A007019 -id:A007019 - cp's OEIS frontend

A000680 a(n) = (2n)!/2^n.

1, 1, 6, 90, 2520, 113400, 7484400, 681080400, 81729648000, 12504636144000, 2375880867360000, 548828480360160000, 151476660579404160000, 49229914688306352000000, 18608907752179801056000000, 8094874872198213459360000000, 4015057936610313875842560000000

Offset: 0

N. J. A. Sloane

Denominators in the expansion of cos(sqrt(2)*x) = 1 - (sqrt(2)*x)^2/2! + (sqrt(2)*x)^4/4! - (sqrt(2)*x)^6/6! + ... = 1 - x^2 + x^4/6 - x^6/90 + ... By Stirling's formula in A000142: a(n) ~ 2^(n+1) * (n/e)^(2n) * sqrt(Pi*n) - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 20 2001
a(n) is also the constant term in the product: Product_{1<=i, j<=n, i!=j} (1 - x_i/x_j)^2. - Sharon Sela (sharonsela(AT)hotmail.com), Feb 12 2002
a(n) is also the number of lattice paths in the n-dimensional lattice [0..2]^n. - T. D. Noe, Jun 06 2002
Representation as the n-th moment of a positive function on the positive half-axis: a(n) = Integral_{x>=0} (x^n*exp(-sqrt(2*x))/sqrt(2*x)), n=0,1,... - Karol A. Penson, Mar 10 2003
Number of permutations of [2n] with no increasing runs of odd length. Example: a(2) = 6 because we have 1234, 13/24, 14/23, 23/14, 24/13 and 34/12 (runs separated by slashes). - Emeric Deutsch, Aug 29 2004
This is also the number of ways of arranging the elements of n distinct pairs, assuming the order of elements is significant and the pairs are distinguishable. When the pairs are not distinguishable, see A001147 and A132101. For example, there are 6 ways of arranging 2 pairs [1,1], [2,2]: {[1122], [1212], [1221], [2211], [2121], [2112]}. - Ross Drewe, Mar 16 2008
n married couples are seated in a row so that every wife is to the left of her husband. The recurrence a(n+1) = a(n)*((2*n + 1) + binomial(2*n+1, 2)) conditions on whether the (n+1)st couple is seated together or separated by at least one other person. - Geoffrey Critzer, Jun 10 2009
a(n) is the number of functions f:[2n]->[n] such that the preimage of {y} has cardinality 2 for every y in [n]. Note that [k] denotes the set {1,2,...,k} and [0] denotes the empty set. - Dennis P. Walsh, Nov 17 2009
a(n) is also the number of n X 2n (0,1)-matrices with row sum 2 and column sum 1. - Shanzhen Gao, Feb 12 2010
Number of ways that 2n people of different heights can be arranged (for a photograph) in two rows of equal length so that every person in the front row is shorter than the person immediately behind them in the back row.
a(n) is the number of functions f:[n]->[n^2] such that, if floor((f(x))^.5) = floor((f(y))^.5), then x = y. For example, with n = 4, the range of f consists of one element from each of the four sets {1,2,3}, {4,5,6,7,8}, {9,10,11,12,13,14,15}, and {16}. Hence there are 1*3*5*7 = 105 ways to choose the range for f, and there are 4! ways to injectively map {1,2,3,4} to the four elements of the range. Thus there are 105*24 = 2520 such functions. Note also that a(n) = n!*(product of the first n odd numbers). - Dennis P. Walsh, Nov 28 2012
a(n) is also the 2*n th difference of n-powers of A000217 (triangular numbers). For example a(2) is the 4th difference of the squares of triangular numbers. - Enric Reverter i Bigas, Jun 24 2013
a(n) is the multinomial coefficient (2*n) over (2, 2, 2, ..., 2) where there are n 2's in the last parenthesis. It is therefore also the number of words of length 2n obtained with n letters, each letter appearing twice. - Robert FERREOL, Jan 14 2018
Number of ways to put socks and shoes on an n-legged animal, if a sock must be put on before a shoe. - Daniel Bishop, Jan 29 2018

