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 1-5 of 5 results.

A130905 Expansion of e.g.f. exp(x^2 / 2) / (1 - x).

Original entry on oeis.org

1, 1, 3, 9, 39, 195, 1185, 8295, 66465, 598185, 5982795, 65810745, 789739335, 10266611355, 143732694105, 2155990411575, 34495848612225, 586429426407825, 10555729709800275, 200558864486205225, 4011177290378833575
Offset: 0

Views

Author

Karol A. Penson, Jun 08 2007

Keywords

Comments

a(n) is also the number of oriented simple graphs on n labeled vertices, such that each weakly connected component with 3 or more vertices is a directed cycle. - Austin Shapiro, Apr 17 2009
The Kn2p sums, p>=1, see A180662 for the definition of these sums, of triangle A193229 lead to this sequence. - Johannes W. Meijer, Jul 21 2011
Compare with A000266 with e.g.f. exp( -x^2 / 2) / (1 - x). - Michael Somos, Jul 24 2011
a(n) is the number of permutations of an n-set where each transposition (two cycle) is counted twice. That is, each transposition is an involution and is its own inverse, but if we imagine each transposition can be oriented in one of two ways, then a permutation with oriented transpositions is just a oriented simple graph. Conversely, an oriented simple graph with restrictions on connected components comes from a permutation with oriented transpositions. - Michael Somos, Jul 25 2011

Examples

			1 + x + 3*x^2 + 9*x^3 + 39*x^4 + 195*x^5 + 1185*x^6 + 8295*x^7 + ...
a(2) = 3 because there are 3 oriented simple graphs on two labeled vertices. a(3) = 9 because for oriented simple graphs on three labeled vertices there is 1 with no edges, 6 with one edge, 0 with two edges, and 2 with three edges which are directed cycles such that each weakly connected component with 3 or more vertices is a directed cycle.

Links

Crossrefs

Cf. A000522, A130906.
Cf. A000266, A355268.

Programs

  • Maple
    A130905 := proc(n) local x: n!*coeftayl(exp(x^2/2)/(1-x), x=0, n) end: seq(A130905(n), n=0..25); # Johannes W. Meijer, Jul 21 2011
  • Mathematica
    CoefficientList[Series[E^(x^2/2)/(1-x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 20 2012 *)
  • PARI
    {a(n) = if( n<0, 0, n! * polcoeff( exp( x^2 / 2 + x * O(x^n)) / (1 - x), n))} /* Michael Somos, Jul 24 2011 */

Formula

E.g.f.: exp(x^2/2) / (1-x) = exp( x^2 / 2 + sum(k>=1, x^k/k ) ).
E.g.f.: 1/E(0) where E(k)=1 - x/(1 - x/(x + (2*k+2)/E(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Sep 20 2012
D-finite with recurrence: a(n) = n*a(n-1) + (n-1)*a(n-2) - (n-2)*(n-1)*a(n-3) . - Vaclav Kotesovec, Oct 20 2012
a(n) ~ n!*exp(1/2) . - Vaclav Kotesovec, Oct 20 2012
E.g.f.: E(0)/(1-x)^2, where E(k)= 1 - x/(1 - x/(x - 2*(k+1)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 10 2013
a(n) = n! * Sum_{k=0..floor(n/2)} 1/(2^k * k!). - Seiichi Manyama, Feb 27 2024

Extensions

Superfluous leading 1 deleted by Johannes W. Meijer, Jul 21 2011

A370696 Expansion of e.g.f. exp(x^3)/(1-x).

Original entry on oeis.org

1, 1, 2, 12, 48, 240, 1800, 12600, 100800, 967680, 9676800, 106444800, 1297296000, 16864848000, 236107872000, 3552515366400, 56840245862400, 966284179660800, 17402007419596800, 330638140972339200, 6612762819446784000, 138878156300083200000
Offset: 0

Views

Author

Seiichi Manyama, Feb 27 2024

Keywords

Crossrefs

Cf. A000522, A355268.
Cf. A130906.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^3)/(1-x)))
  • PARI
    a(n) = n!*sum(k=0, n\3, 1/k!);

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} 1/k!.

A373772 Expansion of e.g.f. exp(x^3 / (6 * (1 - x))) / (1 - x).

Original entry on oeis.org

1, 1, 2, 7, 32, 180, 1210, 9450, 84000, 836920, 9234400, 111742400, 1471023400, 20925905000, 319830310800, 5226116295400, 90906373958400, 1676967192700800, 32697692264036800, 671856896755844800, 14509136903381120000, 328520930667097168000
Offset: 0

Views

Author

Seiichi Manyama, Jun 18 2024

Keywords

Crossrefs

Cf. A130906, A361597, A373773.
Cf. A000930, A361533.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\3, binomial(n-2*k, n-3*k)/(6^k*k!));

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-2*k,n-3*k)/(6^k * k!).

A373773 Expansion of e.g.f. exp(x^3 / (6 * (1 - x)^2)) / (1 - x).

Original entry on oeis.org

1, 1, 2, 7, 36, 240, 1930, 17990, 189840, 2233000, 28949200, 410009600, 6297999400, 104275571400, 1851050401200, 35065930299400, 705993054166400, 15051593241484800, 338705933426660800, 8021585392026606400, 199416162740963168000, 5191567315003621552000
Offset: 0

Views

Author

Seiichi Manyama, Jun 18 2024

Keywords

Crossrefs

Cf. A130906, A361597, A373772.
Cf. A361596, A361598.
Cf. A373757.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\3, binomial(n-k, n-3*k)/(6^k*k!));

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-k,n-3*k)/(6^k * k!).
From Vaclav Kotesovec, Jun 18 2024: (Start)
Recurrence: 6*a(n) = 6*(3*n-2)*a(n-1) - 6*(n-1)*(3*n-4)*a(n-2) + 3*(n-2)*(n-1)*(2*n-3)*a(n-3) - (n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 3^(-1/3) * exp(19/72 - 3^(-2/3)*n^(1/3) + 3^(2/3)*n^(2/3)/2 - n) * n^(n + 1/6). (End)

A130908 E.g.f.: exp(x+x^2/2!+x^3/3!)/(1-x).

Original entry on oeis.org

1, 2, 6, 23, 106, 576, 3622, 26006, 210828, 1910096, 19162096, 211095732, 2534829376, 32962249568, 461527198056, 6923249156336, 110774157354832, 1883174989346016, 33897247428278368, 644048388555567536, 12880972761058252896, 270500465268345299072, 5951010522336442007776, 136873244273143429751328
Offset: 0

Views

Author

Karol A. Penson, Jun 08 2007

Keywords

Crossrefs

Cf. A130905, A130906, A130907.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[Exp[x+x^2/2!+x^3/3!]/(1-x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 01 2014 *)
  • PARI
    x='x+O('x^66); Vec(serlaplace(exp(x+x^2/2!+x^3/3!)/(1-x))) \\ Joerg Arndt, Jun 01 2014

Formula

a(n) ~ n! * exp(5/3). - Vaclav Kotesovec, Aug 04 2014
D-finite with recurrence +2*a(n) +2*(-n-1)*a(n-1) +(n-1)*(n-2)*a(n-3) +(n-1)*(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Aug 20 2021

Extensions

Incorrect initial term 1 removed by Harvey P. Dale, Jun 01 2014
Showing 1-5 of 5 results.

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