cp's OEIS Frontend

A327369 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and exactly k endpoints (vertices of degree 1).

%I A327369 #19 Nov 26 2024 17:55:17
%S A327369 1,1,0,1,0,1,2,0,6,0,15,12,30,4,3,314,320,260,80,50,0,13757,10890,
%T A327369 5445,1860,735,66,15,1142968,640836,228564,64680,16800,2772,532,0,
%U A327369 178281041,68362504,17288852,3666600,702030,115416,17892,1016,105
%N A327369 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and exactly k endpoints (vertices of degree 1).
%H A327369 Andrew Howroyd, <a href="/A327369/b327369.txt">Table of n, a(n) for n = 0..1325</a>
%F A327369 Column-wise binomial transform of A327377.
%F A327369 E.g.f.: 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 A327369 Triangle begins:
%e A327369       1
%e A327369       1     0
%e A327369       1     0     1
%e A327369       2     0     6     0
%e A327369      15    12    30     4     3
%e A327369     314   320   260    80    50     0
%e A327369   13757 10890  5445  1860   735    66    15
%t A327369 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Count[Length/@Split[Sort[Join@@#]],1]==k&]],{n,0,5},{k,0,n}]
%o A327369 (PARI)
%o A327369 Row(n)={ \\ R, U, B are e.g.f. of A055302, A055314, A059167.
%o A327369   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 A327369   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 A327369   my(B=x^2/2 + log(sum(k=0, n, 2^binomial(k, 2)*(x*exp(-x + O(x^n)))^k/k!)));
%o A327369   my(A=exp(x + U + subst(B-x, x, x*(1-y) + R)));
%o A327369   Vecrev(n!*polcoef(A, n), n + 1);
%o A327369 }
%o A327369 { for(n=0, 8, print(Row(n))) } \\ _Andrew Howroyd_, Oct 05 2019
%Y A327369 Row sums are A006125.
%Y A327369 Row sums without the first column are A245797.
%Y A327369 Column k = 0 is A059167.
%Y A327369 Column k = 1 is A277072.
%Y A327369 Column k = 2 is A277073.
%Y A327369 Column k = 3 is A277074.
%Y A327369 Column k = n is A123023.
%Y A327369 Column k = n - 1 is A327370.
%Y A327369 The unlabeled version is A327371.
%Y A327369 The covering version is A327377.
%Y A327369 Cf. A004110, A055302, A055314, A059166, A059167, A100743, A327227, A327228, A327362, A327364.
%K A327369 nonn,tabl
%O A327369 0,7
%A A327369 _Gus Wiseman_, Sep 04 2019
%E A327369 Terms a(28) and beyond from _Andrew Howroyd_, Sep 09 2019