			For n = 2, a(2) = 6 since there are 6 functions f:[4]->[2] with size 2 preimages for both {1} and {2}. In this case, there are binomial(4, 2) = 6 ways to choose the 2 elements of [4] f maps to {1} and the 2 elements of [4] that f maps to {2}. - _Dennis P. Walsh_, Nov 17 2009

G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, 1998.
H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 283.
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 112.
Shanzhen Gao and Kenneth Matheis, Closed formulas and integer sequences arising from the enumeration of (0,1)-matrices with row sum two and some constant column sums. In Proceedings of the Forty-First Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congr. Numer. 202 (2010), 45-53.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
C. B. Tompkins, Methods of successive restrictions in computational problems involving discrete variables. 1963, Proc. Sympos. Appl. Math., Vol. XV pp. 95-106; Amer. Math. Soc., Providence, R.I.

T. D. Noe, Table of n, a(n) for n = 0..100
Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.
R. Florez and L. Junes, A relation between triangular numbers and prime numbers, Integers 12(1) (2012), 83-96.
M. Ghebleh, Antichains of (0, 1)-matrices through inversions, Linear Algebra and its Applications 458 (2014), 503-511.
S. A. Joffe, Calculation of the first thirty-two Eulerian numbers from central differences of zero, Quart. J. Pure Appl. Math. 47 (1914), 103-126. [Accessible only in the USA through the Hathi Trust Digital Library.]
Peter D. Loly and Ian D. Cameron, Frierson's 1907 Parameterization of Compound Magic Squares Extended to Orders 3^L, L = 1, 2, 3, ..., with Information Entropy, arXiv:2008.11020 [math.HO], 2020.
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
Robert A. Proctor, Let's Expand Rota's Twelvefold Way For Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
Dennis Walsh, Counting integer functions with size-2 preimage constraints, preprint.
Eric Weisstein's World of Mathematics, Lattice Path.
Index to divisibility sequences
Index entries for related partition-counting sequences

Cf. A084939, A084940, A084941, A084942, A084943, A084944, A087127, A001147, A132101.
A diagonal of the triangle in A241171.
Main diagonal of A267479, row sums of A267480.
Row n=2 of A089759.
Column n=2 of A187783.
Even bisection of column k=0 of A097591.

A000680 := n->(2*n)!/(2^n);
a[0]:=1:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]*(2*n-1)*n od: seq(a[n], n=0..16); # Zerinvary Lajos, Mar 08 2008
seq(product(binomial(2*n-2*k,2),k=0..n-1),n=0..16); # Dennis P. Walsh, Nov 17 2009

Table[Product[Binomial[2 i, 2], {i, 1, n}], {n, 0, 16}]
polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k -2), n]]; Array[ polygorial[6, #] &, 17, 0] (* Robert G. Wilson v, Dec 26 2016 *)
Table[(2n)!/2^n,{n,0,20}] (* Harvey P. Dale, Sep 21 2020 *)

