A167547 -id:A167547 - cp's OEIS frontend

A167546 The ED1 array read by antidiagonals.

1, 1, 1, 2, 4, 1, 6, 12, 7, 1, 24, 48, 32, 10, 1, 120, 240, 160, 62, 13, 1, 720, 1440, 960, 384, 102, 16, 1, 5040, 10080, 6720, 2688, 762, 152, 19, 1, 40320, 80640, 53760, 21504, 6144, 1336, 212, 22, 1

Offset: 1

Johannes W. Meijer, Nov 10 2009

The coefficients in the upper right triangle of the ED1 array (m > n) were found with the a(n,m) formula while the coefficients in the lower left triangle of the ED1 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).
Our procedure for finding the coefficients in the lower left triangle can be compared with the procedure that De Smit and Lenstra used to fill in the hole in the middle of 'The Print Gallery' by M. C. Escher, see the links. In this lithograph Escher made use of the so-called Droste effect, hence we propose to call this square array of numbers the ED1 array.
For the ED2, ED3 and ED4 arrays see A167560, A167572 and A167584.

			The ED1 array begins with:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1
1, 4, 7, 10, 13, 16, 19, 22, 25, 28
2, 12, 32, 62, 102, 152, 212, 282, 362, 452
6, 48, 160, 384, 762, 1336, 2148, 3240, 4654, 6432
24, 240, 960, 2688, 6144, 12264, 22200, 37320, 59208, 89664
120, 1440, 6720, 21504, 55296, 122880, 245640, 452880, 783144, 1285536

B. de Smit and H.W. Lenstra, The Mathematical Structure of Escher's Print Gallery, Notices of the AMS, Volume 50, Number 4, pp. 446-457, April 2003.
Johannes W. Meijer, The four Escher-Droste arrays, jpg image, Mar 08 2013.
A. Ryabov, P. Chvosta, Tracer dynamics in a single-file system with absorbing boundary, arXiv preprint arXiv:1402.1949 [cond-mat.stat-mech], 2014.

A000012, A016777, 2*A005891, A167547, A167548 and A167549 equal the first sixth rows of the array.
A000142 equals the first column of the array.
A167550 equals the a(n, n+1) diagonal of the array.
A047053 equals the a(n, n) diagonal of the array.
A167558 equals the a(n+1, n) diagonal of the array.
A167551 equals the row sums of the ED1 array read by antidiagonals.
A167552 is a triangle related to the a(n) formulas of rows of the ED1 array.
A167556 is a triangle related to the GF(z) formulas of the rows of the ED1 array.
A167557 is the lower left triangle of the ED1 array.
Cf. A068424 (the (m-1)!/(m-n-1)! factor), A007680 (the (2*n-1)*(n-1)! factor).
Cf. A167560 (ED2 array), A167572 (ED3 array), A167584 (ED4 array).

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-1+m-1)!/(2*m-2)! od; for m from n+1 to mmax do a(n,m):= (2*n-1)*(n-1)! + 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);

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 - 1 + m - 1)!/(2*m - 2)!)]; For[m = n + 1, m <= mmax, m++, a[n, m] = (2*n - 1)*(n - 1)! + 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 *)

a(n,m) = (2*(m-1)!/(m-n-1)!)*Integral_{y>=0} sinh(y*(2*n-1))/cosh(y)^(2*m-1) 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-1)!
which in its turn leads to, see also A167557,
a(n,m) = 4^(m-1)*(m-1)!*(n+m-2)!/(2*m-2)! for m <= n.

A167552 A triangle related to the a(n) formulas of the rows of the ED1 array A167546.

Original entry on oeis.org

1, 3, -2, 5, -5, 2, 7, -7, 14, -8, 9, -6, 63, -66, 24, 11, 0, 209, -264, 308, -144, 13, 13, 559, -689, 2236, -2132, 720, 15, 35, 1281, -1255, 11640, -14980, 14064, -5760, 17, 68, 2618, -1360, 47753, -68068, 145452, -126480, 40320

Offset: 1

Johannes W. Meijer, Nov 10 2009, Nov 23 2009

The a(n) formulas given below correspond to the first ten rows of the ED1 array A167546.

The recurrence relations of the a(n) formulas for the left hand triangle columns, see the cross-references below, lead to the sequences A003148 and A007318.

			Row 1: a(n) = 1.
Row 2: a(n) = 3*n - 2.
Row 3: a(n) = 5*n^2 - 5*n + 2.
Row 4: a(n) = 7*n^3 - 7*n^2 + 14*n - 8.
Row 5: a(n) = 9*n^4 - 6*n^3 + 63*n^2 - 66*n + 24.
Row 6: a(n) = 11*n^5 + 0*n^4 + 209*n^3 - 264*n^2 + 308*n - 144.
Row 7: a(n) = 13*n^6 +13*n^5 +559*n^4 -689*n^3 +2236*n^2 -2132*n +720.
Row 8: a(n) = 15*n^7 + 35*n^6 + 1281*n^5 - 1255*n^4 + 11640*n^3 - 14980*n^2 + 14064*n - 5760.
Row 9: a(n) = 17*n^8 + 68*n^7 + 2618*n^6 - 1360*n^5 + 47753*n^4 - 68068*n^3 + 145452*n^2 - 126480*n + 40320.
Row 10: a(n) = 19*n^9 + 114*n^8 + 4902*n^7 + 684*n^6 + 163419*n^5 - 224694*n^4 + 1048268*n^3 - 1308264*n^2 + 1081632*n - 403200.

A167546 is the ED1 array.
A000012, A016777, 2*A005891, A167547, A167548 and A167549 are the first sixth ED1 array rows.
A098557 and A167553 equal the first two right hand columns of this triangle.
A005408, A167554 and A167555, A168302 and A168303 equal the first five left hand columns of this triangle.
A000142 equals the row sums.
Cf. A003148 and A007318.

cp's OEIS Frontend

A167546 The ED1 array read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula