Displaying 1-10 of 28 results found.
0, 1, 12, 13, 360, 361, 372, 373, 20160, 20161, 20172, 20173, 20520, 20521, 20532, 20533, 1814400, 1814401, 1814412, 1814413, 1814760, 1814761, 1814772, 1814773, 1834560, 1834561, 1834572, 1834573, 1834920, 1834921, 1834932, 1834933, 239500800, 239500801, 239500812, 239500813, 239501160, 239501161, 239501172, 239501173, 239520960, 239520961
COMMENTS
Fixed points of involution A225901. This can be also viewed as a function that reinterprets base-2 representation of n in base-((2n)!/2) where the digits are multiplied with the successive terms of A002674, thus a(0) = 0.
PROG
;; This implements the given recurrence:
(Python)
from sympy import factorial as f
def a007623(n, p=2): return n if n<p else a007623(n//p, p+1)*10 + n%p
def a255411(n):
x=(str(a007623(n)) + '0')
y="".join(str(int(i) + 1) if int(i)>0 else '0' for i in x)[::-1]
return 0 if n==0 else sum([int(y[i])*f(i + 1) for i in range(len(y))])
def a153880(n):
x=(str(a007623(n)) + '0')[::-1]
return 0 if n==0 else sum([int(x[i])*f(i + 1) for i in range(len(x))])
def a(n): return 0 if n==0 else a255411(a153880(a(n//2))) if n%2==0 else 1 + a255411(a153880(a((n - 1)//2)))
Order of alternating group A_n, or number of even permutations of n letters.
(Formerly M2933 N1179)
+10
213
1, 1, 1, 3, 12, 60, 360, 2520, 20160, 181440, 1814400, 19958400, 239500800, 3113510400, 43589145600, 653837184000, 10461394944000, 177843714048000, 3201186852864000, 60822550204416000, 1216451004088320000, 25545471085854720000, 562000363888803840000
COMMENTS
For n >= 3, a(n-1) is also the number of ways that a 3-cycle in the symmetric group S_n can be written as a product of 2 long cycles (of length n). - Ahmed Fares (ahmedfares(AT)my-deja.com), Aug 14 2001
a(n) is the number of Hamiltonian circuit masks for an n X n adjacency matrix of an undirected graph. - Chad Brewbaker, Jan 31 2003 a(n-1) is the number of necklaces one can make with n distinct beads: n! bead permutations, divide by two to represent flipping the necklace over, divide by n to represent rotating the necklace. Related to Stirling numbers of the first kind, Stirling cycles. - Chad Brewbaker, Jan 31 2003 Number of increasing runs in all permutations of [n-1] (n>=2). Example: a(4)=12 because we have 12 increasing runs in all the permutations of [3] (shown in parentheses): (123), (13)(2), (3)(12), (2)(13), (23)(1), (3)(2)(1). - Emeric Deutsch, Aug 28 2004 Minimum permanent over all n X n (0,1)-matrices with exactly n/2 zeros. - Simone Severini, Oct 15 2004 The number of permutations of 1..n that have 2 following 1 for n >= 1 is 0, 1, 3, 12, 60, 360, 2520, 20160, ... . - Jon Perry, Sep 20 2008 Starting (1, 3, 12, 60, ...) = binomial transform of A000153: (1, 2, 7, 32, 181, ...). - Gary W. Adamson, Dec 25 2008 The asymptotic expansion of the higher order exponential integral E(x,m=1,n=3) ~ exp(-x)/x*(1 - 3/x + 12/x^2 - 60/x^3 + 360/x^4 - 2520/x^5 + 20160/x^6 - 81440/x^7 + ...) leads to the sequence given above. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009 Starting (1, 3, 12, 60, ...) = eigensequence of triangle A002260, (a triangle with k terms of (1,2,3,...) in each row given k=1,2,3,...). Example: a(6) = 360, generated from (1, 2, 3, 4, 5) dot (1, 1, 3, 12, 60) = (1 + 2 + 9 + 48 + 300). - Gary W. Adamson, Aug 02 2010 For n>=2: a(n) is the number of connected 2-regular labeled graphs on (n+1) nodes (Cf. A001205). - Geoffrey Critzer, Feb 16 2011. The Fi1 and Fi2 triangle sums of A094638 are given by the terms of this sequence (n>=1). For the definition of these triangle sums see A180662. - Johannes W. Meijer, Apr 20 2011 a(n-1) is, for n>=2, also the number of necklaces with n beads (only C_n symmetry, no turnover) with n-1 distinct colors and signature c[.]