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A255192 Triangle of number of connected subgraphs of K(n,n) with m edges.

%I A255192 #21 Mar 25 2023 07:51:28
%S A255192 1,4,1,81,78,36,9,1,4096,8424,9552,7464,4272,1812,560,120,16,1,390625,
%T A255192 1359640,2696200,3880300,4394600,4059000,3111140,1994150,1070150,
%U A255192 478800,176900,53120,12650,2300,300,25,1,60466176,314452800,939988800,2075760000
%N A255192 Triangle of number of connected subgraphs of K(n,n) with m edges.
%C A255192 m ranges from 2n-1 to n^2.
%C A255192 First column is A068087.
%F A255192 Sum(k>=0, T(n,k)*(-1)^k ) = A136126(2*n-1,n-1) = A092552(n+1), alternating row sums.
%e A255192 Triangle begins:
%e A255192 ----|------------------------------------------------------------
%e A255192 n\m |  1 2 3 4  5  6    7    8    9   10   11   12  13  14 15 16
%e A255192 ----|------------------------------------------------------------
%e A255192 1   |  1
%e A255192 2   |  - - 4 1
%e A255192 3   |  - - - - 81 78   36    9    1
%e A255192 4   |  - - - -  -  - 4096 8424 9552 7464 4272 1812 560 120 16  1
%o A255192 (Python)
%o A255192 from math import comb as binomial
%o A255192 def f(x, a, b, k):
%o A255192     if b == k == 0:
%o A255192         return 1
%o A255192     if b == 0 or k == 0:
%o A255192         return 0
%o A255192     if x == a:
%o A255192         return sum(binomial(a, n) * f(x, x, b - 1, k - n) for n in range(1, a + 1))
%o A255192     return sum(binomial(b, n) * f(x, x, n, k2) * f(n, b, a - x, k - k2)
%o A255192         for n in range(1, b + 1) for k2 in range(0, k + 1) )
%o A255192 def a(n, m):
%o A255192     return f(1, n, n, m)
%o A255192 for n in range(1, 5):
%o A255192     print([a(n, m) for m in range(1, n * n + 1)])
%Y A255192 Cf. A005333 (row sums?).
%K A255192 nonn,tabf
%O A255192 1,2
%A A255192 _Thomas Dybdahl Ahle_, Feb 16 2015