PARI
```
a(n) = (2*n)! / 2^n
```

E.g.f.: 1/(1 - x^2/2) (with interpolating zeros). - Paul Barry, May 26 2003
a(n) = polygorial(n, 6) = (A000142(n)/A000079(n))*A001813(n) = (n!/2^n)*Product_{i=0..n-1} (4*i + 2) = (n!/2^n)*4^n*Pochhammer(1/2, n) = gamma(2*n+1)/2^n. - Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003
a(n) = A087127(n,2*n) = Sum_{i=0..2*n} (-1)^(2*n-i)*binomial(2*n, i)*binomial(i+2, 2)^n. Let T(n,k,j) = ((n - k + j)*(2*n - 2*k + 1))^n*binomial(2*n, 2*k-j+1) then a(n) = Sum{k=0..n} (T(n,k,1) - T(n,k,0)). For example a(12) = A087127(12,24) = Sum_{k=0..12} (T(12,k,1) - T(12,k,0)) = 24!/2^12. - André F. Labossière, Mar 29 2004 [Corrected by Jianing Song, Jan 08 2019]
For even n, a(n) = binomial(2n, n)*(a(n/2))^2. For odd n, a(n) = binomial(2n, n+1)*a((n+1)/2)*a((n-1)/2). For positive n, a(n) = binomial(2n, 2)*a(n-1) with a(0) = 1. - Dennis P. Walsh, Nov 17 2009
a(n) = Product_{i=1..n} binomial(2i, 2).
a(n) = a(n-1)*binomial(2n, 2).
From Peter Bala, Feb 21 2011: (Start)
a(n) = Product_{k = 0..n-1} (T(n) - T(k)), where T(n) = n*(n + 1)/2 is the n-th triangular number.
Compare with n! = Product_{k = 0..n-1} (n - k).
Thus we may view a(n) as a generalized factorial function associated with the triangular numbers A000217. Cf. A010050. The corresponding generalized binomial coefficients a(n)/(a(k)*a(n-k)) are triangle A086645. Also cf. A186432.
a(n) = n*(n + n-1)*(n + n-1 + n-2)*...*(n + n-1 + n-2 + ... + 1).
For example, a(5) = 5*(5+4)*(5+4+3)*(5+4+3+2)*(5+4+3+2+1) = 113400. (End).
G.f.: 1/U(0) where U(k)= x*(2*k - 1)*k + 1 - x*(2*k + 1)*(k + 1)/U(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Oct 28 2012
a(n) = n!*(product of the first n odd integers). - Dennis P. Walsh, Nov 28 2012
a(0) = 1, a(n) = a(n-1)*T(2*n-1), where T(n) is the n-th triangular number. For example: a(4) = a(3)*T(7) = 90*28 = 2520. - Enric Reverter i Bigas, Jun 24 2013
E.g.f.: 1/(1 - x/(1 - 2*x/(1 - 3*x/(1 - 4*x/(1 - 5*x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, May 10 2017
From Amiram Eldar, Jun 25 2020: (Start)
Sum_{n>=0} 1/a(n) = cosh(sqrt(2)).
Sum_{n>=0} (-1)^n/a(n) = cos(sqrt(2)). (End)
D-finite with recurrence a(n) -n*(2*n-1)*a(n-1)=0. - R. J. Mathar, Jan 28 2022
a(n) = n *A007019(n-1), n>0. - R. J. Mathar, Jan 28 2022

A009775 Exponential generating function is tanh(log(1+x)).

Original entry on oeis.org

0, 1, -1, 0, 6, -30, 90, 0, -2520, 22680, -113400, 0, 7484400, -97297200, 681080400, 0, -81729648000, 1389404016000, -12504636144000, 0, 2375880867360000, -49893498214560000, 548828480360160000, 0, -151476660579404160000

Offset: 0

R. H. Hardin

Brandon Humpert and Jeremy L. Martin, The Incidence Hopf Algebra of Graphs, DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), pp. 517-526. [See Example 3.4]

Cf. A052277, A007019, A046979, A007415, A007452, A092820, A217260.

A009775(n)=polcoeff(tanh(log(1+x+O(x^n)*x)),n)*n!  \\ M. F. Hasler, Oct 10 2012

a(0) = 0, a(4n+3) = 0, a(n) = (-1)^[n == 2, 5, 8 mod 8] * n!/2^floor(n/2). - Ralf Stephan, Mar 06 2004
From Peter Bala, Nov 25 2011: (Start)
(1): a(n) = i*n!/2^(n+1)*{(i-1)^(n+1)-(-1-i)^(n+1)} for n>=1.
The function tanh(log(1+x)) is a disguised form of the rational function (x^2+2*x)/(x^2+2*x+2). Observe that
(2): (x^2+2*x)/(x^2+2*x+2) = d/dx[x - atan((x^2+2*x)/(2*x+2))].
Hence, with an offset of 1, the egf for this sequence is
(3): x - atan((x^2+2*x)/(2*x+2)) = x^2/2! - x^3/3! + 6*x^5/5!- 30*x^6/6! + 90*x^7/7! - ....
This sequence is closely related to the series reversion of the function E(x)-1, where E(x) = sec(x)+tan(x) is the egf for the sequence of zigzag numbers A000111. Under the change of variable x -> sec(x)+tan(x)-1 the rational function (x^2+2*x)/(2*x+2) transforms to tan(x). Hence atan((x^2+2*x)/(2*x+2)) is the inverse function of sec(x)+tan(x)-1.
Recurrence relation:
(4): 2*a(n)+2*n*a(n-1)+n*(n-1)*a(n-2) = 0 with a(1) = 1, a(2) = -1.
(End)