^2 c[.]^(n-2). This means that two beads have the same color, and for n=2 the second factor is omitted. Say, cyclic(c[1]c[1]c[2]c[3]..c[n-1]), in short 1123...(n-1), taken cyclically. E.g., n=2: 11, n=3: 112, n=4: 1123, 1132, 1213, n=5: 11234, 11243, 11324, 11342, 11423, 11432, 12134, 12143, 13124, 13142, 14123, 14132. See the next-to-last entry in line n>=2 of the representative necklace partition array A212359. - Wolfdieter Lang, Jun 26 2012 For m >= 3, a(m-1) is the number of distinct Hamiltonian circuits in a complete simple graph with m vertices. See also A001286. - Stanislav Sykora, May 10 2014 In factorial base ( A007623) these numbers have a simple pattern: 1, 1, 1, 11, 200, 2200, 30000, 330000, 4000000, 44000000, 500000000, 5500000000, 60000000000, 660000000000, 7000000000000, 77000000000000, 800000000000000, 8800000000000000, 90000000000000000, 990000000000000000, etc. See also the formula based on this observation, given below. - Antti Karttunen, Dec 19 2015 Also (by definition) the independence number of the n-transposition graph. - Eric W. Weisstein, May 21 2017 Number of permutations of n letters containing an even number of even cycles. - Michael Somos, Jul 11 2018 Equivalent to Brewbaker's and Sykora's comments, a(n - 1) is the number of undirected cycles covering n labeled vertices, hence the logarithmic transform of A002135. - Gus Wiseman, Oct 20 2018 For n >= 2 and a set of n distinct leaf labels, a(n) is the number of binary, rooted, leaf-labeled tree topologies that have a caterpillar shape (column k=1 of A306364). - Noah A Rosenberg, Feb 11 2019 Also the clique covering number of the n-Bruhat graph. - Eric W. Weisstein, Apr 19 2019 a(n) is the number of lattices of the form [s,w] in the weak order on S_n, for a fixed simple reflection s. - Bridget Tenner, Jan 16 2020 For n > 3, a(n) = p_1^e_1*...*p_m^e_m, where p_1 = 2 and e_m = 1. There exists p_1^x where x <= e_1 such that p_1^x*p_m^e_m is a primitive Zumkeller number ( A180332) and p_1^e_1*p_m^e_m is a Zumkeller number ( A083207). Therefore, for n > 3, a(n) = p_1^e_1*p_m^e_m*r, where r is relatively prime to p_1*p_m, is also a Zumkeller number. - Ivan N. Ianakiev, Mar 11 2020 For n>1, a(n) is the number of permutations of [n] that have 1 and 2 as cycle-mates, that is, 1 and 2 are contained in the same cycle of a cyclic representation of permutations of [n]. For example, a(4) counts the 12 permutations with 1 and 2 as cycle-mates, namely, (1 2 3 4), (1 2 4 3), (1 3 2 4), (1 3 4 2), (1 4 2 3), (1 4 3 2), (1 2 3) (4), (1 3 2) (4), (1 2 4 )(3), (1 4 2)(3), (1 2)(3 4), and (1 2)(3)(4). Since a(n+2)=row sums of A162608, our result readily follows. - Dennis P. Walsh, May 28 2020
REFERENCES
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 87-8, 20. (a), c_n^e(t=1).
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).
LINKS
Olivier Bodini, Antoine Genitrini, and Mehdi Naima, Ranked Schröder Trees, arXiv:1808.08376 [cs.DS], 2018. S-Z Song, S-G Hwang, S-H Rim, and G-S Cheon, Extremes of permanents of (0,1)-matrices, Special issue on the Combinatorial Matrix Theory Conference (Pohang, 2002). Linear Algebra Appl. 373 (2003), 197-210.
