A000165 -id:A000165 - cp's OEIS frontend

A021012 Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).

1, 1, -1, 2, -4, 2, 6, -18, 18, -6, 24, -96, 144, -96, 24, 120, -600, 1200, -1200, 600, -120, 720, -4320, 10800, -14400, 10800, -4320, 720, 5040, -35280, 105840, -176400, 176400, -105840, 35280, -5040, 40320, -322560, 1128960, -2257920, 2822400, -2257920, 1128960, -322560, 40320, 362880, -3265920

Offset: 0

N. J. A. Sloane

Triangle T(n,k), read by rows: given by [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA [ -1, -1, -2, -2, -3, -3, -4, -4, -5, -5, ...], where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 14 2005

			Triangle begins:
   1;
   1,  -1;
   2,  -4,   2;
   6, -18,  18,  -6;
  24, -96, 144, -96, 24;
  ...
x^3 = 6*LaguerreL(0,x) - 18*LaguerreL(1,x) + 18*LaguerreL(2,x) - 6*LaguerreL(3,x).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 799.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Index entries for sequences related to Laguerre polynomials

Columns include (essentially) A000142, A001563, A001804, A001805, A001806, A001807.
Cf. A000165 (row sum of absolute values).
Cf. A136572.

[[(-1)^k*Factorial(n)*Binomial(n,k): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Feb 06 2018

row[n_] := Table[ a[n, k], {k, 0, n}] /. SolveAlways[ x^n == Sum[ a[n, k]*LaguerreL[k, x], {k, 0, n}], x] // First; (* or, after Vladeta Jovovic: *) row[n_] := Table[(-1)^k*n!*Binomial[n, k], {k, 0, n}]; Table[ row[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, Oct 05 2012 *)

for(n=0,10, for(k=0,n, print1((-1)^k*n!*binomial(n,k), ", "))) \\ G. C. Greubel, Feb 06 2018

T(n, k) = (-1)^k*n!*binomial(n, k). - Vladeta Jovovic, May 11 2003
Sum_{k>=0} T(n, k)*T(m, k) = (n+m)!. - Philippe Deléham, Feb 14 2005
Unsigned sequence = A136572 * A007318 - Gary W. Adamson, Jan 07 2008
A136572*PS, where PS is a triangle with PS[n,k] = (-1)^k*A007318[n,k]. PS = 1/PS. - Gerald McGarvey, Aug 20 2009

More terms from Vladeta Jovovic, May 11 2003

A109692 Triangle of coefficients in expansion of (1+x)(1+3x)(1+5x)(1+7x)...*(1+(2n-1)x).

Original entry on oeis.org

1, 1, 1, 1, 4, 3, 1, 9, 23, 15, 1, 16, 86, 176, 105, 1, 25, 230, 950, 1689, 945, 1, 36, 505, 3480, 12139, 19524, 10395, 1, 49, 973, 10045, 57379, 177331, 264207, 135135, 1, 64, 1708, 24640, 208054, 1038016, 2924172, 4098240, 2027025

Offset: 0

Philippe Deléham, Aug 08 2005

Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 2, 3, 4, 5, 6, 7, 8, 9, ...] where DELTA is the operator defined in A084938.

T(n,k), 0 <= k <= n, is the number of elements in the Coxeter group B_n with absolute length k. - Jose Bastidas, Jul 14 2023

			Triangle T(n,k) begins:
  1;
  1,  1;
  1,  4,   3;
  1,  9,  23,   15;
  1, 16,  86,  176,   105;
  1, 25, 230,  950,  1689,   945;
  1, 36, 505, 3480, 12139, 19524, 10395;
  ...

