A167572 -id:A167572 - cp's OEIS frontend

A167580 A triangle related to the a(n) formulas of the rows of the ED3 array A167572.

1, 6, -1, 20, 0, 3, 56, 28, 98, -15, 144, 192, 1080, -48, 105, 352, 880, 7568, 2024, 6534, -945, 832, 3328, 40976, 31616, 132444, -8112, 10395, 1920, 11200, 187488, 274480, 1593960, 286900, 972162, -135135, 4352, 34816, 761600, 1784320, 13962848

Offset: 1

Johannes W. Meijer, Nov 10 2009

The a(n) formulas given below correspond to the first ten rows of the ED3 array A167572.

The recurrence relations of the a(n) formulas for the left hand triangle columns, see the cross-references below, lead to the sequences A013609, A003148, A081277 and A079628.

			Row 1: a(n) = 1.
Row 2: a(n) = 6*n - 1.
Row 3: a(n) = 20*n^2 + 0*n + 3.
Row 4: a(n) = 56*n^3 + 28*n^2 + 98*n - 15.
Row 5: a(n) = 144*n^4 + 192*n^3 + 1080*n^2 - 48*n + 105.
Row 6: a(n) = 352*n^5 + 880*n^4 + 7568*n^3 + 2024*n^2 + 6534*n - 945.
Row 7: a(n) = 832*n^6 + 3328*n^5 + 40976*n^4 + 31616*n^3 + 132444*n^2 - 8112*n + 10395.
Row 8: a(n) = 1920*n^7 + 11200*n^6 + 187488*n^5 + 274480*n^4 + 1593960*n^3 + 286900*n^2 + 972162*n - 135135.
Row 9: a(n) = 4352*n^8 + 34816*n^7 + 761600*n^6 + 1784320*n^5 + 13962848*n^4 + 7874944*n^3 + 29641200*n^2 - 2080800*n + 2027025.
Row 10: a(n) = 9728*n^9 + 102144*n^8 + 2830848*n^7 + 9645312*n^6 + 98382912*n^5 + 106720416*n^4 + 522283552*n^3 + 69265488*n^2 + 255468870*n - 34459425.

A167572 is the ED3 array.
A000012, A016969, A167573, A167574 and A167575 equal the first five rows of the ED3 array.
A014480, A167581, A167582, A168305 and A168306 equal the first five left hand triangle columns.
A001147 equals the first right hand triangle column.
A167576 equals the row sums.
Cf. A013609, A003148, A079628 and A081277.

Comment and links added by Johannes W. Meijer, Nov 23 2009

A167576 The first column of the ED3 array A167572.

Original entry on oeis.org

1, 5, 23, 167, 1473, 16413, 211479, 3192975, 54010305, 1030249845, 21566327895, 497334999735, 12405876372225, 335591130336525, 9716331072597975, 301633179343890975, 9941514351641143425, 348336799875365041125

Offset: 1

Johannes W. Meijer, Nov 10 2009

Basically a(n) measures the difference between the Euler factorial n! and the Luschny factorial L(n) at half-integer values. For the Luschny factorial see the link. The formula given in the Maple section is a variant of a formula given by Cyril Damamme in A135457. - Peter Luschny, Jul 18 2015

			G.f. = x + 5*x^2 + 23*x^3 + 167*x^4 + 1473*x^5 + 16413*x^6 + ...

G. C. Greubel, Table of n, a(n) for n = 1..400
Peter Luschny, Is the Gamma function misdefined? Hadamard versus Euler - Who found the better Gamma function?

Equals the first column of the ED3 array A167572.
Equals the first right hand column of A167583.
Other columns are A167577 and A167578.
Cf. A097801 (the 2*(-1)^n*(2*n-5)!! factor).
Cf. A007509 and A025547 (the sum((-1)^(k+n)/(2*k+1), k=0..n-1) factor).
Cf. A024199 and A135457.

L := x -> (1+x*(Psi(1-x/2)-Psi(1/2-x/2)))/(-x)!:
a := x -> (L(x-1/2)-(x-1/2)!)*2^(x-1)*sqrt(Pi):
seq(simplify(a(n)),n=1..18); # Peter Luschny, Jul 18 2015
a := proc(n) option remember: if n=1 then 1 else (2*n-1)*a(n-1)+2*(-1)^n*doublefactorial(2*n-5) fi: end: seq(a(n),n=1..18); # Johannes W. Meijer, Jul 20 2015

a[ n_] := If[ n < 1, 0, (2 n - 3)!! ((-1)^n - I (4 n - 2) Sum[ I^k / k, {k, 1, 2 n - 1, 2}])]; (* Michael Somos, Jul 20 2015 *)
a[ n_] := If[ n < 1, 0, (2 n - 3)!! ((-1)^n + (4 n - 2) Sum[ KroneckerSymbol[ -4, k]/ k, {k, 2 n - 1}])]; (* Michael Somos, Jan 31 2019 *)

