A167584 The ED4 array read by antidiagonals.
1, 2, 1, 13, 6, 1, 76, 41, 10, 1, 789, 372, 93, 14, 1, 7734, 4077, 1020, 169, 18, 1, 110937, 53106, 13269, 2212, 269, 22, 1, 1528920, 795645, 198990, 33165, 4140, 393, 26, 1, 28018665, 13536360, 3383145, 563850, 70485, 6996, 541, 30, 1
Offset: 1
Examples
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)
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
Crossrefs
Programs
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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 *)
Formula
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)
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