Seiichi Manyama, Rows n = 0..139, flattened

Cf. A039758 (signed version). A028338 transposed.
Diagonals: A000012, A000290, A024196, A024197, A024198; A001147, A004041, A028339, A028340, A028341.
Row sums: A000165.
Central terms: A293318.
Cf. A161198 (transposed scaled triangle version).

nmax:=8; mmax:=nmax: for n from 0 to nmax do a(n, n) := doublefactorial(2*n-1) od: for n from 0 to nmax do a(n, 0):=1 od: for n from 2 to nmax do for m from 1 to n-1 do a(n, m) := a(n-1,m) + (2*n-1)*a(n-1,m-1) od; od: seq(seq(a(n, m), m=0..n), n=0..nmax); # Johannes W. Meijer, Jun 08 2009, revised Nov 25 2012

T(n,m) = T(n-1,m) + (2*n-1)*T(n-1,m-1) with T(n,n) = (2*n-1)!! and T(n,0) = 1. - Johannes W. Meijer, Jun 08 2009

A120778 Numerators of partial sums of Catalan numbers scaled by powers of 1/4.

Original entry on oeis.org

1, 5, 11, 93, 193, 793, 1619, 26333, 53381, 215955, 436109, 3518265, 7088533, 28539857, 57414019, 1846943453, 3711565741, 14911085359, 29941580393, 240416274739, 482473579583, 1936010885087, 3883457090629, 62306843256889

Offset: 0

Wolfdieter Lang, Jul 20 2006

For denominators see A120777.
From the expansion of 0 = sqrt(1-1) = 1 - (1/2)*Sum_{k>=0} C(k)/4^k one has r:=lim_{n->infinity} r(n) = 2, with the partial sums r(n) defined below.
The series a(n)/A046161(n+1) is absolutely convergent to 1. - Ralf Steiner, Feb 16 2017
If n >= 1 it appears a(n-1) is equal to the difference between the denominator and the numerator of the ratio (2n)!!/(2n-1)!!. In particular a(n-1) is the difference between the denominator and the numerator of the ratio A001147(2n-1)/A000165(2n). See examples. - Anthony Hernandez, Feb 05 2020
From Peter Bala, Feb 16 2022: (Start)
Sum_{k = 0..n-1} Catalan(k)/4^k = (1/4^n)*(2*n)*binomial(2*n,n) *( 1 - 1/(1*2)*(n-1)/(n+1) - 1/(2*3)*(n-1)*(n-2)/((n+1)*(n+2)) - 1/(3*4)*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) - 1/(4*5)*(n-1)*(n-2)*(n-3)*(n-4)/((n+1)*(n+2)*(n+3)*(n+4)) - ... ). Cf. A082687 and A101028.
This identity allows us to extend the definition of Sum_{k = 0..n} Catalan(k)/4^k to non-integral values of n. (End)

			Rationals r(n): [1, 5/4, 11/8, 93/64, 193/128, 793/512, 1619/1024, 26333/16384, ...].
From _Anthony Hernandez_, Feb 05 2020: (Start)
For n = 4. The 4th even number is 8, and 8!!/(8-1)!! = 128/35, so a(4-1) = a(3) = 128 - 35 = 93.
For n = 7. The 7th even number is 14, and 14!!/(14-1)!! = 2048/429, so a(7-1) = a(6) = 2048 - 429 = 1619. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Wolfdieter Lang, Rationals r(n) and limit 2

Factors of A160481. Cf. A120777 (denominators), A082687, A101028, A141244.

[Numerator(2*(1-Binomial(2*n+2,n+1)/4^(n+1))): n in [0..25]]; // Vincenzo Librandi, Feb 17 2017

a := n -> 2^(2*(n+1) - add(i, i=convert(n+1, base, 2)))* (1-((n+1/2)!)/(sqrt(Pi)*(n+1)!)): seq(simplify(a(n)), n=0..23); # Peter Luschny, Dec 21 2017

f[n_] := f[n] = Numerator[(4/Pi) (n + 1) Integrate[x^n*ArcSin[Sqrt[x]], {x, 0, 1}]]; Array[f, 23, 0] (* Robert G. Wilson v, Jan 03 2011 *)
a[n_] := 2^(2(n+1) - DigitCount[n+1,2,1])(1 - ((n+1/2)!)/(Sqrt[Pi](n+1)!));
Table[a[n], {n, 0, 23}] (* Peter Luschny, Dec 21 2017 *)

