This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A167584 #37 May 10 2025 03:18:39
%S A167584 1,2,1,13,6,1,76,41,10,1,789,372,93,14,1,7734,4077,1020,169,18,1,
%T A167584 110937,53106,13269,2212,269,22,1,1528920,795645,198990,33165,4140,
%U A167584 393,26,1,28018665,13536360,3383145,563850,70485,6996,541,30,1
%N A167584 The ED4 array read by antidiagonals.
%C A167584 The coefficients in the upper right triangle of the ED4 array (m>n) were found with the a(n,m) formula while the coefficients in the lower left triangle of the ED4 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 A167584 For the ED1, ED2 and ED3 arrays see A167546, A167560 and A167572.
%C A167584 The Madhava-Gregory-Leibniz series representation for Pi/4 is the case m = 0 of the following more general result: for m = 0,1,2,... there holds 1/(2*m)! * Pi/4 = Sum_{k >= 0} ( (-1)^(m+k) * 1/Product_{j = -m .. m} (2*k + 1 + 2*j) ). The entries of this table are given by truncating these series to n-1 terms and then scaling by certain double factorials -- see the formula below. - _Peter Bala_, Nov 06 2016
%H A167584 G. C. Greubel, <a href="/A167584/b167584.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A167584 Johannes W. Meijer, The four Escher-Droste arrays, <a href="/A167584/a167584.jpg">jpg image</a>, Mar 08 2013.
%H A167584 Wikipedia, <a href="https://en.wikipedia.org/wiki/Double_factorial">Double factorial</a>
%F A167584 a(n,m) = ((2*m-3)!!/(2*(2*m-2*n-3)!!))*Integral_{y=0..oo} sinh(y*(2*n))/(cosh(y))^(2*m-1) dy for m>n.
%F A167584 The (n-1)-differences of the n-th array row lead to the recurrence relation
%F A167584 Sum_{k=0..n-1} (-1)^k*binomial(n-1,k)*a(n,m-k) = 2^(n-1)*n!
%F A167584 From _Peter Bala_, Nov 06 2016: (Start)
%F A167584 T(n,m) = ((2*m - 3)!!/(2*(2*m - 2*n - 3)!!)) * Sum_{k = 0..n-1} (-1)^(k+1)*binomial(2*n - k - 1, k)*2^(2*n - 2*k - 1)*1/(2*n - 2*m - 2*k + 1), for n and m >= 0.
%F A167584 Note the double factorial for a negative odd integer N is defined in terms of the gamma function as N!! = 2^((N+1)/2)*Gamma(N/2 + 1)/sqrt(Pi).
%F A167584 T(n, m) = (2*m - 3)!! * (2*n + 2*m - 3)!! * Sum_{k = 0..n-1} ( (-1)^(m + k + 1) / Product_{j = -(m-1) .. m-1} (2*k + 1 + 2*j) ).
%F A167584 Using this result we can extend the table to nonpositive values of m (the column index). Column 0 is a signed version of A001193. We have for m <= 0, T(n,m) = (2*n - 2*|m| - 3)!!/(2*|m| + 1)!! * Sum_{k = 0..n-1} (-1)^k*Product_{j = -|m|..|m|} (2*k + 1 + 2*j).
%F A167584 Recurrence: T(n, m) = (4*m - 2)*T(n-1, m) + (2*n + 2*m - 5)*(2*n - 2*m - 1)*T(n-2, m).
%F A167584 For a fixed value of n, the entries in row n are polynomial in the value of the column index m. The first few polynomials are [1, 4*m - 2, 12*m^2 - 8*m + 9, 32*m^3 - 16*m^2 + 120*m - 60, 80*m^4 + 952*m^2 - 768*m + 525, ...]. (End)
%e A167584 The ED4 array begins with:
%e A167584 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
%e A167584 2, 6, 10, 14, 18, 22, 26, 30, 34, 38
%e A167584 13, 41, 93, 169, 269, 393, 541, 713, 909, 1129
%e A167584 76, 372, 1020, 2212, 4140, 6996, 10972, 16260, 23052, 31540
%e A167584 789, 4077, 13269, 33165, 70485, 133869, 233877, 382989, 595605, 888045
%e A167584 7734, 53106, 198990, 563850, 1339110, 2812194, 5389566, 9619770, 16216470, 26081490
%e A167584 ...
%e A167584 From _Peter Bala_, Nov 06 2016: (Start)
%e A167584 Table extended to nonpositive values of m:
%e A167584 n\m| -4 -3 -2 -1 0
%e A167584 -----------------------------------
%e A167584 0 | 0 0 0 0 0
%e A167584 1 | 1 1 1 1 1
%e A167584 2 | -18 -14 -10 -6 -2
%e A167584 3 | 233 141 73 29 9
%e A167584 4 | -2844 -1428 -620 -228 -60
%e A167584 5 | 39309 17877 7149 2325 525
%e A167584 ...
%e A167584 Column 0: (-1)^(n+1)*(2*n - 3)!!*n. See A001193;
%e A167584 Column -1: (-1)^n*(2*n - 5)!!/3!!*n*(7 - 4*n^2);
%e A167584 Column -2: (-1)^n*(2*n - 7)!!/5!!*n(-149 + 120*n^2 - 16*n^4);
%e A167584 Column -3: (-1)^n*(2*n - 9)!!/7!!*n*(6483 - 6076*n^2 + 1232*n^4 - 64*n^6);
%e A167584 Column -4: (-1)^n*(2*n - 11)!!/9!!*n*(-477801 + 489136*n^2 - 120288*n^4 + 9984*n^6 - 256*n^8). (End)
%p A167584 T := proc (n, m) option remember;
%p A167584 if n = 0 then 0
%p A167584 elif n = 1 then 1
%p A167584 else (4*m-2)*T(n-1,m)+(2*n+2*m-5)*(2*n-2*m-1)*T(n-2,m)
%p A167584 end if;
%p A167584 end proc:
%p A167584 #square array read by antidiagonals
%p A167584 seq(seq(T(n-m,m), m = 1..n-1), n = 1..10);
%p A167584 # _Peter Bala_, Nov 06 2016
%t A167584 T[0, k_] := 0; T[1, k_] := 1; T[n_, k_] := T[n, k] = (4*k - 2)*T[n - 1, k] + (2*n + 2*k - 5)*(2*n - 2*k - 1)*T[n - 2, k]; Table[T[n - k, k], {n, 2, 12}, {k, 1, n - 1}] (* _G. C. Greubel_, Jan 20 2017 *)
%Y A167584 A000012, A016825, A167585, A167586 and A167587 equal the first five rows of the array.
%Y A167584 A024199, A167588 and A167589 equal the first three columns of the array.
%Y A167584 A167590 equals the row sums of the ED4 array read by antidiagonals.
%Y A167584 A167591 is a triangle related to the a(n) formulas of the rows of the ED4 array.
%Y A167584 A167594 is a triangle related to the GF(z) formulas of the rows of the ED4 array.
%Y A167584 Cf. A002866 (the 2^(n-1)*n! factor).
%Y A167584 Cf. A167546 (ED1 array), A167560 (ED2 array), A167572 (ED3 array). Cf. A001193, A003881.
%K A167584 nonn,tabl
%O A167584 1,2
%A A167584 _Johannes W. Meijer_, Nov 10 2009