A167565 -id:A167565 - cp's OEIS frontend

A167566 The third left hand column of triangle A167565.

2, 16, 67, 202, 497, 1064, 2058, 3684, 6204, 9944, 15301, 22750, 32851, 46256, 63716, 86088, 114342, 149568, 192983, 245938, 309925, 386584, 477710, 585260, 711360, 858312, 1028601, 1224902, 1450087, 1707232, 1999624, 2330768, 2704394

Offset: 3

Johannes W. Meijer, Nov 10 2009

G. C. Greubel, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Equals the third left hand column of triangle A167565.

Other left hand columns are A000027, A000292, A167567 and A168304.

Table[(7*n^5 - 30*n^4 + 45*n^3 - 30*n^2 + 8*n)/5!, {n,3,100}] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1}, {2, 16, 67, 202, 497, 1064}, 100] (* G. C. Greubel, Jun 16 2016 *)

Vec((1*z^2 + 4*z + 2)/(1-z)^6 + O(z^50)) \\ Michel Marcus, Jul 05 2017

a(n) = n*(7*n^4 - 30*n^3 + 45*n^2 - 30*n + 8)/120 \\ Charles R Greathouse IV, Jul 14 2017

From Johannes W. Meijer, Nov 23 2009: (Start)
a(n) = (7*n^5 - 30*n^4 + 45*n^3 - 30*n^2 + 8*n)/5!.
G.f.: (1*z^2 + 4*z + 2)/(1-z)^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) = 7. (End)
a(n) = A024166(n-2) + A000389(n+2). - J. M. Bergot, Jul 04 2017

A167567 The fourth left hand column of triangle A167565.

Original entry on oeis.org

0, 14, 124, 601, 2120, 6096, 15168, 33858, 69432, 132990, 240812, 415987, 690352, 1106768, 1721760, 2608548, 3860496, 5595006, 7957884, 11128205, 15323704, 20806720, 27890720, 36947430, 48414600, 62804430, 80712684

Offset: 4

Johannes W. Meijer, Nov 10 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 A167565.

Other left hand columns are A000027, A000292, A167566 and A168304.

LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {0, 14, 124, 601,
  2120, 6096, 15168, 33858}, 100] (* G. C. Greubel, Jun 16 2016 *)

a(n) = (27*n^7 - 287*n^6 + 1113*n^5 - 1925*n^4 + 1428*n^3 - 308*n^2 - 48*n)/7!.
G.f.: (1*z^3 + 12*z^2 + 14*z + 0)/(1-z)^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) = 27.

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

A168304 The fifth left hand column of triangle A167565.

Original entry on oeis.org

24, 368, 2736, 13712, 53121, 171258, 480711, 1210572, 2793219, 5996562, 12117677, 23257104, 42696758, 75408396, 128723898, 213203256, 343741122, 540958044, 832928118, 1257300704, 1863880095, 2717733590, 3902905305, 5526820260, 7725470805

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 A167565.

Other left hand columns are A000027, A000292, A167566 and A167567.

[(321*n^9-4500*n^8+25506*n^7-75096*n^6+121905*n^5- 104580*n^4+2736*n^2+37164*n^3-3456*n)/362880: n in [5..40]]; // Vincenzo Librandi, Jul 18 2016

LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{24, 368, 2736, 13712, 53121, 171258, 480711, 1210572, 2793219, 5996562},50] (* G. C. Greubel, Jul 17 2016 *)

a(n) = (321*n^9 - 4500*n^8 + 25506*n^7 - 75096*n^6 + 121905*n^5 - 104580*n^4 + 2736*n^2 + 37164*n^3 - 3456*n)/9!.
G.f.: (z^4 + 32*z^3 + 136*z^2 + 128*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) = 321.

A005359 a(n) = n! if n is even, otherwise 0 (from Taylor series for cos x).

Original entry on oeis.org

1, 0, 2, 0, 24, 0, 720, 0, 40320, 0, 3628800, 0, 479001600, 0, 87178291200, 0, 20922789888000, 0, 6402373705728000, 0, 2432902008176640000, 0, 1124000727777607680000, 0, 620448401733239439360000, 0

Offset: 0

N. J. A. Sloane, Russ Cox

nonn

Normally sequences like this are not included, since with the alternating 0's deleted it is already in the database.
Stirling transform of a(n)=[0,2,0,24,0,720,...] is A052841(n)=[0,2,6,38,270,...]. - Michael Somos, Mar 04 2004
Stirling transform of a(n-1)=[1,0,2,0,24,0,...] is A000670(n-1)=[1,1,3,13,75,...]. - Michael Somos, Mar 04 2004
Stirling transform of a(n-1)=[0,0,2,0,24,0,...] is A052875(n-1)=[0,0,2,12,74,...]. - Michael Somos, Mar 04 2004
Stirling transform of (-1)^n*A052811(n)=[0,2,-6,46,-340,...] is a(n)=[0,2,0,24,0,...]. - Michael Somos, Mar 04 2004
Also n-th derivative of arctanh(x) at x=0. - Michel Lagneau, Aug 13 2012
Binomial convolution square of A177145 (with offset 0) because each permutation in S_{2n} uniquely determines a bi-partition of its elements into even and odd cycles and these are both enumerated by A177145. - Michael Somos, Mar 19 2019

Douglas Hofstadter, "Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought".

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Michael Somos, Number of permutations with all cycles of even length, answer to Mathematics Stack Exchange question 3152701, Mar 18 2019.
Index entries for sequences related to factorial numbers

From Johannes W. Meijer, Nov 12 2009: (Start)
Equals the first right hand column of A167565.
Equals the first left hand column of A167568.
(End)
Cf. A177145.
Bisection (even part) gives A010050.

BB:={E=Prod(Z,Z),S=Union(Epsilon,Prod(S,E))}: ZL:=[S,BB, labeled]: > seq(count(ZL,size=n),n=0..25); # Zerinvary Lajos, Apr 22 2007
a:=n->n!+(-1)^n*n!: seq(a(n)/2, n=0..25); # Zerinvary Lajos, Mar 25 2008

Riffle[Range[0,30,2]!,0] (* Harvey P. Dale, Nov 16 2011 *)
a[ n_] := If[n >= 0 && EvenQ[n], n!, 0]; (* Michael Somos, Mar 19 2019 *)

{a(n) = if(n<0, 0, if(n%2, 0, n!))}; /* Michael Somos, Mar 04 2004 */

E.g.f. 1/(1-x^2) = d/dx log(sqrt((1+x)/(1-x))). a(2n)=(2n)!, a(2n+1)=0. - Michael Somos, Mar 04 2004
a(n) = Product_{k=0..n/2-1} binomial(n-2k,2)*2^(n/2) for even n. - Geoffrey Critzer, Jun 05 2016
From Ilya Gutkovskiy, Jun 05 2016: (Start)
D-finite with recurrence a(n) = n*(n - 1)*a(n-2), a(0)=1, a(1)=0.
a(n) = n!*((-1)^n + 1)/2. (End)

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.

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

cp's OEIS Frontend

A167566 The third left hand column of triangle A167565.

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A167567 The fourth left hand column of triangle A167565.

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A168304 The fifth left hand column of triangle A167565.

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A005359 a(n) = n! if n is even, otherwise 0 (from Taylor series for cos x).

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A167560 The ED2 array read by ascending 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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A003148 a(n+1) = a(n) + 2n(2n+1)a(n-1), with a(0) = a(1) = 1.