a(n) = numerator(r(n)), with the rationals r(n):=Sum_{k = 0..n} C(k)/4^k with C(k) := A000108(k) (Catalan numbers). Rationals r(n) are taken in lowest terms.
r(n) = (4/Pi)*(n+1)*Integral_{x = 0..1} x^n*arcsin(sqrt(x)) dx. - Groux Roland, Jan 03 2011
r(n) = 2*(1 - binomial(2*n+2,n+1)/4^(n+1)). - Groux Roland, Jan 04 2011
a(n) = A141244(2n+2) = A141244(2n+3) (conjectural). - Greg Martin, Aug 16 2014, corrected by M. F. Hasler, Aug 18 2014
From Peter Luschny, Dec 21 2017: (Start)
a(n) = numerator(1 - ((n+1/2)!)/(sqrt(Pi)*(n+1)!)).
a(n) = 2^(2*(n+1) - HammingWeight(n+1))*(1 - ((n+1/2)!)/(sqrt(Pi)*(n+1)!)). (End)

A055786 Numerators of Taylor series expansion of arcsin(x). Also arises from arccos(x), arccsc(x), arcsec(x), arcsinh(x).

Original entry on oeis.org

1, 1, 3, 5, 35, 63, 231, 143, 6435, 12155, 46189, 88179, 676039, 1300075, 5014575, 9694845, 100180065, 116680311, 2268783825, 1472719325, 34461632205, 67282234305, 17534158031, 514589420475, 8061900920775, 5267108601573

Offset: 0

N. J. A. Sloane, Jul 13 2000

Note that the sequence is not monotonic.

			arcsin(x) is usually written as x + x^3/(2*3) + 1*3*x^5/(2*4*5) + 1*3*5*x^7/(2*4*6*7) + ..., which is x + 1/6*x^3 + 3/40*x^5 + 5/112*x^7 + 35/1152*x^9 + 63/2816*x^11 + ... (A055786/A002595) when reduced to lowest terms.
arccos(x) = Pi/2 - (x + (1/6)*x^3 + (3/40)*x^5 + (5/112)*x^7 + (35/1152)*x^9 + (63/2816)*x^11 + ...) (A055786/A002595).
arccsc(x) = 1/x + 1/(6*x^3) + 3/(40*x^5) + 5/(112*x^7) + 35/(1152*x^9) + 63/(2816*x^11) + ... (A055786/A002595).
arcsec(x) = Pi/2 -(1/x + 1/(6*x^3) + 3/(40*x^5) + 5/(112*x^7) + 35/(1152*x^9) + 63/(2816*x^11) + ...) (A055786/A002595).
arcsinh(x) = x - (1/6)*x^3 + (3/40)*x^5 - (5/112)*x^7 + (35/1152)*x^9 - (63/2816)*x^11 + ... (A055786/A002595).
i*Pi/2 - arccosh(x) = i*x + (1/6)*i*x^3 + (3/40)*i*x^5 + (5/112)*i*x^7 + (35/1152)*i*x^9 + (63/2816)*i*x^11 + (231/13312)*i*x^13 + (143/10240)*i*x^15 + (6435/557056)*i*x^17 + ... (A055786/A002595).
0, 1, 0, 1/6, 0, 3/40, 0, 5/112, 0, 35/1152, 0, 63/2816, 0, 231/13312, 0, 143/10240, 0, 6435/557056, 0, 12155/1245184, 0, 46189/5505024, 0, ... = A055786/A002595.
a(4) = 35 = 3*5*7*9 / gcd( 3*5*7*9, (2*4*6*8) * (2*4+1))

Bronstein-Semendjajew, Taschenbuch der Mathematik, 7th German ed. 1965, ch. 4.2.6
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
H. B. Dwight, Tables of Integrals and Other Mathematical Data, Macmillan, NY, 1968, Chap. 3.

T. D. Noe, Table of n, a(n) for n=0..200
Eric Weisstein's World of Mathematics, Inverse Cosecant.
Eric Weisstein's World of Mathematics, Inverse Cosine.
Eric Weisstein's World of Mathematics, Inverse Secant.
Eric Weisstein's World of Mathematics, Inverse Sine.
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Cosecant.
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Cosine.
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Sine.
Herbert S. Wilf, Generatingfunctionology, Academic Press, NY, 1994. See p. 54.

Cf. A002595.
a(n) / A002595(n) = A001147(n) / ( A000165(n) * (2*n+1))
Cf. A162443 where BG1[-3,n] = (-1)*A002595(n-1)/A055786(n-1) for n >= 1. - Johannes W. Meijer, Jul 06 2009

[Numerator( (n+1)*Binomial(2*n+2,n+1)/(2^(2*n+1)*(2*n+1)^2) ): n in [0..25]]; // G. C. Greubel, Jan 25 2020

seq( numer( (n+1)*binomial(2*n+2,n+1)/(2^(2*n+1)*(2*n+1)^2) ), n=0..25); # G. C. Greubel, Jan 25 2020

Numerator/@Select[CoefficientList[Series[ArcSin[x],{x,0,60}],x], #!=0&]  (* Harvey P. Dale, Apr 29 2011 *)

vector(25, n, numerator(2*n*binomial(2*n,n)/(4^n*(2*n-1)^2)) ) \\ G. C. Greubel, Jan 25 2020

[numerator( (n+1)*binomial(2*n+2,n+1)/(2^(2*n+1)*(2*n+1)^2) ) for n in (0..25)] # G. C. Greubel, Jan 25 2020

a(n) / A052469(n) = A001147(n) / ( A000165(n) *2*n ). E.g., a(6) = 77 = 1*3*5*7*9*11 / gcd( 1*3*5*7*9*11, 2*4*6*8*10*12*12 ).

a(n) = numerator((2*n)!/(2^(2*n)*(n)!^2*(2*n+1))). - Johannes W. Meijer, Jul 06 2009

Edited by Johannes W. Meijer, Jul 06 2009

A068102 a(n) = n! * 2^n * Sum_{i=1..n} 1/(i*2^i).

Original entry on oeis.org

1, 5, 32, 262, 2644, 31848, 446592, 7150512, 128749536, 2575353600, 56661408000, 1359913708800, 35358235430400, 990036819072000, 29701191750451200, 950439443688806400, 32314962008209305600, 1163338987982963097600, 44206887945726303436800, 1768275639474152546304000

Offset: 1

Benoit Cloitre, Apr 14 2002

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A000254, A069015, A087547, A000165, A001008

I:=[1,5]; [n le 2 select I[n]  else (3*n-1)*Self(n-1)-2*(n-1)^2*Self(n-2): n in [1..25] ]; // Vincenzo Librandi, Feb 19 2015

seq(add(n!/i*2^(n-i), i=1..n), n=1..100); # Robert Israel, Aug 14 2014

a[n_] := FullSimplify[n! (2^n Log[2] - LerchPhi[1/2, 1, 1 + n]/2)]; Array[a,10] (* Vladimir Reshetnikov, Jan 21 2011 *)

a(b):=n!*sum(binomial(k,n-k)*(-1)^(n-k)*binomial(2*k,k)*(H(2*k)-H(k)),k,floor(n/2),n); /* Vladimir Kruchinin, Feb 04 2023 */

first(n)=my(v=vector(n),t=1); v[1]=1; for(k=2,n, v[k]=2*k*v[k-1] + t; t*=k); v \\ Charles R Greathouse IV, Aug 22 2022

