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A167546 The ED1 array read by antidiagonals.

%I A167546 #26 Jul 22 2025 07:28:59
%S A167546 1,1,1,2,4,1,6,12,7,1,24,48,32,10,1,120,240,160,62,13,1,720,1440,960,
%T A167546 384,102,16,1,5040,10080,6720,2688,762,152,19,1,40320,80640,53760,
%U A167546 21504,6144,1336,212,22,1
%N A167546 The ED1 array read by antidiagonals.
%C A167546 The coefficients in the upper right triangle of the ED1 array (m > n) were found with the a(n,m) formula while the coefficients in the lower left triangle of the ED1 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 A167546 Our procedure for finding the coefficients in the lower left triangle can be compared with the procedure that De Smit and Lenstra used to fill in the hole in the middle of 'The Print Gallery' by M. C. Escher, see the links. In this lithograph Escher made use of the so-called Droste effect, hence we propose to call this square array of numbers the ED1 array.
%C A167546 For the ED2, ED3 and ED4 arrays see A167560, A167572 and A167584.
%H A167546 B. de Smit and H.W. Lenstra, <a href="https://www.ams.org/journals/notices/200304/200304FullIssue.pdf">The Mathematical Structure of Escher's Print Gallery</a>, Notices of the AMS, Volume 50, Number 4, pp. 446-457, April 2003.
%H A167546 Johannes W. Meijer, The four Escher-Droste arrays, <a href="/A167546/a167546.jpg">jpg image</a>, Mar 08 2013.
%H A167546 A. Ryabov, P. Chvosta, <a href="http://arxiv.org/abs/1402.1949">Tracer dynamics in a single-file system with absorbing boundary</a>, arXiv preprint arXiv:1402.1949 [cond-mat.stat-mech], 2014.
%F A167546 a(n,m) = (2*(m-1)!/(m-n-1)!)*Integral_{y>=0} sinh(y*(2*n-1))/cosh(y)^(2*m-1) for m > n.
%F A167546 The (n-1)-differences of the n-th array row lead to the recurrence relation
%F A167546 Sum_{k=0..n-1} (-1)^k*binomial(n-1,k)*a(n,m-k) = (2*n-1)*(n-1)!
%F A167546 which in its turn leads to, see also A167557,
%F A167546 a(n,m) = 4^(m-1)*(m-1)!*(n+m-2)!/(2*m-2)! for m <= n.
%e A167546 The ED1 array begins with:
%e A167546 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
%e A167546 1, 4, 7, 10, 13, 16, 19, 22, 25, 28
%e A167546 2, 12, 32, 62, 102, 152, 212, 282, 362, 452
%e A167546 6, 48, 160, 384, 762, 1336, 2148, 3240, 4654, 6432
%e A167546 24, 240, 960, 2688, 6144, 12264, 22200, 37320, 59208, 89664
%e A167546 120, 1440, 6720, 21504, 55296, 122880, 245640, 452880, 783144, 1285536
%p A167546 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-1+m-1)!/(2*m-2)! od; for m from n+1 to mmax do a(n,m):= (2*n-1)*(n-1)! + 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 A167546 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 - 1 + m - 1)!/(2*m - 2)!)]; For[m = n + 1, m <= mmax, m++, a[n, m] = (2*n - 1)*(n - 1)! + 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 A167546 A000012, A016777, 2*A005891, A167547, A167548 and A167549 equal the first sixth rows of the array.
%Y A167546 A000142 equals the first column of the array.
%Y A167546 A167550 equals the a(n, n+1) diagonal of the array.
%Y A167546 A047053 equals the a(n, n) diagonal of the array.
%Y A167546 A167558 equals the a(n+1, n) diagonal of the array.
%Y A167546 A167551 equals the row sums of the ED1 array read by antidiagonals.
%Y A167546 A167552 is a triangle related to the a(n) formulas of rows of the ED1 array.
%Y A167546 A167556 is a triangle related to the GF(z) formulas of the rows of the ED1 array.
%Y A167546 A167557 is the lower left triangle of the ED1 array.
%Y A167546 Cf. A068424 (the (m-1)!/(m-n-1)! factor), A007680 (the (2*n-1)*(n-1)! factor).
%Y A167546 Cf. A167560 (ED2 array), A167572 (ED3 array), A167584 (ED4 array).
%K A167546 easy,nonn,tabl
%O A167546 1,4
%A A167546 _Johannes W. Meijer_, Nov 10 2009