Extended with signs by Olivier Gérard, Mar 15 1997

A007415 Expand sin x / exp x = x-x^2+x^3/3-x^5/30+... and invert nonzero coefficients.

Original entry on oeis.org

0, 1, -1, 3, 0, -30, 90, -630, 0, 22680, -113400, 1247400, 0, -97297200, 681080400, -10216206000, 0, 1389404016000, -12504636144000, 237588086736000, 0, -49893498214560000, 548828480360160000, -12623055048283680000, 0, 3786916514485104000000

Offset: 0

N. J. A. Sloane

sign

J. Roberts, Lure of the Integers, Annotated scanned copy of pp. 81, 86 with notes.

Absolute values are essentially the same as A046979, where zeros are replaced by ones.
a(4n+2) = -(-1)^n*A052277(n), a(2n+1) = (-1)^[n/2]*A007019(n).
Cf. A046981, A007452, A009775, A092820.

a:= n-> (p-> `if`(p=0,0,1/p))(coeff(series(sin(x)/exp(x), x, n+1), x, n)):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 17 2017

1/CoefficientList[Sin[x]/Exp[x] + O[x]^26, x] /. ComplexInfinity -> 0 // Quiet (* Jean-François Alcover, Feb 26 2019 *)

a(n) = [n mod 4 > 0] * (-1)^(n+1+[n/4]) * n!/2^[n/2]. - Ralf Stephan, Mar 06 2004

E.g.f.: sin(x)/exp(x) = x-x^2/(G(0)+x); G(k)=2k+1-x+x*(2k+1)/(4k+3-x+x^2*(4k+3)/( (2k+2)*(4k+5)-x^2+x*(2k+2)*(4k+5)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 20 2011

A046979 Denominators of Taylor series for exp(x)*sin(x).

Original entry on oeis.org

1, 1, 1, 3, 1, 30, 90, 630, 1, 22680, 113400, 1247400, 1, 97297200, 681080400, 10216206000, 1, 1389404016000, 12504636144000, 237588086736000, 1, 49893498214560000, 548828480360160000, 12623055048283680000, 1, 3786916514485104000000, 49229914688306352000000

Offset: 0

N. J. A. Sloane

			1*x + 1*x^2 + 1/3*x^3 - 1/30*x^5 - 1/90*x^6 - 1/630*x^7 + 1/22680*x^9 + 1/113400*x^10 + ...

G. W. Caunt, Infinitesimal Calculus, Oxford Univ. Press, 1914, p. 477.

T. D. Noe, Table of n, a(n) for n=0..100
Herbert S. Wilf, Generatingfunctionology, Academic Press, NY, 1994. See p. 54.

Cf. A046978, A046981, A052277, A007019, A007452, A009775, A092820.

Essentially the same as absolute values of A007415.

a:= n-> denom(coeff(series(sin(x)/exp(x), x, n+1), x, n)):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 17 2017

Denominator[CoefficientList[Series[Exp[x]Sin[x],{x,0,30}],x] ] (* Harvey P. Dale, Feb 14 2015 *)

a(n) = if (n % 4, n!/2^floor(n/2), 1); \\ Michel Marcus, Oct 12 2015

a(4n) = 1, a(n) = n!/2^floor(n/2).

A046981 Denominators of Taylor series for exp(x)*cos(x).

Original entry on oeis.org

1, 1, 1, 3, 6, 30, 1, 630, 2520, 22680, 1, 1247400, 7484400, 97297200, 1, 10216206000, 81729648000, 1389404016000, 1, 237588086736000, 2375880867360000, 49893498214560000, 1, 12623055048283680000, 151476660579404160000, 3786916514485104000000

Offset: 0

N. J. A. Sloane

			1+1*x-1/3*x^3-1/6*x^4-1/30*x^5+1/630*x^7+1/2520*x^8+1/22680*x^9-...

G. W. Caunt, Infinitesimal Calculus, Oxford Univ. Press, 1914, p. 477.

T. D. Noe, Table of n, a(n) for n=0..100

Cf. A046980, A046979, A052277, A007019, A007415, A009775, A092820.

Absolute values are essentially the same as A007452.

Denominator/@CoefficientList[Series[Exp[x]Cos[x],{x,0,30}],x] (* Harvey P. Dale, Jun 28 2011 *)

a(4n+2) = 1, a(n) = n!/2^[n/2].

A267724 T(n,k)=Number of nXk arrays containing k copies of 0..n-1 with every element equal to at least one horizontal neighbor and the top left element equal to 0.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 3, 2, 6, 0, 1, 5, 30, 6, 24, 0, 1, 12, 58, 630, 24, 120, 0, 1, 26, 588, 1398, 22680, 120, 720, 0, 1, 60, 2388, 93672, 55896, 1247400, 720, 5040, 0, 1, 143, 17456, 609264, 33807456, 3353880, 97297200, 5040, 40320, 0, 1, 353, 102874

Offset: 1

R. H. Hardin, Jan 20 2016

Table starts
.0......1.....1...........1.........1...........1.........1........1.........1
.0......1.....1...........3.........5..........12........26.......60.......143
.0......2.....2..........30........58.........588......2388....17456....102874
.0......6.....6.........630......1398.......93672....609264.20702760.279681294
.0.....24....24.......22680.....55896....33807456.331295040
.0....120...120.....1247400...3353880.22933860480
.0....720...720....97297200.281725200
.0...5040..5040.10216206000
.0..40320.40320
.0.362880

			Some solutions for n=4 k=4
..0..0..2..2....0..0..0..0....0..0..1..1....0..0..2..2....0..0..3..3
..2..2..0..0....3..3..3..3....3..3..2..2....1..1..0..0....1..1..2..2
..1..1..3..3....2..2..1..1....0..0..2..2....3..3..2..2....1..1..2..2
..3..3..1..1....2..2..1..1....3..3..1..1....3..3..1..1....0..0..3..3

R. H. Hardin, Table of n, a(n) for n = 1..70

Column 2 and column 3 are A000142(n-1).

Column 4 is A007019(n-1).

A007452 Expand cos x / exp x and invert nonzero coefficients.

Original entry on oeis.org

1, -1, 0, 3, -6, 30, 0, -630, 2520, -22680, 0, 1247400, -7484400, 97297200, 0, -10216206000, 81729648000, -1389404016000, 0, 237588086736000, -2375880867360000, 49893498214560000, 0, -12623055048283680000

Offset: 0

N. J. A. Sloane

sign

Absolute values are essentially the same as A046981, where zeros are replaced by ones.

Cf. A052277, A007019, A046979, A007415, A009775, A092820.

a(4n+2) = 0, a(n) = (-1)^[n == 1, 4, 7 mod 8] * n!/2^floor(n/2). - Ralf Stephan, Mar 06 2004

A090932 a(n) = n! / 2^floor(n/2).

