A167552 -id:A167552 - cp's OEIS frontend

A167554 The second left hand column of triangle A167552.

-2, -5, -7, -6, 0, 13, 35, 68, 114, 175, 253, 350, 468, 609, 775, 968, 1190, 1443, 1729, 2050, 2408, 2805, 3243, 3724, 4250, 4823, 5445, 6118, 6844, 7625, 8463, 9360, 10318, 11339, 12425, 13578, 14800, 16093, 17459, 18900, 20418

Offset: 2

Johannes W. Meijer, Nov 10 2009

G. C. Greubel, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Equals the second left hand column of A167552.

Other left hand columns are A005408, A167555, A168302 and A168303.

Table[(1/6)*(2*n^3 - 15*n^2 + 19*n - 6), {n,2,100}] (* or *) LinearRecurrence[{4,-6,4,-1}, {-2, -5, -7, -6}, 100] (* G. C. Greubel, Jun 15 2016 *)

a(n) = (2*n^3 - 15*n^2 + 19*n - 6)/3!.
G.f.: (z^2 + 3*z - 2)/(z-1)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) - 3*a(n-1) + 3*a(n-2) - a(n-3) = 2*1.

A167555 The third left hand column of triangle A167552.

Original entry on oeis.org

2, 14, 63, 209, 559, 1281, 2618, 4902, 8568, 14168, 22385, 34047, 50141, 71827, 100452, 137564, 184926, 244530, 318611, 409661, 520443, 654005, 813694, 1003170, 1226420, 1487772, 1791909, 2143883, 2549129, 3013479, 3543176

Offset: 3

Johannes W. Meijer, Nov 10 2009

G. C. Greubel, Table of n, a(n) for n = 3..1002
Index entries for linear recurrences with constant coefficients, signature (6, -15, 20, -15, 6, -1).

Equals the third left hand column of triangle A167552.

Other left hand columns are A005408, A167554, A168302 and A168303.

LinearRecurrence[{6,-15,20,-15,6,-1},{2,14,63,209,559,1281},40] (* or *) Table[1/120*(66*n+95*n^2+40*n^3+25*n^4+14*n^5),{n,40}] (* Harvey P. Dale, Mar 26 2012 *)

a(n) = (14*n^5 - 115*n^4 + 400*n^3 - 665*n^2 + 486*n - 120)/5!.
G.f.: (z^3 + 9*z^2 + 2*z + 2)/(z-1)^6.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
a(n) - 5*a(n-1) + 10*a(n-2) - 10*a(n-3) + 5*a(n-4) - a(n-5) = 2*7.

A168302 The fourth left hand column of triangle A167552.

Original entry on oeis.org

-8, -66, -264, -689, -1255, -1360, 684, 8502, 28842, 73150, 159588, 315549, 580723, 1010768, 1681640, 2694636, 4182204, 6314574, 9307264, 13429515, 19013709, 26465824, 36276980, 49036130, 65443950, 86327982, 112659084

Offset: 4

Johannes W. Meijer, Nov 23 2009

G. C. Greubel, Table of n, a(n) for n = 4..1000
Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Equals the fourth left hand column of triangle A167552.

Other left hand columns are A005408, A167554, A167555 and A168303.

LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{-8, -66, -264, -689, -1255, -1360, 684, 8502},50] (* G. C. Greubel, Jul 17 2016 *)

a(n) = (54*n^7 - 1057*n^6 + 7245*n^5 - 24535*n^4 + 45801*n^3 - 47488*n^2 + 25020*n - 5040)/7!.
G.f.: (z^4 + 23*z^3 + 40*z^2 - 2*z - 8)/(z-1)^8.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8).
a(n) - 7*a(n-1) + 21*a(n-2) - 35*a(n-3) + 35*a(n-4) - 21*a(n-5) + 7*a(n-6) - a(n-7) = 2*27.

A168303 The fifth left hand column of triangle A167552.

Original entry on oeis.org

24, 308, 2236, 11640, 47753, 163419, 485121, 1284987, 3101175, 6927921, 14502059, 28718989, 54217878, 98183330, 171418854, 289756194, 475873962, 761609034, 1190854830, 1823151902, 2738088199, 4040638965, 5867589455, 8395197525, 11848267665

Offset: 5

Johannes W. Meijer, Nov 23 2009

G. C. Greubel, Table of n, a(n) for n = 5..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Equals the fifth left hand column of triangle A167552.

Other left hand columns are A005408, A167554, A167555 and A168302.

[(642*n^9-13833*n^8+132840*n^7-726642*n^6+ 2439738*n^5-5133177*n^4+6699660*n^3-5194188*n^2+ 2157840*n- 362880)/362880: n in [5..40]]; // Vincenzo Librandi, Jul 18 2016

LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{24, 308, 2236, 11640, 47753, 163419, 485121, 1284987, 3101175, 6927921},50] (* G. C. Greubel, Jul 17 2016 *)

a(n) = (642*n^9 - 13833*n^8 + 132840*n^7 - 726642*n^6 + 2439738*n^5 - 5133177*n^4 + 6699660*n^3 - 5194188*n^2 + 2157840*n - 362880)/9!
G.f.: (z^5 + 53*z^4 + 260*z^3 + 236*z^2 + 68*z + 24)/(1-z)^10.
a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10).
a(n) - 9*a(n-1) + 36*a(n-2) - 84*a(n-3) + 126*a(n-4) - 126*a(n-5) + 84*a(n-6) - 36*a(n-7) + 9*a(n-8) - a(n-9) = 2*321.

A167553 The second right hand column of triangle A167552.

Original entry on oeis.org

3, -5, 14, -66, 308, -2132, 14064, -126480, 1081632, -11925792, 125458560, -1636387200, 20447873280, -307814964480

Offset: 2

Johannes W. Meijer, Nov 10 2009

sign

Equals the second right hand column of triangle A167552.

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.

A003148 a(n+1) = a(n) + 2n(2n+1)a(n-1), with a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 7, 27, 321, 2265, 37575, 390915, 8281665, 114610545, 2946939975, 51083368875, 1542234996225, 32192256321225, 1114841223671175, 27254953356505875, 1064057291370698625, 29845288035840902625, 1296073464766972266375, 41049997128507054562875

Offset: 0

N. J. A. Sloane

Numerators of sequence of fractions with e.g.f. 1/((1-x)*(1+x)^(1/2)). The denominators are successive powers of 2.
a(n) is the coefficient of x^n in arctan(sqrt(2*x/(1-x)))/sqrt(2*x*(1-x)) multiplied by (2*n+1)!!.
This sequence is the linking pin between the a(n) formulas of the ED1, ED2, ED3 and ED4 array rows, see A167552, A167565, A167580 and A167591. - Johannes W. Meijer, Nov 23 2009

			arctan(sqrt(2*x/(1-x)))/sqrt(2*x*(1-x)) = 1 + 1/3*x + 7/15*x^2 + 9/35*x^3 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
P. S. Bruckman, An interesting sequence of numbers derived from various generating functions, Fib. Quart., 10 (1972), 169-181.
R. J. Mathar, Numerical Representation of the Incomplete Gamma Function of Complex Argument, arXiv:math/0306184 [math.NA], 2003-2004; cf. Eq. 22.

Cf. A002943, A046161, A049606, A077568, A084543, A091520, A123746.

Cf. A167552, A167565, A167580, A167591.

a003148 n = a003148_list !! n
a003148_list = 1 : 1 : zipWith (+) (tail a003148_list)
                          (zipWith (*) (tail a002943_list) a003148_list)
-- Reinhard Zumkeller, Nov 22 2011

[n le 2 select 1 else Self(n-1) + 2*(n-2)*(2*n-3)*Self(n-2): n in [1..30]]; // G. C. Greubel, Nov 04 2022

# double factorial of odd "l" df := proc(l) local n; n := iquo(l,2); RETURN( factorial(l)/2^n/factorial(n)); end: x := 1; for n from 1 to 15 do if n mod 2 = 0 then x := 2*n*x+df(2*n-1); else x := 2*n*x-df(2*n-1); fi; print(x); od; quit

a[n_] := a[n] = (-1)^n*(2n - 1)!! + 2n*a[n - 1]; a[0] = 1; Table[ a[n], {n, 0, 14}] (* Jean-François Alcover, Dec 01 2011, after R. J. Mathar *)
a[ n_] := If[ n < 0, 0, (2 n + 1)!! Hypergeometric2F1[ -n, 1/2, 3/2, 2]]; (* Michael Somos, Apr 20 2018 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ 1 / ((1 - 2 x) Sqrt[1 + 2 x]), {x, 0, n}]]; (* Michael Somos, Apr 20 2018 *)
RecurrenceTable[{a[0]==a[1]==1,a[n+1]==a[n]+2n(2n+1)a[n-1]},a,{n,20}] (* Harvey P. Dale, Jul 27 2019 *)

Vec(serlaplace(1/(sqrt(1+2*x + O(x^20))*(1-2*x)))) \\ Andrew Howroyd, Feb 05 2018

@CachedFunction
def a(n): return 1 if (n<2) else a(n-1) + 2*(n-1)*(2*n-1)*a(n-2) # a = A003148
[a(n) for n in range(31)] # G. C. Greubel, Nov 04 2022

a(n) = (-1)^n*(2n-1)!! + 2*n*a(n-1) with (2n-1)!! = 1*3*5*..*(2n-1) the double factorial. - R. J. Mathar, Jun 12 2003
a(n) = ((2*n+1)!!/4) * Integral_{-Pi..Pi} cos(x)^n * cos(x/2) dx. - R. J. Mathar, Jun 30 2003
a(n) = (2n+1)!! 2F1(-n, 1/2;3/2;2). - R. J. Mathar, Jun 30 2003
In terms of the (terminating) Gauss hypergeometric function/series, 2F1(., .; .; 2), a(n) is a special case of the family of integer sequences defined by a(m, n) = ((2*n+2*m+1)!!/(2*m+1)) * 2F1(-n, m+1/2; m+3/2; 2), for m >= 0, n >= 0. An integral form can be seen as a(m, n) = ((2*n+2*m+1)!!/4) * Integral_{-Pi..Pi} (sin(x/2))^(2*m) * (cos(x))^n * cos(x/2) dx. A recurrence property is 4*(n+1)*a(m, n) = (2*m-1)*a(m-1, n+1) + (-1)^n*(2*n+2*m+1)!!. Sequences that have these properties are a(0, n) = this sequence, a(1, n) = A077568, a(2, n) = A084543. - R. J. Mathar, Jun 30 2003
E.g.f.: 1/(sqrt(1+2*x)*(1-2*x)). - Vladeta Jovovic, Oct 12 2003
a(n) = (2^n)*n!*A123746(n)/A046161(n) = (2^n)*n!*Sum_{k=0..n} binomial(2*k,k)*(-1/4)^k. From the e.g.f. - Wolfdieter Lang, Oct 06 2008
a(n) = A049606(n)*A123746(n). - Johannes W. Meijer, Nov 23 2009
a(n) = A091520(n) * n! / 2^n. - Michael Somos, Mar 17 2011

a(16)-a(20) from Andrew Howroyd, Feb 05 2018

A098557 Expansion of e.g.f. (1/2)(1+x)log((1+x)/(1-x)).

Original entry on oeis.org

0, 1, 2, 2, 8, 24, 144, 720, 5760, 40320, 403200, 3628800, 43545600, 479001600, 6706022400, 87178291200, 1394852659200, 20922789888000, 376610217984000, 6402373705728000, 128047474114560000, 2432902008176640000, 53523844179886080000, 1124000727777607680000

Offset: 0

Paul Barry, Sep 14 2004

G. C. Greubel, Table of n, a(n) for n = 0..450

From Johannes W. Meijer, Nov 12 2009: (Start)
Cf. A109613 (odd numbers repeated).
Equals the first left hand column of A167552.
Equals the first right hand column of A167556.
A098557(n)*A064455(n) equals the second right hand column of A167556(n).
(End)
Cf. A052558, A073742.

[0,1] cat [Factorial(n-1) + Factorial(n-2)*(1+(-1)^n)/2: n in [2..30]]; // G. C. Greubel, Jan 17 2018

Join[{0,1}, Table[(n-1)! + (n-2)!*(1+(-1)^n)/2, {n,2,30}]] (* or *) With[{nmax = 50}, CoefficientList[Series[(1/2)*(1 + x)*Log[(1 + x)/(1 - x)], {x,0,nmax}], x]*Range[0,nmax]!] (* G. C. Greubel, Jan 17 2018 *)

for(n=0, 30, print1(if(n==0,0, if(n==1, 1, (n-1)! + (n-2)!*(1 + (-1)^n)/2)), ", ")) \\ G. C. Greubel, Jan 17 2018

a(n+1) = n! + (n-1)! * (1-(-1)^n)/2.
a(n+2) = 2*A052558(n).
conjecture: -a(n) +a(n-1) +(n-1)*(n-3)*a(n-2)=0. - R. J. Mathar, Nov 14 2011
G.f.: 1-G(0), where G(k)= 1 + x*(2*k-1)/(1 - x*(2*k+2)/(x*(2*k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 11 2013
Sum_{n>=1} 1/a(n) = sinh(1) + 1 = A073742 + 1. - Amiram Eldar, Jan 22 2023

cp's OEIS Frontend

A167554 The second left hand column of triangle A167552.

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A167555 The third left hand column of triangle A167552.

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A168302 The fourth left hand column of triangle A167552.

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A168303 The fifth left hand column of triangle A167552.

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A167553 The second right hand column of triangle A167552.

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A167546 The ED1 array read by antidiagonals.

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A003148 a(n+1) = a(n) + 2n*(2n+1)*a(n-1), with a(0) = a(1) = 1.

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A098557 Expansion of e.g.f. (1/2)*(1+x)*log((1+x)/(1-x)).

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A003148 a(n+1) = a(n) + 2n(2n+1)a(n-1), with a(0) = a(1) = 1.

A098557 Expansion of e.g.f. (1/2)(1+x)log((1+x)/(1-x)).