FORMULA
a(n) = numerator(n!/2) and A141044(n) = denominator(n!/2). D-finite with recurrence: a(0) = a(1) = a(2) = 1; a(n) = n*a(n-1) for n>2. - Chad Brewbaker, Jan 31 2003 [Corrected by N. J. A. Sloane, Jul 25 2008] a(0) = 0, a(1) = 1; a(n) = Sum_{k=1..n-1} k*a(k). - Amarnath Murthy, Oct 29 2002 Stirling transform of a(n+1) = [1, 3, 12, 160, ...] is A083410(n) = [1, 4, 22, 154, ...]. - Michael Somos, Mar 04 2004 a(n) = 0^n + Sum_{k=0..n} (-1)^(n-k-1)*T(n-1, k)*cos(Pi*(n-k-1)/2)^2.
E.g.f.: (2 - x^2)/(2 - 2*x).
E.g.f. of a(n+2), n>=0, is 1/(1-x)^3.
a(n+1) = (-1)^n * A136656(n,1), n>=1. a(n) = n!/2 for n>=2 (proof from the e.g.f). - Wolfdieter Lang, Apr 30 2010 a(n) = (n-2)! * t(n-1), n>1, where t(n) is the n-th triangular number ( A000217). - Gary Detlefs, May 21 2010 O.g.f.: 1 + x*Sum_{n>=0} n^n*x^n/(1 + n*x)^n. - Paul D. Hanna, Sep 13 2011 a(n) = if n < 2 then 1, otherwise Pochhammer(n,n)/binomial(2*n,n). - Peter Luschny, Nov 07 2011 a(n) = Sum_{k=0..floor(n/2)} s(n,n-2*k) where s(n,k) are Stirling number of the first kind, A048994. - Mircea Merca, Apr 07 2012 a(n-1), n>=3, is M_1([2,1^(n-2)])/n = (n-1)!/2, with the M_1 multinomial numbers for the given n-1 part partition of n. See the second to last entry in line n>=3 of A036038, and the above necklace comment by W. Lang. - Wolfdieter Lang, Jun 26 2012 G.f.: A(x) = 1 + x + x^2/(G(0)-2*x) where G(k) = 1 - (k+1)*x/(1 - x*(k+3)/G(k+1)); (continued fraction). - Sergei N. Gladkovskii, Dec 26 2012. G.f.: 1 + x + (Q(0)-1)*x^2/(2*(sqrt(x)+x)), where Q(k) = 1 + (k+2)*sqrt(x)/(1 - sqrt(x)/(sqrt(x) + 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013 G.f.: 1 + x + (x*Q(x)-x^2)/(2*(sqrt(x)+x)), where Q(x) = Sum_{n>=0} (n+1)!*x^n*sqrt(x)*(sqrt(x) + x*(n+2)). - Sergei N. Gladkovskii, May 15 2013 G.f.: 1 + x/2 + (Q(0)-1)*x/(2*(sqrt(x)+x)), where Q(k) = 1 + (k+1)*sqrt(x)/(1 - sqrt(x)/(sqrt(x) + 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013 G.f.: 1 + x + x^2*G(0)/2, where G(k) = 1 + 1/(1 - x/(x + 1/(k+3)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013 G.f.: 1+x + x^2*W(0), where W(k) = 1 - x*(k+3)/( x*(k+3) - 1/(1 - x*(k+1)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013 a(0)=a(1)=1; after which, for even n: a(n) = (n/2) * (n-1)!, and for odd n: a(n) = (n-1)/2 * ((n-1)! + (n-2)!). [The formula was empirically found after viewing these numbers in factorial base, A007623, and is easily proved by considering formulas from Lang (Apr 30 2010) and Detlefs (May 21 2010) shown above.] For n >= 1, a(2*n+1) = a(2*n) + A153880(a(2*n)). [Follows from above.] (End) The o.g.f. A(x) satisfies the Riccati equation x^2*A'(x) + (x - 1)*A(x) + 1 - x^2 = 0.
G.f.: A(x) = 1 + x + x^2/(1 - 3*x/(1 - x/(1 - 4*x/(1 - 2*x/(1 - 5*x/(1 - 3*x/(1 - ... - (n + 2)*x/(1 - n*x/(1 - ... ))))))))) (apply Stokes, 1982).
A(x) = 1 + x + x^2/(1 - 2*x - x/(1 - 3*x/(1 - 2*x/(1 - 4*x/(1 - 3*x/(1 - 5*x/(1 - ... - n*x/(1 - (n+2)*x/(1 - ... ))))))))). (End)
H(x) = (1 - (1 + x)^(-2)) / 2 = x - 3*x^2/2! + 12*x^3/3! - ..., an e.g.f. for the signed sequence here (n!/2!), ignoring the first two terms, is the compositional inverse of G(x) = (1 - 2*x)^(-1/2) - 1 = x + 3*x^2/2! + 15*x^3/3! + ..., an e.g.f. for A001147. Cf. A094638. H(x) is the e.g.f. for the sequence (-1)^m * m!/2 for m = 2,3,4,... . Cf. A001715 for n!/3! and A001720 for n!/4!. Cf. columns of A094587, A173333, and A213936 and rows of A138533. - Tom Copeland, Dec 27 2019 Sum_{n>=0} 1/a(n) = 2*(e-1).
Sum_{n>=0} (-1)^n/a(n) = 2/e. (End)
EXAMPLE
G.f. = 1 + x + x^2 + 3*x^3 + 12*x^4 + 60*x^5 + 360*x^6 + 2520*x^7 + ...
MATHEMATICA
a[n_]:= If[n > 2, n!/2, 1]; Array[a, 21, 0]
a[ n_]:= If[n<0, 0, n! SeriesCoefficient[(2-x^2)/(2-2x), {x, 0, n}]]; (* Michael Somos, May 22 2014 *) a[ n_]:= If[n<0, 0, n! SeriesCoefficient[1 +Sinh[-Log[1-x]], {x, 0, n}]]; (* Michael Somos, May 22 2014 *) Table[GroupOrder[AlternatingGroup[n]], {n, 0, 20}] (* Eric W. Weisstein, May 21 2017 *)
PROG
(PARI) {a(n) = if( n<2, n>=0, n!/2)};
(PARI) a(n)=polcoeff(1+x*sum(m=0, n, m^m*x^m/(1+m*x+x*O(x^n))^m), n) \\ Paul D. Hanna(Scheme, using memoization-macro definec for which an implementation can be found in http://oeis.org/wiki/Memoization )
(definec ( A001710 n) (cond ((<= n 2) 1) (else (* n ( A001710 (- n 1)))))) (Python)
from math import factorial
(SageMath)
def A001710(n): return (factorial(n) +int(n<2))//2
CROSSREFS
Cf. A000142, A000153, A000255, A001147, A001286, A001720, A002135, A002260, A007623, A007717, A049444, A049459, A093468, A094587, A094638, A138533, A153880, A173333, A213936, A215771, A319225, A319226, A320655. a(n+1)= A046089(n, 1), n >= 1 (first column of triangle), A161739 (q(n) sequence).
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Aug 20 2001
1, 2, 24, 720, 40320, 3628800, 479001600, 87178291200, 20922789888000, 6402373705728000, 2432902008176640000, 1124000727777607680000, 620448401733239439360000, 403291461126605635584000000, 304888344611713860501504000000, 265252859812191058636308480000000
COMMENTS
Denominators in the expansion of cos(x): cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + x^8/8! - ...
Contribution from Peter Bala, Feb 21 2011: (Start) We may compare the representation a(n) = Product_{k = 0..n-1} (n*(n+1)-k*(k+1)) with n! = Product_{k = 0..n-1} (n-k). Thus we may view a(n) as a generalized factorial function associated with the oblong numbers A002378. Cf. A000680. The associated generalized binomial coefficients a(n)/(a(k)*a(n-k)) are triangle A086645, cf. A186432. (End) Also, this sequence is the denominator of cosh(x) = (e^x + e^(-x))/2 = 1 + x^2/2! + x^4/4! + x^6/6! + ... - Mohammad K. Azarian, Jan 19 2012 Also (2n+1)-th derivative of arccoth(x) at x = 0. - Michel Lagneau, Aug 18 2012 Product of the partition parts of 2n+1 into exactly two positive integer parts, n > 0. Example: a(3) = 720, since 2(3)+1 = 7 has 3 partitions with exactly two positive integer parts: (6,1), (5,2), (4,3). Multiplying the parts in these partitions gives: 6! = 720. - Wesley Ivan Hurt, Jun 03 2013
REFERENCES
H. B. Dwight, Tables of Integrals and Other Mathematical Data, Macmillan, NY, 1968, p. 88.
Isaac Newton, De analysi, 1669; reprinted in D. Whiteside, ed., The Mathematical Works of Isaac Newton, vol. 1, Johnson Reprint Co., 1964; see p. 20.
FORMULA
E.g.f.: arctanh(x) = Sum_{k>=0} a(k) * x^(2*k+1)/ (2*k+1)!.
E.g.f.: 1/(1-x^2) = Sum_{k>=0} a(k) * x^(2*k) / (2*k)!. - Paul Barry, Sep 14 2004 a(n) = Product_{k = 1..n} (2*k*n-k*(k-1)). - Peter Bala, Feb 21 2011 G.f.: G(0) where G(k) = 1 + 2*x*(2*k+1)*(4*k+1)/(1 - 4*x*(k+1)*(4*k+3)/(4*x*(k+1)*(4*k+3) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 18 2012 a(n) ~ 2*sqrt(Pi)*4^n*n^(2*n+1/2)/exp(2*n).
Sum_{n>=0} 1/a(n) = cosh(1) = A073743. (End)
EXAMPLE
G.f. = 1 + 2*x + 24*x^2 + 720*x^3 + 40320*x^4 + 3628800*x^5 + ...
PROG
(Sage) [stirling_number1(2*n+1, 1) for n in range(0, 22)] # Zerinvary Lajos, Nov 26 2009
AUTHOR
Joe Keane (jgk(AT)jgk.org)
a(n) = 4^n*(2*n+1)!.
(Formerly M5174 N2246)
+10
16
1, 24, 1920, 322560, 92897280, 40874803200, 25505877196800, 21424936845312000, 23310331287699456000, 31888533201572855808000, 53572735778642397757440000, 108431217215972213061058560000
COMMENTS
There is a formula for numerical integration (see MATLAB Central file ID# 71037):
Integral_{x=0..1} f(x) dx = 2*Sum_{m=1..M} Sum_{n>=0} 1/((2*M)^(2*n + 1)*(2*n + 1)!)*f^(2*n)(x)|_x = (m - 1/2)/M, where the notation f^(2*n)(x)|_x = (m - 1/2)/M is the (2*n)-th derivative of the function f(x) at the points x = (m - 1/2)/M.
When we choose M = 1, then the corresponding coefficients are generated as 2*1/(2^(2*n + 1)*(2*n + 1)!) = 1/(4^n*(2*n + 1)!).
Therefore, this sequence also occurs in the denominator of the numerical integration formula at M = 1. (End)
Denominators in the expansion of 2*sinh(x/2) = x + x^3/24 + x^5/1920 + x^7/322560 + ....
If f(x) is a polynomial in x then the central difference f(x+1/2) - f(x-1/2) = 2*sinh(D/2)(f(x)) = D(f(x)) + (1/24)*D^3(f(x)) + (1/1920)*D^5(f(x)) + ..., where D denotes the differential operator d/dx. Formulas for higher central differences in terms of powers of the operator D can be obtained from the expansion of powers of the function 2*sinh(x/2). For example, the expansion (2*sinh(x/2))^2 = x^2 + (1/12)*x^4 + (1/360)*x^6 + .. leads to the second central difference formula f(x+1) - 2*f(x) + f(x-1) = D^2(f(x)) + (1/12)*D^4(f(x)) + (1/360)* D^6(f(x)) + .... See A002674. (End)
REFERENCES
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).
FORMULA
a(n) = 16^n * Pochhammer(1,n) * Pochhammer(3/2,n). - Roger L. Bagula, Apr 26 2013 Sum_{n>=0} 1/a(n) = 2*sinh(1/2).
Sum_{n>=0} (-1)^n/a(n) = 2*sin(1/2). (End)
MATHEMATICA
a[n_] := 4^n*(2*n + 1)!; Array[a, 12, 0] (* Amiram Eldar, Apr 09 2022 *)
CROSSREFS
A bisection of A002866 and (apart from initial term) also a bisection of A007346.
a(n) = binomial(2*n+1,n)*(n+1)^2.
(Formerly M4855 N2075)
+10
15
1, 12, 90, 560, 3150, 16632, 84084, 411840, 1969110, 9237800, 42678636, 194699232, 878850700, 3931426800, 17450721000, 76938289920, 337206098790, 1470171918600, 6379820115900, 27569305764000, 118685861314020, 509191949220240, 2177742427450200, 9287309860732800
COMMENTS
Coefficients for numerical differentiation.