{a(n) = if( n<1, 0, prod(k=1, n-1, 2*k - 1) * ((-1)^n - (4*n - 2) * sum(k=1, n, (-1)^k / (2*k - 1))))}; /* Michael Somos, Jul 20 2015 */

a(n) = (-1)^n*(2*n-3)!!*(1 + (4*n-2)*Sum_{k=0..n-1} (-1)^(k+n)/(2*k+1)).
a(n) = (2*n-1)*a(n-1) + 2*(-1)^n*(2*n-5)!! with a(1) = 1.
a(n) = 4*a(n-1) + (4*n^2 - 16*n + 15)*a(n-2) with a(1) = 1 and a(2) = 5 [Superseeker].
0 = a(n)*a(n+1)*(-440*a(n+2) - 220*a(n+3) + 55*a(n+4)) + a(n)*a(n+2)*(536*a(n+2) - 118*a(n+3) - 4*a(n+4)) + a(n)*a(n+3)*(-4*a(n+3) + a(n+4)) + a(n+1)^2*(-220*a(n+2) - 32*a(n+3) + 8*a(n+4)) + a(n+1)*a(n+2)*(+71*a(n+2) + 4*a(n+3) - 2*a(n+4)) + a(n+2)^2*(-4*a(n+2) + a(n+3)) if n>0. - Michael Somos, Jul 19 2015
a(n) = (-1 + (n-1/2)*LerchPhi(-1,1,n+1/2) + (-n+1/2)*LerchPhi(-1,1,-n+1/2))/(1-2*n)!!. - Johannes W. Meijer, Jul 20 2015
a(n) = A024199(n) + A135457(n). - Cyril Damamme, Jul 22 2015
a(n) = ((-1)^n/(2*n - 1) + Pi/2 - (-1)^n LerchPhi(-1, 1, n + 1/2)) (2*n - 1)!!. - Michael Somos, Jan 31 2019

A167577 The second column of the ED3 array A167572.

Original entry on oeis.org

1, 11, 83, 741, 8169, 106107, 1592235, 27062325, 514246545, 10798366635, 248374594755, 6209158112325, 167651197407225, 4861802228946075, 150717766502187675, 4973638859450709525, 174078640829054894625

Offset: 1

Johannes W. Meijer, Nov 10 2009

G. C. Greubel, Table of n, a(n) for n = 1..150

Equals the second column of the ED3 array A167572.
Other columns are A167576 and A167578.
Cf. A007509 and A025547 (the sum((-1)^(k+n)/(2*k+1), k=0..n-1) factor).

Table[(1/2)*(-1)^n*(2*n - 5)!!*((4*n^2 - 6*n - 2) + (16*n^3 - 24*n^2 - 4*n + 6)*Sum[(-1)^(k + n)/(2*k + 1), {k, 0, n - 1}]), {n, 1,50}] (* G. C. Greubel, Jun 16 2016 *)

a(n) = (1/2)*(-1)^n*(2*n-5)!!*((4*n^2-6*n-2)+(16*n^3-24*n^2-4*n+6)*sum((-1)^(k+n)/ (2*k+1), k=0..n-1)).

A167578 The third column of the ED3 array A167572.

Original entry on oeis.org

1, 17, 183, 2043, 26529, 398025, 6765975, 128556675, 2699661825, 62092533825, 1552309291575, 41912411683275, 1215458905032225, 37679245697871225, 1243414695550433175, 43519523831289457875, 1610222144582102522625

Offset: 1

Johannes W. Meijer, Nov 10 2009

G. C. Greubel, Table of n, a(n) for n = 1..150

Equals the third column of the ED3 array A167572.
Other columns are A167576 and A167577.
Cf. A007509 and A025547 (the sum((-1)^(k+n)/(2*k+1), k=0..n-1) factor).

Table[(1/4)*(-1)^(n)*(2*n - 7)!!*((8*n^4 - 20*n^3 - 22*n^2 + 55*n + 12) + (32*n^5 - 80*n^4 - 80*n^3 + 200*n^2 + 18*n - 45)*(Sum[(-1)^(k + n)/(2*k + 1), {k, 0, n - 1}])), {n, 1, 50}] (* G. C. Greubel, Jun 16 2016 *)

a(n) = (1/4)*(-1)^(n)*(2*n-7)!!*((8*n^4-20*n^3-22*n^2+55*n+12)+(32*n^5-80*n^4-80*n^3+200*n^2+18*n-45)*(sum((-1)^(k+n)/(2*k+1),k=0..n-1))).

A167574 The fourth row of the ED3 array A167572.

Original entry on oeis.org

167, 741, 2043, 4409, 8175, 13677, 21251, 31233, 43959, 59765, 78987, 101961, 129023, 160509, 196755, 238097, 284871, 337413, 396059, 461145, 533007, 611981, 698403, 792609, 894935, 1005717, 1125291, 1253993, 1392159, 1540125

Offset: 1

Johannes W. Meijer, Nov 10 2009

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Equals the fourth row of the ED3 array A167572.

LinearRecurrence[{4,-6,4,-1},{167, 741, 2043, 4409},100] (* G. C. Greubel, Jun 16 2016 *)

a(n) = 56*n^3 + 28*n^2 + 98*n - 15.
G.f.: (15*z^3 + 81*z^2 + 73*z + 167)/(1-z)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - G. C. Greubel, Jun 16 2016

A167575 The fifth row of the ED3 array A167572.

Original entry on oeis.org

1473, 8169, 26529, 66345, 140865, 266793, 464289, 756969, 1171905, 1739625, 2494113, 3472809, 4716609, 6269865, 8180385, 10499433, 13281729, 16585449, 20472225, 25007145, 30258753, 36299049, 43203489, 51050985, 59923905

Offset: 1

Johannes W. Meijer, Nov 10 2009

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Equals the fifth row of the ED3 array A167572.

LinearRecurrence[{5, -10, 10, -5, 1}, {1473, 8169, 26529, 66345, 140865}, 100] (* G. C. Greubel, Jun 16 2016 *)

a(n) = 144*n^4 + 192*n^3 + 1080*n^2 - 48*n + 105.
G.f.: (105*z^4 + 660*z^3 + 414*z^2 + 804*z + 1473)/(1-z)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - G. C. Greubel, Jun 16 2016

A167583 A triangle related to the GF(z) formulas of the rows of the ED3 array A167572.

Original entry on oeis.org

1, 1, 5, 3, 14, 23, 15, 81, 73, 167, 105, 660, 414, 804, 1473, 945, 6825, 2850, 7578, 7629, 16413, 10395, 85050, 19425, 99420, 61389, 111882, 211479, 135135, 1237005, 59535, 1642725, 429525, 1461375, 1518525, 3192975, 2027025, 20540520, -2619540

Offset: 1

Johannes W. Meijer, Nov 10 2009

The GF(z) formulas given below correspond to the first ten rows of the ED3 array A167572. The polynomials in their numerators lead to the triangle given above.

			Row 1: GF(z) = 1/(1-z).
Row 2: GF(z) = (z + 5)/(1-z)^2.
Row 3: GF(z) = (3*z^2 + 14*z + 23)/(1-z)^3.
Row 4: GF(z) = (15*z^3 + 81*z^2 + 73*z + 167)/(1-z)^4.
Row 5: GF(z) = (105*z^4 + 660*z^3 + 414*z^2 + 804*z + 1473)/(1-z)^5.
Row 6: GF(z) = (945*z^5 + 6825*z^4 + 2850*z^3 + 7578*z^2 + 7629*z + 16413)/(1-z)^6.
Row 7: GF(z) = (10395*z^6 + 85050*z^5 + 19425*z^4 + 99420*z^3 + 61389*z^2 + 111882*z + 211479)/(1-z)^7.
Row 8: GF(z) = (135135*z^7 + 1237005*z^6 + 59535*z^5 + 1642725*z^4 + 429525*z^3 + 1461375*z^2 + 1518525*z + 3192975)/(1-z)^8.
Row 9: GF(z) = (2027025*z^8 + 20540520*z^7 - 2619540*z^6 + 32228280*z^5 - 2479050*z^4 + 27797400*z^3 + 15813900*z^2 + 28153800*z + 54010305)/(1-z)^9.
Row 10: GF(z) = (34459425*z^9 + 383107725*z^8 - 115135020*z^7 + 722119860*z^6 - 283607730*z^5 + 703347750*z^4 + 89576100*z^3 + 470110500*z^2 + 495868185*z + 1030249845)/(1-z)^10.

A167572 is the ED3 array.
A001147 equals the first left hand column.
A167576 equals the first right hand column.
A014481 equals the row sums.

A167579 The row sums of the ED3 array A167572 read by antidiagonals.

Original entry on oeis.org

1, 6, 35, 268, 2421, 26978, 349063, 5258520, 89121833, 1697446846, 35576664363, 819585397028, 20461925870365

Offset: 1

Johannes W. Meijer, Nov 10 2009

nonn

A167572 is the ED3 array.

A167546 The ED1 array read by antidiagonals.

Original entry on oeis.org

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.

A167560 The ED2 array read by ascending antidiagonals.

Original entry on oeis.org

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

Johannes W. Meijer, Nov 10 2009

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.

			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

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

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).

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);
# alternative
A167560 := proc(n,m)
    option remember ;
    if m > n then
        n!+add( (-1)^(k-1)*binomial(n-1,k)*procname(n,m-k),k=1..n-1) ;
    else
        4^(m-1)*(m-1)!*(n+m-1)!/(2*m-1)! ;
    end if;
end proc:
seq( seq(A167560(d-m,m),m=1..d-1),d=2..12) ; # R. J. Mathar, Jun 28 2024

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 *)

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.

cp's OEIS Frontend

A167580 A triangle related to the a(n) formulas of the rows of the ED3 array A167572.

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A167576 The first column of the ED3 array A167572.

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A167577 The second column of the ED3 array A167572.

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A167578 The third column of the ED3 array A167572.

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A167574 The fourth row of the ED3 array A167572.

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A167575 The fifth row of the ED3 array A167572.

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Mathematica

Formula

A167583 A triangle related to the GF(z) formulas of the rows of the ED3 array A167572.

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A167579 The row sums of the ED3 array A167572 read by antidiagonals.

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A167546 The ED1 array read by antidiagonals.

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Maple