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%I #19 Jan 12 2023 11:41:09
%S 1,2,1,6,4,1,24,16,6,1,120,80,32,8,1,720,480,192,54,10,1,5040,3360,
%T 1344,384,82,12,1,40320,26880,10752,3072,680,116,14,1,362880,241920,
%U 96768,27648,6144,1104,156,16,1
%N The ED2 array read by ascending antidiagonals
%C The coefficients in the upper right triangle of the ED2 array (m>n) were found with the a(n,m) formula while the coefficients in the lower left triangle of the ED2 array (m<=n) were found with the recurrence relation, see below. We use for the array rows the letter n (>=1) and for the array columns the letter m (>=1).
%C The ED2 array is related to the EG1 matrix, see A162005, because sum(EG1(2*m-1,n) * z^(2*m-1), m=1..infinity) = ((2*n-1)!/(4^(n-1)*(n-1)!^2))*int(sinh(y*(2*z))/cosh(y)^(2*n), y=0..infinity).
%C For the ED1, ED3 and ED4 arrays see A167546, A167572 and A167584.
%H Johannes W. Meijer, <a href="/A167560/a167560.jpg">The four Escher-Droste arrays</a>, Mar 08 2013.
%F a(n,m) = ((m-1)!/((m-n-1)!))*int(sinh(y*(2*n))/(cosh(y))^(2*m),y=0..infinity) for m>n.
%F The (n-1)-differences of the n-th array row lead to the recurrence relation
%F sum((-1)^k*binomial(n-1,k)*a(n-1,m-k),k=0..n-1) = n!
%F which in its turn leads to, see A167569,
%F a(n,m) = 4^(m-1)*(m-1)!*(n+m-1)!/(2*m-1)! if m<=n.
%e The ED2 array begins with:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
%e 2, 4, 6, 8, 10, 12, 14, 16, 18, 20
%e 6, 16, 32, 54, 82, 116, 156, 202, 254, 312
%e 24, 80, 192, 384, 680, 1104, 1680, 2432, 3384, 4560
%e 120, 480, 1344, 3072, 6144, 11160, 18840, 30024, 45672, 66864
%e 720, 3360, 10752, 27648, 61440, 122880, 226800, 392832, 646128, 1018080
%p nmax:=10; mmax:=10; for n from 1 to nmax do for m from 1 to n do a(n,m) := 4^(m-1)*(m-1)!*(n+m-1)!/(2*m-1)! od; for m from n+1 to mmax do a(n,m):= n! + sum((-1)^(k-1)*binomial(n-1,k)*a(n,m-k),k=1..n-1) od; od: for n from 1 to nmax do for m from 1 to n do d(n,m):=a(n-m+1,m) od: od: T:=1: for n from 1 to nmax do for m from 1 to n do a(T):= d(n,m): T:=T+1: od: od: seq(a(n),n=1..T-1);
%t nmax = 10; mmax = 10; For[n = 1, n <= nmax, n++, For[m = 1, m <= n, m++, a[n, m] = 4^(m - 1)*(m - 1)!*((n + m - 1)!/(2*m - 1)!)]; For[m = n + 1, m <= mmax, m++, a[n, m] = n! + Sum[(-1)^(k - 1)*Binomial[n - 1, k]*a[n, m - k], {k, 1, n - 1}]]; ]; For[n = 1, n <= nmax, n++, For[m = 1, m <= n, m++, d[n, m] = a[n - m + 1, m]]; ]; t = 1; For[n = 1, n <= nmax, n++, For[m = 1, m <= n, m++, a[t] = d[n, m]; t = t + 1]]; Table[a[n], {n, 1, t - 1}] (* _Jean-François Alcover_, Dec 20 2011, translated from Maple *)
%Y A000012, A005843 (n>=1), 2*A104249 (n>=1), A167561, A167562 and A167563 equal the first sixth rows of the array.
%Y A000142 equals the first column of the array.
%Y A047053 equals the a(n, n) diagonal of the array.
%Y 2*A034177 equals the a(n+1, n) diagonal of the array.
%Y A167570 equals the a(n+2, n) diagonal of the array,
%Y A167564 equals the row sums of the ED2 array read by antidiagonals.
%Y A167565 is a triangle related to the a(n) formulas of the rows of the ED2 array.
%Y A167568 is a triangle related to the GF(z) formulas of the rows of the ED2 array.
%Y A167569 is the lower left triangle of the ED2 array.
%Y Cf. A162005 (EG1 triangle).
%Y Cf. A167546 (ED1 array), A167572 (ED3 array), A167584 (ED4 array).
%K easy,nonn,tabl
%O 1,2
%A _Johannes W. Meijer_, Nov 10 2009
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