A167560 -id:A167560 - cp's OEIS frontend

A167572 The ED3 array read by antidiagonals.

1, 5, 1, 23, 11, 1, 167, 83, 17, 1, 1473, 741, 183, 23, 1, 16413, 8169, 2043, 323, 29, 1, 211479, 106107, 26529, 4409, 503, 35, 1, 3192975, 1592235, 398025, 66345, 8175, 723, 41, 1, 54010305, 27062325, 6765975, 1127655, 140865, 13677, 983, 47, 1

Offset: 1

Johannes W. Meijer, Nov 10 2009

The coefficients in the upper right triangle of the ED3 array (m>n) were found with the a(n,m) formula while the coefficients in the lower left triangle of the ED3 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 ED4 arrays see A167546, A167560 and A167584.

			The ED3 array begins with:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1
5, 11, 17, 23, 29, 35, 41, 47, 53, 59
23, 83, 183, 323, 503, 723, 983, 1283, 1623, 2003
167, 741, 2043, 4409, 8175, 13677, 21251, 31233, 43959, 59765
1473, 8169, 26529, 66345, 140865, 266793, 464289, 756969, 1171905, 1739625
16413, 106107, 398025, 1127655, 2678325, 5623443, 10768737, 19194495, 32297805, 51834795

Johannes W. Meijer, The four Escher-Droste arrays, jpg image, Mar 08 2013.

A000012, A016969, A167573, A167574 and A167575 equal the first five rows of the array.
A167576, A167577 and A167578 equal the first three columns of the array.
A167579 equals the row sums of the ED3 array read by antidiagonals.
A167580 is a triangle related to the a(n) formulas of the rows of the ED3 array.
A167583 is a triangle related to the GF(z) formulas of the rows of the ED3 array.
Cf. A014481 (the 2^(n-1)*(n-1)!*(2*n-1) factor).
Cf. A167546 (ED1 array), A167560 (ED2 array), A167584 (ED4 array).

a(n,m) = ((2*m-1)!!/ (2*m-2*n-1)!!)*int(sinh(y*(2*n-1))/(cosh(y))^(2*m),y=0..infinity) for m>n.
The (n-1)-differences of the n-th array row lead to the recurrence relation
sum((-1)^k*binomial(n-1,k)*a(n,m-k),k=0..n-1) = 2^(n-1)*(n-1)!*(2*n-1).

A167584 The ED4 array read by antidiagonals.

Original entry on oeis.org

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

Johannes W. Meijer, Nov 10 2009

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

			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)

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

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.

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 *)

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)

A167571 The row sums of triangle A167569.

Original entry on oeis.org

1, 6, 54, 680, 11160, 226800, 5504688, 155433600, 5006655360, 181193760000, 7279267564800, 321452513141760, 15477343917235200, 806956698772224000, 45293698702796544000, 2723112069367173120000, 174590385277914250444800

Offset: 1

Johannes W. Meijer, Nov 10 2009

A167569 is the lower left triangle of the ED2 array A167560.

Vincenzo Librandi, Table of n, a(n) for n = 1..100

A167560 is the ED2 array.

Equals the row sums of triangle A167569.

[Factorial(n)*(4^n - 1)/(2*n + 1) : n in [1..20]]; // Wesley Ivan Hurt, Sep 05 2025

Table[n! (4^n - 1) / (2 n + 1), {n, 1, 20}] (* Vincenzo Librandi, Jun 23 2013 *)

a(n) = Sum_{m=1..n} 4^(m-1)*(m-1)!*(n+m-1)!/(2*m-1)!.
From Peter Bala, Jun 21 2013: (Start)
a(n) = n!*(4^n - 1)/(2*n + 1).
E.g.f.: 1/(4*sqrt(x))*log((1 - 3*x + 2*x^(3/2))/(1 - 3*x - 2*x^(3/2))). (End)

cp's OEIS Frontend

A167572 The ED3 array read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A167584 The ED4 array read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A167571 The row sums of triangle A167569.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula