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A342557 T(n,m) is the number of unlabeled connected graphs without endpoints on m nodes with n edges, where T(n,m), m <= n, is a triangle read by rows.

%I A342557 #45 Feb 07 2024 23:27:35
%S A342557 0,0,0,0,0,1,0,0,0,1,0,0,0,1,1,0,0,0,1,3,1,0,0,0,0,3,5,1,0,0,0,0,2,11,
%T A342557 8,1,0,0,0,0,1,15,31,12,1,0,0,0,0,1,12,63,71,16,1,0,0,0,0,0,8,89,231,
%U A342557 144,21,1,0,0,0,0,0,5,97,513,707,274,27,1
%N A342557 T(n,m) is the number of unlabeled connected graphs without endpoints on m nodes with n edges, where T(n,m), m <= n, is a triangle read by rows.
%C A342557 The number of nonzero terms in the n-th row is A083920(n-1). - _Hugo Pfoertner_, Feb 01 2024
%H A342557 Andrew Howroyd, <a href="/A342557/b342557.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50; first 24 rows from Hugo Pfoertner)
%F A342557 Bivariate inverse Euler transform of A369932. - _Andrew Howroyd_, Feb 07 2024
%e A342557 The triangle begins
%e A342557   0;
%e A342557   0, 0;
%e A342557   0, 0, 1;
%e A342557   0, 0, 0, 1;
%e A342557   0, 0, 0, 1, 1;
%e A342557   0, 0, 0, 1, 3,  1;
%e A342557   0, 0, 0, 0, 3,  5,  1;
%e A342557   0, 0, 0, 0, 2, 11,  8,  1;
%e A342557   0, 0, 0, 0, 1, 15, 31, 12,  1;
%e A342557   0, 0, 0, 0, 1, 12, 63, 71, 16, 1;
%o A342557 (PARI) \\ Needs G() defined in A369932.
%o A342557 InvEulerMTS(p)={my(n=serprec(p, x)-1, q=log(p), vars=variables(p)); sum(i=1, n, moebius(i)*substvec(q + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i)}
%o A342557 T(n)={my(r=Vec(InvEulerMTS(substvec(G(n),[x,y],[y,x])))); vector(#r-1, i, Vecrev(Pol(r[i+1]/y),i)) }
%o A342557 { my(A=T(12)); for(i=1, #A, print(A[i])) } \\ _Andrew Howroyd_, Feb 07 2024
%Y A342557 Cf. A004108 (column sums), A342556 (row sums).
%Y A342557 Cf. A083920, A369932 (not necessarily connected).
%K A342557 nonn,tabl
%O A342557 1,20
%A A342557 _Hugo Pfoertner_, May 21 2021