cp's OEIS Frontend

A327377 Triangle read by rows where T(n,k) is the number of labeled simple graphs covering n vertices with exactly k endpoints (vertices of degree 1).

%I A327377 #13 Nov 26 2024 17:54:34
%S A327377 1,0,0,0,0,1,1,0,3,0,10,12,12,4,3,253,260,160,60,35,0,12068,9150,4230,
%T A327377 1440,480,66,15,1052793,570906,195048,53200,12600,2310,427,0,
%U A327377 169505868,63523656,15600032,3197040,585620,95088,14056,1016,105
%N A327377 Triangle read by rows where T(n,k) is the number of labeled simple graphs covering n vertices with exactly k endpoints (vertices of degree 1).
%C A327377 A graph is covering if there are no isolated vertices.
%H A327377 Andrew Howroyd, <a href="/A327377/b327377.txt">Table of n, a(n) for n = 0..1325</a>
%F A327377 Column-wise inverse binomial transform of A327369.
%F A327377 E.g.f.: exp(-x)*exp(x + U(x,y) + B(x*(1-y) + R(x,y))), where R(x,y) is the e.g.f. of A055302, U(x,y) is the e.g.f. of A055314 and B(x) + x is the e.g.f. of A059167. - _Andrew Howroyd_, Oct 05 2019
%e A327377 Triangle begins:
%e A327377       1
%e A327377       0     0
%e A327377       0     0     1
%e A327377       1     0     3     0
%e A327377      10    12    12     4     3
%e A327377     253   260   160    60    35     0
%e A327377   12068  9150  4230  1440   480    66    15
%o A327377 (PARI)
%o A327377 Row(n)={ \\ R, U, B are e.g.f. of A055302, A055314, A059167.
%o A327377   my(U=sum(n=2, n, x^n*sum(k=1, n, stirling(n-2, n-k, 2)*y^k/k!)) + O(x*x^n));
%o A327377   my(R=sum(n=1, n, x^n*sum(k=1, n, stirling(n-1, n-k, 2)*y^k/k!)) + O(x*x^n));
%o A327377   my(B=x^2/2 + log(sum(k=0, n, 2^binomial(k, 2)*(x*exp(-x + O(x^n)))^k/k!)));
%o A327377   my(A=exp(-x + O(x*x^n))*exp(x + U + subst(B-x, x, x*(1-y) + R)));
%o A327377   Vecrev(n!*polcoef(A, n), n + 1);
%o A327377 }
%o A327377 { for(n=0, 8, print(Row(n))) } \\ _Andrew Howroyd_, Oct 05 2019
%Y A327377 Row sums are A006129.
%Y A327377 Column k = 0 is A100743.
%Y A327377 Column k = n is A123023.
%Y A327377 Row sums without the first column are A327227.
%Y A327377 The non-covering version is A327369.
%Y A327377 The unlabeled version is A327372.
%Y A327377 Cf. A004110, A006125, A055302, A055314, A059167, A245797, A327362, A327364, A327370.
%K A327377 nonn,tabl
%O A327377 0,9
%A A327377 _Gus Wiseman_, Sep 05 2019
%E A327377 Terms a(28) and beyond from _Andrew Howroyd_, Oct 05 2019