E.g.f.: -log(1-x)/(1-2*x). - Vladeta Jovovic, Feb 07 2003
a(n+1) = 2*(n+1)*a(n) + n!, a(0)=0. - Jaume Oliver Lafont, Sep 15 2009
a(n) = 2^n*n!*(log(2) - 2*Integral_{x=0..1} x^(2*n+1)/(1+x^2)^(n+1) dx). Thus a(n)/(2^n*n!) -> log(2) as n -> inf. Cf. A087547. - Peter Bala, Jun 21 2013
a(n) = (3*n-1)*a(n-1) - 2*(n-1)^2*a(n-2). - Vaclav Kotesovec, Aug 13 2013
The sequence b(n) = 2^n*n! = A000165(n) also satisfies the above second-order recurrence of Kotesovec. This leads to the generalized continued fraction expansion lim_{n->oo} a(n)/b(n) = log(2) = 1/(2 - 2/(5 - 8/(8 - 18/(11 - ... - 2*(n - 1)^2/((3*n - 1) - ... ))))). - Peter Bala, Feb 18 2015
a(n)/n! is the linear term of the sum of the n-th row of a Pascal-like triangle T in which T(n,k) = binomial(x+n, k). - Greg Dresden and Ivan Kuznetsov, Aug 22 2022
a(n) = n!*Sum_{k=floor(n/2)..n} (-1)^(n-k)*C(k,n-k)*C(2*k,k)*(H(2*k)-H(k)),  where H(n) are the harmonic numbers. - Vladimir Kruchinin, Feb 04 2023

A051581 a(n) = (2*n+7)!!/7!!, related to A001147 (odd double factorials).

Original entry on oeis.org

1, 9, 99, 1287, 19305, 328185, 6235515, 130945815, 3011753745, 75293843625, 2032933777875, 58955079558375, 1827607466309625, 60311046388217625, 2110886623587616875, 78102805072741824375, 3046009397836931150625

Offset: 0

Wolfdieter Lang

Row m=7 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..400
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Cf. A000165, A001147(n+1), A002866(n+1).

Cf. A051577, A051578, A051579, A051580 (rows m=0..6), A051582, A051583.

List([0..20], n-> Product([0..n-1], j-> 2*j+9) ); # G. C. Greubel, Nov 12 2019

[1] cat [(&*[2*j+9: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Nov 12 2019

df:=doublefactorial; seq(df(2*n+7)/df(7), n = 0..20); # G. C. Greubel, Nov 12 2019

Table[2^n*Pochhammer[9/2, n], {n,0,20}] (* G. C. Greubel, Nov 12 2019 *)

vector(20, n, prod(j=1,n-1, 2*j+7) ) \\ G. C. Greubel, Nov 12 2019

[product( (2*j+9) for j in (0..n-1)) for n in (0..20)] # G. C. Greubel, Nov 12 2019

a(n) = (2*n+7)!!/7!!.
E.g.f.: 1/(1-2*x)^(9/2).
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x/(x + 1/(2*k+9)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 02 2013
From Peter Bala, May 26 2017: (Start)
a(n+1) = (2*n + 9)*a(n) with a(0) = 1.
O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 9*x)*A(x) - 1 with A(0) = 1.
G.f. as an S-fraction: A(x) = 1/(1 - 9*x/(1 - 2*x/(1 - 11*x/(1 - 4*x/(1 - 13*x/(1 - 6*x/(1 - ... - (2*n + 7)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).
Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 9*x/(1 - 11*x/(1 - 2*x/(1 - 13*x/(1 - 4*x/(1 - 15*x/(1 - 6*x/(1 - ... - (2*n + 9)*x/(1 - 2*n*x/(1 - ...)))))))))). (End)
From Amiram Eldar, Dec 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 105 * sqrt(e*Pi/2) * erf(1/sqrt(2)) - 147, where erf is the error function.
Sum_{n>=0} (-1)^n/a(n) = 77 - 105 * sqrt(Pi/(2*e)) * erfi(1/sqrt(2)), where erfi is the imaginary error function. (End)

A051583 a(n) = (2*n+9)!!/9!!, related to A001147 (odd double factorials).

Original entry on oeis.org

1, 11, 143, 2145, 36465, 692835, 14549535, 334639305, 8365982625, 225881530875, 6550564395375, 203067496256625, 6701227376468625, 234542958176401875, 8678089452526869375, 338445488648547905625, 13876265034590464130625

Offset: 0

Wolfdieter Lang

Row m=9 of the array A(3; m,n) := (2*n+m)!!/m!!, m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..398
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Cf. A000165, A001147(n+1), A002866(n+1), A178647.

Cf. A051577, A051578, A051579, A051580, A051581, A051582 (rows m=0..8).

List([0..20], n-> Product([0..n-1], j-> 2*j+11) ); # G. C. Greubel, Nov 12 2019

[1] cat [(&*[2*j+11: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Nov 12 2019

seq(2^n*pochhammer(11/2,n), n = 0..20); # G. C. Greubel, Nov 12 2019

(2*Range[0,20]+9)!!/945 (* Harvey P. Dale, Apr 10 2019 *)
Table[2^n*Pochhammer[11/2, n], {n,0,20}] (* G. C. Greubel, Nov 12 2019 *)

vector(20, n, prod(j=0,n-2, 2*j+11) ) \\ G. C. Greubel, Nov 12 2019

[product( (2*j+11) for j in (0..n-1)) for n in (0..20)] # G. C. Greubel, Nov 12 2019

a(n) = (2*n+9)!!/9!!.
E.g.f.: 1/(1-2*x)^(11/2).
From Peter Bala, May 26 2017: (Start)
a(n+1) = (2*n + 11)*a(n) with a(0) = 1.
O.g.f. satisfies the Riccati differential equation 2*x^2*A(x)' = (1 - 11*x)*A(x) - 1 with A(0) = 1.
G.f. as an S-fraction: A(x) = 1/(1 - 11*x/(1 - 2*x/(1 - 13*x/(1 - 4*x/(1 - 15*x/(1 - 6*x/(1 - ... - (2*n + 9)*x/(1 - 2*n*x/(1 - ...))))))))) (by Stokes 1982).
Reciprocal as an S-fraction: 1/A(x) = 1/(1 + 11*x/(1 - 13*x/(1 - 2*x/(1 - 15*x/(1 - 4*x/(1 - 17*x/(1 - 6*x/(1 - ... - (2*n + 11)*x/(1 - 2*n*x/(1 - ...)))))))))). (End)
From Amiram Eldar, Dec 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 945 * sqrt(e*Pi/2) * erf(1/sqrt(2)) - 1332, where erf is the error function.
Sum_{n>=0} (-1)^n/a(n) = 945 * sqrt(Pi/(2*e)) * erfi(1/sqrt(2)) - 684, where erfi is the imaginary error function. (End)

A066318 Number of necklaces with n labeled beads of 2 colors.

Original entry on oeis.org

2, 4, 16, 96, 768, 7680, 92160, 1290240, 20643840, 371589120, 7431782400, 163499212800, 3923981107200, 102023508787200, 2856658246041600, 85699747381248000, 2742391916199936000, 93241325150797824000, 3356687705428721664000, 127554132806291423232000

Offset: 1

Christian G. Bower, Dec 13 2001

nonn

In the normal probability distribution with mean 0 and standard deviation 1, the expected value E[|x|^(2n-1)] = a(n)/sqrt(2*Pi), while E[|x|^(2n)] = E[x^(2n)] = A001147(n). - Stanislav Sykora, Jan 15 2017

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 66 (2.1.27,29).

Vincenzo Librandi, Table of n, a(n) for n = 1..400
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
Wikipedia, Normal distribution, formula for E(|x|^p).
Index entries for sequences related to necklaces.

Apart from initial term, same as A032184.