Original entry on oeis.org

1, 1, 1, 3, 6, 30, 90, 630, 2520, 22680, 113400, 1247400, 7484400, 97297200, 681080400, 10216206000, 81729648000, 1389404016000, 12504636144000, 237588086736000, 2375880867360000, 49893498214560000, 548828480360160000, 12623055048283680000, 151476660579404160000

Offset: 0

Jon Perry, Feb 26 2004

nonn

Number of permutations of the n-th row of Pascal's triangle.
Can be seen as the multiplicative equivalent to the generalized pentagonal numbers. - Peter Luschny, Oct 13 2012
a(n) is the number of permutations of [n] in which all ascents start at an even position. For example, a(3) = 3 counts 213, 312, 321. - David Callan, Nov 25 2021

			From _Rigoberto Florez_, Apr 07 2017: (Start)
a(5) = 5!/2^2 = 120/4 = 30.
a(6) = 6!/2^3 = 1*6*15 = 90.
a(7) = 7!/2^3 = 3*10*21 = 630. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Rigoberto Flórez and Leandro Junes, A relation between triangular numbers and prime numbers, Integers, Vol. 12, No. 1 (2012), pp. 83-96.

Cf. A052277, A007019.

The function appears in several expansions: A009775, A046979, A046981, A007415, A007452.

[Factorial(n) / 2^Floor(n/2): n in [0..25]]; // Vincenzo Librandi, May 14 2011

a:= n-> n!/2^floor(n/2): seq(a(n), n=0..40);

Table[n!/2^Floor[n/2], {n, 0, 21}] (* Michael De Vlieger, Jul 25 2016 *)
nxt[{n_,a_,b_}]:={n+1,b,a Binomial[n,2]}; NestList[nxt,{2,1,1},30][[All,2]] (* Harvey P. Dale, Aug 26 2022 *)

PARI
```
a(n)=n!/2^floor(n/2)
```

from math import factorial
def A090932(n): return factorial(n)>>(n>>1) # Chai Wah Wu, Jan 18 2023

@CachedFunction
def A090932(n):
    if n == 0 : return 1
    fact = n//2 if is_even(n) else n
    return fact * A090932(n-1)
[A090932(n) for n in (0..21)] # Peter Luschny, Oct 13 2012

a(n) = binomial(n-1, 2) * a(n-2).
E.g.f.: (1+x)/(1-1/2*x^2).
E.g.f.: G(0)  where G(k) = 1 + x/(1 - x/(x + 2/G(k+1) )) ; (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 27 2012
G.f.: G(0), where G(k)= 1 + (2*k+1)*x/(1 - x*(k+1)/(x*(k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 28 2013
a(n) = (n+1)!/A093968(n+1). - Anton Zakharov, Jul 25 2016
a(n) ~ sqrt(2*Pi*n)*exp(-n)*n^n/2^floor(n/2). - Ilya Gutkovskiy, Jul 25 2016
From Rigoberto Florez, Apr 07 2017: (Start)
if n=2k, n! / 2^k = t(1)t(3)t(5)...t(2k-1),
if n=2k+1, n! / 2^k = t(2)t(4)t(6)...t(2k),
if n=2k, n! / 2^k = (t(k)-t(0))*(t(k)-t(1))*...*(t(k)-t(k-1)),
with t(i)= i-th triangular number. (End)
From Amiram Eldar, Feb 25 2022: (Start)
Sum_{n>=0} 1/a(n) = cosh(sqrt(2)) + sinh(sqrt(2))/sqrt(2).
Sum_{n>=0} (-1)^n/a(n) = cosh(sqrt(2)) - sinh(sqrt(2))/sqrt(2). (End)

Edited by Ralf Stephan, Sep 07 2004

cp's OEIS Frontend

A000680 a(n) = (2n)!/2^n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A009775 Exponential generating function is tanh(log(1+x)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

Extensions

A007415 Expand sin x / exp x = x-x^2+x^3/3-x^5/30+... and invert nonzero coefficients.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A046979 Denominators of Taylor series for exp(x)*sin(x).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A046981 Denominators of Taylor series for exp(x)*cos(x).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A267724 T(n,k)=Number of nXk arrays containing k copies of 0..n-1 with every element equal to at least one horizontal neighbor and the top left element equal to 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A007452 Expand cos x / exp x and invert nonzero coefficients.

Views

Author

Keywords

Crossrefs

Formula

A090932 a(n) = n! / 2^floor(n/2).

Views

Author

Keywords

Comments

Examples

Links