A319365
Expansion of e.g.f. exp(x^4/4)/(1 - x).
Original entry on oeis.org
1, 1, 2, 6, 30, 150, 900, 6300, 51660, 464940, 4649400, 51143400, 614968200, 7994586600, 111924212400, 1678863186000, 26865216378000, 456708678426000, 8220756211668000, 156194368021692000, 3123907159441068000, 65602050348262428000, 1443245107661773416000, 33194637476220788568000
Offset: 0
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m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x^4/4)/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // Vincenzo Librandi, Dec 28 2018
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f:= gfun:-rectoproc({(n+1)*(n+2)*(n+3)*(n+4)*a(n)-(n+2)*(n+3)*(n+4)*a(n+1)-(n+5)*a(n+4)+a(n+5)},seq(a(i)=[1,1,2,6,30][i+1],i=0..4)},a(n),remember):
map(f, [$0..30]); # Robert Israel, Dec 28 2018
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nmax = 23; CoefficientList[Series[Exp[x^4/4]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
A316666
Number of simple relaxed compacted binary trees of right height at most one with no sequences on level 1 and no final sequences on level 0.
Original entry on oeis.org
1, 0, 1, 3, 15, 87, 597, 4701, 41787, 413691, 4512993, 53779833, 695000919, 9680369943, 144560191149, 2303928046437, 39031251610227, 700394126116851, 13270625547477177, 264748979672169681, 5547121478845459983, 121784530649198053263, 2795749225338111831429, 66981491857058929294653
Offset: 0
- G. C. Greubel, Table of n, a(n) for n = 0..448
- Antoine Genitrini, Bernhard Gittenberger, Manuel Kauers and Michael Wallner, Asymptotic Enumeration of Compacted Binary Trees, arXiv:1703.10031 [math.CO], 2017.
- Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017.
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m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (3*Exp(-x) + x-2)/(1-x)^2 )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Dec 12 2018
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aseq := n-> 3*round((n+2)*n!/exp(1))-(n+2)*n!: bseq := n-> (n+2)*n!- 2* round((n+2)*n!/exp(1)): s := (a,b,n)-> a*aseq(n) + b*bseq( n): seq(s(1,0,n),n = 0..20); # Gary Detlefs, Dec 11 2018
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terms = 24;
CoefficientList[(3E^-z+z-2)/(1-z)^2 + O[z]^terms, z] Range[0, terms-1]! (* Jean-François Alcover, Sep 14 2018 *)
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Vec(serlaplace((3*exp(-x + O(x^25)) + x - 2)/(1 - x)^2)) \\ Andrew Howroyd, Jul 10 2018
A373770
Expansion of e.g.f. exp(x^2 / (2 * (1 - x))) / (1 - x).
Original entry on oeis.org
1, 1, 3, 12, 63, 405, 3075, 26880, 265545, 2922885, 35447895, 469396620, 6736095135, 104102463465, 1723322736135, 30416726340000, 570089983287825, 11306156398562025, 236514323713142475, 5204122351983254700, 120139520273298100575, 2903216115946088267325
Offset: 0
A373771
Expansion of e.g.f. exp(x^2 / (2 * (1 - x)^3)) / (1 - x).
Original entry on oeis.org
1, 1, 3, 18, 147, 1425, 15855, 200130, 2838465, 44767485, 777046095, 14705245170, 301014595035, 6621102973485, 155640761791515, 3891902825660850, 103115436832433025, 2884715829245475225, 84950805438277854075, 2626194012669689512050
Offset: 0
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
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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 *)
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x='x+O('x^66); Vec(serlaplace(exp(x+x^2/2!+x^3/3!)/(1-x))) \\ Joerg Arndt, Jun 01 2014
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