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