Cf. A000165, A000796, A001147, A019775, A019727.

a_n:=List([1..10], n->Factorial(n-1)*2^n); # Stefano Spezia, Nov 17 2018

[Factorial(n-1)*2^n: n in [1..20]]; // Vincenzo Librandi, Sep 23 2011

with(combstruct):A:=[N,{N=Cycle(Union(Z$2))},labeled]: seq(count(A,size=n),n=1..18); # Zerinvary Lajos, Oct 07 2007
# alternative Maple program:
a:= n-> 2*doublefactorial(2*n-2):
seq(a(n), n=1..20);  # Alois P. Heinz, Jun 22 2017

mx = 18; Rest[ Range[0, mx]! CoefficientList[ Series[ Log[1/(1 - 2 x)], {x, 0, mx}], x]] (* Robert G. Wilson v, Sep 22 2011 *)
Table[(n-1)!*2^n,{n,20}] (* Harvey P. Dale, Dec 15 2011 *)

a(n):=(n-1)!*2^n$ makelist(a(n), n, 1, 10);  /* Stefano Spezia, Nov 21 2018 */

apply( A066318=n->(n-1)!<M. F. Hasler, Jan 15 2017

import math
for n in range(1,10): print(math.factorial(n-1)*2**n, end=', ') # Stefano Spezia, Nov 17 2018

[2^n*factorial(n-1) for n in (1..20)] # G. C. Greubel, Nov 21 2018

a(n) = (n-1)!*2^n.
E.g.f.: log(1/(1-2*x)).
Let gd(x,n) = (d^n/dx^n)(exp(-(1/2)*x^2)*sqrt(2)/(2*sqrt(Pi))) = (-1)^((1/2)*n)*(x^2)^((1/2)*n)*2^(-(1/2)*n+1/2)*(exp(I*Pi*n)+1)/(4*sqrt(Pi)*GAMMA(1+(1/2)*n)) be the n-th derivative of the standard Gaussian distribution. Evaluating gd(x,n) at x=1 gives gd(1,n) = 2^(-(1/2)*n+1/2)*(exp(I*Pi*n)+1)*(-1)^((1/2)*n)/(4*sqrt(Pi)*GAMMA(1+(1/2)*n)). A066318 is the denominator of the even summands of the Taylor series expansion of the Gaussian distribution evaluated at x=1. a(n)=denom(gd(1, 2*n))/sqrt(Pi). - Stephen Crowley, May 16 2009
a(n) = 2*(n-1)*a(n-1). - R. J. Mathar, Sep 10 2012
G.f.: G(0), where G(k)= 1  + 1/(1 - 1/(1 + 1/(2*k+2)/x/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 14 2013
a(n) = 2 * (2*n-2)!! = 2 * A000165(n-1). - Alois P. Heinz, Jun 22 2017
a(n) = (sqrt(Pi)/Gamma((2*n+3)/2))*Product_{k=0..n-1} binomial(2*(n-k)+1,2). - Stefano Spezia, Nov 17 2018
From Amiram Eldar, Mar 11 2022: (Start)
Sum_{n>=1} 1/a(n) = sqrt(e)/2 (A019775).
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/(2*sqrt(e)). (End)

A067624 a(n) = 2^(2n)(2*n)!.

Original entry on oeis.org

1, 8, 384, 46080, 10321920, 3715891200, 1961990553600, 1428329123020800, 1371195958099968000, 1678343852714360832000, 2551082656125828464640000, 4714400748520531002654720000, 10409396852733332453861621760000

Offset: 0

Benoit Cloitre, Feb 02 2002

nonn

For n >= 1, a(n) equals the absolute value of the determinant of the 4n X 4n matrix with i's along the superdiagonal (where i is the imaginary unit), and 2, 3, 4, ... 4*n along the subdiagonal, and 0's everywhere else. (See Mathematica code below.) - John M. Campbell, Jun 04 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000165.

