OFFSET
1,2
COMMENTS
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).
From Peter Bala, Nov 06 2016: (Start)
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. (End)
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Johannes W. Meijer, The four Escher-Droste arrays, jpg image, Mar 08 2013.
Wikipedia, Double factorial
FORMULA
a(n,m) = ((2*m-3)!!/(2*(2*m-2*n-3)!!))*int(sinh(y*(2*n))/ (cosh(y))^(2*m-1),y=0..infinity) for m>n.
The (n-1)-differences of the n-th array row lead to the recurrence relation
Sum_{k=0..n-1} (-1)^k*binomial(n-1,k)*a(n,m-k) = 2^(n-1)*n!
From Peter Bala, Nov 06 2016: (Start)
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.
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).
T(n, m) = (2*m - 3)!! * (2*n + 2*m - 3)!! * Sum_{k = 0..n-1} ( (-1)^(m + k + 1) * 1/Product_{j = -(m-1) .. m-1 )} (2*k + 1 + 2*j) ).
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).
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).
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)
EXAMPLE
The ED4 array begins with:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1
2, 6, 10, 14, 18, 22, 26, 30, 34, 38
13, 41, 93, 169, 269, 393, 541, 713, 909, 1129
76, 372, 1020, 2212, 4140, 6996, 10972, 16260, 23052, 31540
789, 4077, 13269, 33165, 70485, 133869, 233877, 382989, 595605, 888045
7734, 53106, 198990, 563850, 1339110, 2812194, 5389566, 9619770, 16216470, 26081490
...
From Peter Bala, Nov 06 2016: (Start)
Table extended to nonpositive values of m:
n\m| -4 -3 -2 -1 0
-----------------------------------
0 | 0 0 0 0 0
1 | 1 1 1 1 1
2 | -18 -14 -10 -6 -2
3 | 233 141 73 29 9
4 | -2844 -1428 -620 -228 -60
5 | 39309 17877 7149 2325 525
...
Column 0: (-1)^(n+1)*(2*n - 3)!!*n. See A001193;
Column -1: (-1)^n*(2*n - 5)!!/3!!*n*(7 - 4*n^2);
Column -2: (-1)^n*(2*n - 7)!!/5!!*n(-149 + 120*n^2 - 16*n^4);
Column -3: (-1)^n*(2*n - 9)!!/7!!*n*(6483 - 6076*n^2 + 1232*n^4 - 64*n^6);
Column -4: (-1)^n*(2*n - 11)!!/9!!*n*(-477801 + 489136*n^2 - 120288*n^4 + 9984*n^6 - 256*n^8). (End)
MAPLE
T := proc (n, m) option remember;
if n = 0 then 0
elif n = 1 then 1
else (4*m-2)*T(n-1, m)+(2*n+2*m-5)*(2*n-2*m-1)*T(n-2, m)
end if;
end proc:
#square array read by antidiagonals
seq(seq(T(n-m, m), m = 1..n-1), n = 1..10);
# Peter Bala, Nov 06 2016
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Johannes W. Meijer, Nov 10 2009
STATUS
approved