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A350635 Triangle read by rows: T(n,k) is the number of n-element unlabeled P-series with k connected components.

%I A350635 #34 Feb 16 2022 15:40:06
%S A350635 1,1,1,3,1,1,7,4,1,1,15,10,4,1,1,31,28,11,4,1,1,63,67,31,11,4,1,1,127,
%T A350635 167,80,32,11,4,1,1,255,388,213,83,32,11,4,1,1,511,908,534,226,84,32,
%U A350635 11,4,1,1,1023,2053,1343,580,229,84,32,11,4,1,1
%N A350635 Triangle read by rows: T(n,k) is the number of n-element unlabeled P-series with k connected components.
%H A350635 Andrew Howroyd, <a href="/A350635/b350635.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%F A350635 G.f.: -1 + exp(Sum_{k>=1} y^k*B(x^k)/k) where B(x) = x*(1 - 2*x + 2*x^2)/((1 - x)*(1 - 2*x)). - _Andrew Howroyd_, Jan 13 2022
%e A350635 Triangle begins:
%e A350635     1;
%e A350635     1,   1;
%e A350635     3,   1,  1;
%e A350635     7,   4,  1,  1;
%e A350635    15,  10,  4,  1,  1;
%e A350635    31,  28, 11,  4,  1, 1;
%e A350635    63,  67, 31, 11,  4, 1, 1;
%e A350635   127, 167, 80, 32, 11, 4, 1, 1;
%e A350635   ...
%o A350635 (PARI)
%o A350635 B(x) = x*(1 - 2*x + 2*x^2)/((1 - x)*(1 - 2*x))
%o A350635 T(n)=[Vecrev(p/y) | p<-Vec(-1 + exp(sum(k=1, n, y^k*B(x^k)/k + O(x*x^n))))]
%o A350635 { my(A=T(8)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Jan 13 2022
%Y A350635 Row sums give A349276.
%Y A350635 Column 1 is A255047(n-1).
%Y A350635 Cf.  A263864 (all posets), A349488 (disconnected).
%K A350635 nonn,tabl
%O A350635 1,4
%A A350635 _Salah Uddin Mohammad_, Jan 09 2022