A093351 -id:A093351 - cp's OEIS frontend

A093352 Number of labeled n-vertex graphs without a 2-component.

1, 1, 1, 5, 55, 959, 31883, 2076383, 267530657, 68644357201, 35172312944057, 36025019516955853, 73784654524456043287, 302228644804839247744495, 2475873364061564307502565395, 40564787473148108729970731074007, 1329227698679709317077126629247388161

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Apr 26 2004

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..50

Cf. A093351, A006129.

A093352 := n -> n!*add((-1)^q/2^q/q!*2^binomial(n-2*q,2)/(n-2*q)!, q=0..floor(n/2)); # Marko Riedel, Apr 05 2022

nn = 15; g = Sum[2^Binomial[n, 2] x^n/n!, {n, 0, nn}];
Range[0, nn]! CoefficientList[Series[g/Exp[x^2/2], {x, 0, nn}], x] (* Geoffrey Critzer, Aug 27 2013 *)

E.g.f.: exp(-x^2/2)*Sum_{n>=0} 2^binomial(n,2)*x^n/n!.

a(n) = n!*Sum_{q=0..floor(n/2)} ((-1)^q/(2^q*q!))*(2^C(n-2*q,2)/(n-2*q)!). - Marko Riedel, Apr 05 2022

A093377 Number of labeled n-vertex graphs without 2-components and without isolated vertices (1-components).

Original entry on oeis.org

1, 0, 0, 4, 38, 728, 26864, 1871576, 251762204, 66308767200, 34497665550400, 35641856042561008, 73354660691960203016, 301272244237002052739424, 2471648864359822034978330304, 40527681073171940835893232576032

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Apr 28 2004

nonn

Also number of unlabeled n-block ordered r-bicoverings, cf. A060053. - Vladeta Jovovic, May 13 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..65

Cf. A006129, A093351, A093352.

nn=20;g=Sum[2^Binomial[n,2]x^n/n!,{n,0,nn}];Range[0,nn]!CoefficientList[Series[Exp[ Log[g]-x-x^2/2!],{x,0,nn}],x]  (* Geoffrey Critzer, Apr 15 2013 *)

N=66; x='x+O('x^N);
egf=exp(-x-x^2/2)*sum(i=0,N, 2^binomial(i, 2)*x^i/i!);
Vec(serlaplace(egf))
/* Joerg Arndt, Jul 06 2011 */

E.g.f.: exp(-x-x^2/2)*Sum_{n>=0} 2^binomial(n, 2)*x^n/n!.

Inverse binomial transform of A093352().

A093376 Number of labeled n-vertex graphs with 2-components and without isolated vertices(1-components).

Original entry on oeis.org

0, 0, 1, 0, 3, 40, 585, 15708, 760277, 67656960, 11346344145, 3648840170580, 2277220167519825, 2780207051899228224, 6675377730807161508041, 31633731603265107184483860, 296598264243770222308892404305

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Apr 28 2004

nonn

Cf. A006129, A093351, A093352.

E.g.f. exp(-x)*(1-exp(-x^2/2))*Sum(2^binomial(n, 2)*x^n/n!, n=0..infinity). Inverse binomial transform of A093351().

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A093352 Number of labeled n-vertex graphs without a 2-component.

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A093377 Number of labeled n-vertex graphs without 2-components and without isolated vertices (1-components).

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A093376 Number of labeled n-vertex graphs with 2-components and without isolated vertices(1-components).

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