A047863 -id:A047863 - cp's OEIS frontend

A174122 Partial sums of A001831.

1, 2, 5, 18, 105, 946, 12589, 240482, 6526289, 250119330, 13512676053, 1027978959346, 110155994874553, 16631509898085074, 3540687511804739837, 1063409634646294541250, 450894476653951603096737

Offset: 0

Jonathan Vos Post, Mar 08 2010

nonn

Partial sums of number of labeled graded partially ordered sets with n elements. The subsequence of primes in this partial sum begins: 2, 5, 12589.

Cf. A001831, A002031, A047863, A052332, A001833, A048194, A052296, A135753, A135754.

a(n) = SUM[i=0..n] A001831(i) = SUM[i=0..n] SUM[j=0..i] ((-1)^j*C(n,j)*A047863(j)).

A228892 Triangular array read by rows. T(n,k) is the number of 2-colored labeled graphs on n nodes with exactly k connected components; n>=1, 1<=k<=n.

Original entry on oeis.org

2, 2, 4, 6, 12, 8, 38, 60, 48, 16, 390, 500, 360, 160, 32, 6062, 6180, 3840, 1680, 480, 64, 134526, 109228, 56280, 22400, 6720, 1344, 128, 4172198, 2673468, 1120784, 384720, 109760, 24192, 3584, 256, 178449270, 89708004, 29975400, 8579424, 2187360, 475776, 80640, 9216, 512, 10508108222, 4108881300

Offset: 1

Geoffrey Critzer, Sep 07 2013

A 2-colored labeled graph is a simple labeled graph in which each vertex is painted black or white and no two vertices of the same color are connected.
Row sums are A047863.
T(n,k) = A228859(n,k)*2^k.

			     2;
     2,    4;
     6,   12,    8;
    38,   60,   48,   16;
   390,  500,  360,  160,  32;
  6062, 6180, 3840, 1680, 480, 64;
  ...

nn=6;f[x_,y_]:=Sum[Exp[x 2^n] x^n/n!,{n,0,nn}];Map[Select[#,#>0&]&,Map[Table[#[[i]]->#[[i]]2^(i-1),{i,1,Length[#]}][[All,2]]&,Drop[Range[0,nn]!CoefficientList[Series[f[x,y]^(y/2),{x,0,nn}],{x,y}],1]]]//Grid

E.g.f.: A(x)^y where A(x) is the e.g.f. for A047863.

cp's OEIS Frontend

A174122 Partial sums of A001831.

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A228892 Triangular array read by rows. T(n,k) is the number of 2-colored labeled graphs on n nodes with exactly k connected components; n>=1, 1<=k<=n.

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