Appears in A162445, A061549 and A120738. - Johannes W. Meijer, Jul 06 2009

[2^(2*n)*Factorial(2*n): n in [0..15]]; // Vincenzo Librandi, Feb 18 2018

for n from 0 to 30 by 2 do printf(`%d,`,2^(n)*(n)!) od: # James Sellers, Feb 11 2002
A067624 := n -> 2^(2*n)*(2*n)!: seq(A067624(n), n=0..12); # Johannes W. Meijer, Jan 05 2017

Table[Abs[Det[Array[KroneckerDelta[#1 + 1, #2]*I &, {4*n, 4*n}] + Array[KroneckerDelta[#1 - 1, #2]*#1 &, {4*n, 4*n}]]], {n, 1, 20}] (* John M. Campbell, Jun 04 2011 *)
Table[2^(2 n) (2 n)!, {n, 0, 30}] (* Vincenzo Librandi, Feb 18 2018 *)

a(n) = A000165(2*n) where A000165(k) are the double factorial numbers 2^k*k!=(2k)!!. - Corrected by Johannes W. Meijer, Jul 05 2009
a(n) = (4*n)!! = 2^(2*n)*(2*n)!. - Johannes W. Meijer, Jul 06 2009
sqrt((1+cos(x))/2) = Sum_{n>=0} (-1)^n * x^(2*n) / a(n).
a(n) = (A280442(n)/A046161(n))/(A223067(n)/A223068(n)). - Johannes W. Meijer, Jan 05 2017
From Amiram Eldar, Jul 12 2020: (Start)
Sum_{n>=0} 1/a(n) = cosh(1/2).
Sum_{n>=0} (-1)^n/a(n) = cos(1/2). (End)

More terms from James Sellers, Feb 11 2002

A112368 a(n) = Sum_{i=0..n} 2^i*i!.

Original entry on oeis.org

1, 3, 11, 59, 443, 4283, 50363, 695483, 11017403, 196811963, 3912703163, 85662309563, 2047652863163, 53059407256763, 1481388530277563, 44331262220901563, 1415527220320869563, 48036189795719781563, 1726380042510080613563, 65503446445655792229563, 2616586102571484256869563

Offset: 0

N. J. A. Sloane, Dec 02 2005

nonn

a(n) is divisible by 73 for all n >= 72, hence this sequence contains only a finite number of primes. - Giovanni Resta, Mar 11 2017

Partial sums of A000165. One less than A004400. one more than A112369. - Michael Somos, Sep 27 2017

			G.f. = 1 + 3*x + 11*x^2 + 59*x^3 + 443*x^4 + 4283*x^5 + 50363*x^6 + 695483*x^7 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..403

Cf. A000165, A004400, A112369.

s = 1; lst = {s}; Do[s += n!!; AppendTo[lst, s], {n, 2, 38, 2}]; lst (* Zerinvary Lajos, Jul 13 2009 *)
a[ n_] := Sum[ 2^k k!, {k, 0, n}]; (* Michael Somos, Sep 27 2017 *)

{a(n) = sum(k=0, n, 2^k * k!)}; /* Michael Somos, Sep 27 2017 */

0 = +a(n)*(+2*a(n+1) - 3*a(n+2) + a(n+3)) + a(n+1)*(-a(n+1) + a(n+2) - a(n+3)) + a(n+2)*(a(n+2)) for all n>=0. - Michael Somos, Sep 27 2017

cp's OEIS Frontend

A021012 Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).

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A109692 Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).

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A120778 Numerators of partial sums of Catalan numbers scaled by powers of 1/4.

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A055786 Numerators of Taylor series expansion of arcsin(x). Also arises from arccos(x), arccsc(x), arcsec(x), arcsinh(x).

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A068102 a(n) = n! * 2^n * Sum_{i=1..n} 1/(i*2^i).

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Maxima

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A051581 a(n) = (2*n+7)!!/7!!, related to A001147 (odd double factorials).

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A109692 Triangle of coefficients in expansion of (1+x)(1+3x)(1+5x)(1+7x)...*(1+(2n-1)x).

A067624 a(n) = 2^(2n)(2*n)!.