A167560 The ED2 array read by ascending antidiagonals

1, 2, 1, 6, 4, 1, 24, 16, 6, 1, 120, 80, 32, 8, 1, 720, 480, 192, 54, 10, 1, 5040, 3360, 1344, 384, 82, 12, 1, 40320, 26880, 10752, 3072, 680, 116, 14, 1, 362880, 241920, 96768, 27648, 6144, 1104, 156, 16, 1

Offset

1,2

Comments

The coefficients in the upper right triangle of the ED2 array (m>n) were found with the a(n,m) formula while the coefficients in the lower left triangle of the ED2 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).

The ED2 array is related to the EG1 matrix, see A162005, because sum(EG1(2*m-1,n) * z^(2*m-1), m=1..infinity) = ((2*n-1)!/(4^(n-1)*(n-1)!^2))*int(sinh(y*(2*z))/cosh(y)^(2*n), y=0..infinity).

For the ED1, ED3 and ED4 arrays see A167546, A167572 and A167584.

Links

Table of n, a(n) for n=1..45.

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

Formula

a(n,m) = ((m-1)!/((m-n-1)!))*int(sinh(y*(2*n))/(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-1,m-k),k=0..n-1) = n!

which in its turn leads to, see A167569,

a(n,m) = 4^(m-1)*(m-1)!*(n+m-1)!/(2*m-1)! if m<=n.

Example

The ED2 array begins with:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1
2, 4, 6, 8, 10, 12, 14, 16, 18, 20
6, 16, 32, 54, 82, 116, 156, 202, 254, 312
24, 80, 192, 384, 680, 1104, 1680, 2432, 3384, 4560
120, 480, 1344, 3072, 6144, 11160, 18840, 30024, 45672, 66864
720, 3360, 10752, 27648, 61440, 122880, 226800, 392832, 646128, 1018080

Maple

nmax:=10; mmax:=10; for n from 1 to nmax do for m from 1 to n do a(n, m) := 4^(m-1)*(m-1)!*(n+m-1)!/(2*m-1)! od; for m from n+1 to mmax do a(n, m):= n! + sum((-1)^(k-1)*binomial(n-1, k)*a(n, m-k), k=1..n-1) od; od: for n from 1 to nmax do for m from 1 to n do d(n, m):=a(n-m+1, m) od: od: T:=1: for n from 1 to nmax do for m from 1 to n do a(T):= d(n, m): T:=T+1: od: od: seq(a(n), n=1..T-1);

Mathematica

nmax = 10; mmax = 10; For[n = 1, n <= nmax, n++, For[m = 1, m <= n, m++, a[n, m] = 4^(m - 1)*(m - 1)!*((n + m - 1)!/(2*m - 1)!)]; For[m = n + 1, m <= mmax, m++, a[n, m] = n! + Sum[(-1)^(k - 1)*Binomial[n - 1, k]*a[n, m - k], {k, 1, n - 1}]]; ]; For[n = 1, n <= nmax, n++, For[m = 1, m <= n, m++, d[n, m] = a[n - m + 1, m]]; ]; t = 1; For[n = 1, n <= nmax, n++, For[m = 1, m <= n, m++, a[t] = d[n, m]; t = t + 1]]; Table[a[n], {n, 1, t - 1}] (* Jean-François Alcover, Dec 20 2011, translated from Maple *)

Crossrefs

A000012, A005843 (n>=1), 2*A104249 (n>=1), A167561, A167562 and A167563 equal the first sixth rows of the array.

A000142 equals the first column of the array.

A047053 equals the a(n, n) diagonal of the array.

2*A034177 equals the a(n+1, n) diagonal of the array.

A167570 equals the a(n+2, n) diagonal of the array,

A167564 equals the row sums of the ED2 array read by antidiagonals.

A167565 is a triangle related to the a(n) formulas of the rows of the ED2 array.

A167568 is a triangle related to the GF(z) formulas of the rows of the ED2 array.

A167569 is the lower left triangle of the ED2 array.

Cf. A162005 (EG1 triangle).

Cf. A167546 (ED1 array), A167572 (ED3 array), A167584 (ED4 array).

Sequence in context: A080245 A080247 A078937 * A132159 A112356 A135885

Adjacent sequences: A167557 A167558 A167559 * A167561 A167562 A167563

Keyword

easy,nonn,tabl

Author

Johannes W. Meijer, Nov 10 2009

Status

approved