (exp(x) - 1)^k / (k * (1 - (exp(x) - 1)^k)).

Original entry on oeis.org

1, 2, 12, 62, 420, 3782, 40572, 463262, 5708820, 80773622, 1319927532, 23675250062, 447145154820, 8830952572262, 185694817024092, 4246473212654462, 105754322266866420, 2811068529133151702, 78039884046777282252, 2243558766132057764462
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 12 2019

Keywords

Crossrefs

Cf. A000009, A000593, A000629, A008277, A088311, A265024, A305550, A330353, A330388.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Sum[(-1)^(k + 1) (Exp[x] - 1)^k/(k (1 - (Exp[x] - 1)^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS2[n, k] (k - 1)! Sum[Mod[d, 2] d, {d, Divisors[k]}], {k, 1, n}], {n, 1, 20}]
nmax = 20; Rest[CoefficientList[Series[Sum[Log[1 + (Exp[x] - 1)^k], {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Dec 15 2019 *)

Formula

E.g.f.: -Sum_{k>=1} log(1 - (exp(x) - 1)^(2*k - 1)).
E.g.f.: A(x) = log(B(x)), where B(x) = e.g.f. of A305550.
exp(Sum_{n>=1} a(n) * log(1 + x)^n / n!) = g.f. of A000009.
a(n) = Sum_{k=1..n} Stirling2(n,k) * (k - 1)! * A000593(k).
E.g.f.: Sum_{k>=1} log(1 + (exp(x) - 1)^k). - Vaclav Kotesovec, Dec 15 2019
a(n) ~ n! * Pi^2 / (24 * (log(2))^(n+1)). - Vaclav Kotesovec, Dec 15 2019

A346964 Expansion of e.g.f. Product_{k>=1} exp(x^k) * (1 + x^k).

Original entry on oeis.org

1, 1, 2, 7, 40, 257, 2086, 19567, 207572, 2451745, 32226922, 462314711, 7178502112, 120315808417, 2157566463950, 41277697722367, 838883560646476, 18020304830796737, 408135672764386642, 9723868266912217255, 242827969365094823192, 6345340713682009241281
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 09 2021

Keywords

Crossrefs

Cf. A000262, A088311, A330200.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Product[Exp[x^k] * (1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[x/(1 - x)] * Product[(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
nmax = 20; CoefficientList[Series[Product[(1 + x^k) / (1 - x^k)^(EulerPhi[k]/k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
Table[n!*Sum[LaguerreL[k, -1, -1]*PartitionsQ[n-k],{k,0,n}], {n,0,20}]

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * A000262(k) * A088311(n-k).
E.g.f.: Product_{k>=1} (1 + x^k) / (1 - x^k)^(A000010(k)/k).
a(n) ~ 2^(-3/2) * (4 + Pi^2/3)^(1/4) * exp(sqrt((4 + Pi^2/3)*n) - n - 1/2) * n^(n - 1/4).
Previous Showing 11-20 of 22 results. Next

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