A125205 Irregular triangle read by rows T(n,k) (n>=1, 0<=k<=n(n-1)/2) giving the total number of connected components in all subgraphs (V,E') with |E'|=k of the complete labeled graph K_n=(V,E).

1, 2, 1, 3, 6, 3, 1, 4, 18, 30, 24, 15, 6, 1, 5, 40, 135, 250, 295, 282, 215, 120, 45, 10, 1, 6, 75, 420, 1385, 3015, 4800, 6365, 7170, 6705, 5065, 3009, 1365, 455, 105, 15, 1, 7, 126, 1050, 5355, 18690, 47880, 96796, 166890, 251370, 329945, 373947, 362292, 297115

Offset

1,2

Links

Table of n, a(n) for n=1..54.

Formula

G.f.: Sum_{n,k} T(n,k)*x^n/n!*y^k=(F(x,y)-1)*exp(F(x,y)-1)=G(x,y)*log(G(x,y)) where G(x,y)=Sum_{n=0..oo} (1+y)^(n(n-1)/2)*x^n/n! and F(x,y)=1+log(G(x,y)) is g.f. of A062734.

Example

Triangle begins:
  1;
  2,  1;
  3,  6,   3,   1;
  4, 18,  30,  24,  15,   6,   1;
  5, 40, 135, 250, 295, 282, 215, 120, 45, 10, 1;
  ...
T(3,1) = 6 since there are three different subgraphs of K_3 with one edge and each subgraph has two connected components.

Prog

(PARI) { G=sum(n=0, 6, (1+y)^(n*(n-1)/2)*x^n/n!); K=G*log(G); for(n=1, 6, print(Vecrev(n!*polcoeff(K, n, x)))) }

Crossrefs

Cf. A062734.

Cf. A125206 (row-reversed version), A125207 (row sums).

Sequence in context: A006895 A202204 A289815 * A125206 A221918 A193897

Adjacent sequences: A125202 A125203 A125204 * A125206 A125207 A125208

Keyword

nonn,tabf

Author

Max Alekseyev, Nov 23 2006

Status

approved