1,2

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).

For the ED1, ED2 and ED3 arrays see A167546, A167560 and A167572.

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

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

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.

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) / 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)

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)

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

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 *)

A000012, A016825, A167585, A167586 and A167587 equal the first five rows of the array.

A024199, A167588 and A167589 equal the first three columns of the array.

A167590 equals the row sums of the ED4 array read by antidiagonals.

A167591 is a triangle related to the a(n) formulas of the rows of the ED4 array.

A167594 is a triangle related to the GF(z) formulas of the rows of the ED4 array.

Cf. A002866 (the 2^(n-1)*n! factor).

Cf. A167546 (ED1 array), A167560 (ED2 array), A167572 (ED3 array). Cf. A001193, A003881.

Sequence in context: A292947 A143663 A064079 * A112226 A192795 A113097

Adjacent sequences: A167581 A167582 A167583 * A167585 A167586 A167587

nonn,tabl

Johannes W. Meijer, Nov 10 2009

approved