Take the first n integers 1,2,3..n and find all combinations with repetitions allowed for the first n of them. Find the sum of each of these combinations to get this sequence. Example for 1 and 2: 1,2,1+1,1+2,2+2 gives sum of 12=a(2). - J. M. Bergot, Mar 08 2016 Let cos(x) = 1 -x^2/2 +x^4/4!-x^6/6!.. = Sum_i (-1)^i x^(2i)/(2i)! be the standard power series of the cosine, and y = 2*(1-cos(x)) = 4*sin^2(x/2) = x^2 -x^4/12 +x^6/360 ...= Sum_i 2*(-1)^(i+1) x^(2i)/(2i)! be a closely related series. Then this sequence represents the reversion x^2 = Sum_i 1/a(i) *y^(i+1). - R. J. Mathar, May 03 2022
REFERENCES
C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 514.
J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 92.
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).
LINKS
R. Shenton and A. W. Kemp, An S-fraction and ln^2(1+x), Journal of Computational and Applied Mathematics, 26 (1989) 367-370 North-Holland.
FORMULA
G.f.: (1 + 2x)/(1 - 4x)^(5/2).
a(n-1) = sum(i_1 + i_2 + ... + i_n) where the sum is over 0 <= i_1 <= i_2 <= ... <= i_n <= n; a(n) = (n+1)^2 C(2n+1, n). - David Callan, Nov 20 2003 Asymptotics: a(n)-> (1/64) * (128*n^2+176*n+41) * 4^n * n^(-1/2)/(sqrt(Pi)), for n->infinity. - Karol A. Penson, Aug 05 2013 E.g.f.: ((1 + 2*x)*(1 + 8*x)*BesselI(0,2*x) + 2*x*(3 + 8*x)*BesselI(1,2*x))*exp(2*x).
Sum_{n>=0} 1/a(n) = Pi^2/9 = A100044. (End) With x = y^2/(1 + y) we have log^2(1 + y) = Sum_{n >= 0} (-1)^n*x^(n+1)/a(n). See Shenton and Kemp.
Series reversion ( Sum_{n >= 0} (-1)^n*x^(n+1)/a(n) ) = Sum_{n >= 1} 2*x^n/(2*n)! = Sum_{n >= 1} x^n/ A002674(n). (End) D-finite with recurrence n^2*a(n) -2*(n+1)*(2*n+1)*a(n-1)=0. - R. J. Mathar, Feb 08 2021
MAPLE
seq((n+1)^2*(binomial(2*n+2, n+1))/2, n=0..29); # Zerinvary Lajos, May 31 2006
MATHEMATICA
Table[Binomial[2n+1, n](n+1)^2, {n, 0, 20}] (* Harvey P. Dale, Mar 23 2011 *)
PROG
(PARI) a(n)=binomial(2*n+1, n)*(n+1)^2
(PARI) x='x+O('x^99); Vec((1+2*x)/(1-4*x)^(5/2)) \\ Altug Alkan, Jul 09 2016 (Python)
from sympy import binomial
def a(n): return binomial(2*n + 1, n)*(n + 1)**2 # Indranil Ghosh, Apr 18 2017
Denominators of central difference coefficients M_{3}^(2n+1).
(Formerly M4588 N1957)
+10
10
1, 8, 1920, 193536, 154828800, 1167851520, 892705701888000, 1428329123020800, 768472460034048000, 4058540589291090739200, 196433364521688791777280000, 5957759187690780937420800000, 30447485794244997427545243648000000, 341011840895543971188506728857600000
COMMENTS
Denominators in the expansion of (2*sinh(x/2))^3 = x^3 + (1/8)*x^5 + (13/1920)*x^7 + (41/193536)*x^9 + ....
Let f(x) be a polynomial in x. The expansion of (2*sinh(x/2))^3 leads to a formula for the third central differences: f(x+3/2) - 3*f(x+1/2) + 3*f(x-1/2) - f(x-3/2) = (2*sinh(D/2))^3(f(x)) = D^3(f(x)) + (1/8)*D^5(f(x)) + (13/1920)* D^7(f(x)) + ..., where D denotes the differential operator d/dx. (End)
REFERENCES
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).
FORMULA
a(n) = denominator(3! * m(3, 2 * n + 1) / (2 * n + 1)!) where m(k, k) = 1; m(k, q) = 0 for k = 0, k > q, or k + q odd; m(1, q) = 1/2^(q-1) for odd q; m(2, q) = 1 for even q; m(k, q+2) = m(k-2, q) + (k/2)^2 * m(k, q) otherwise. [From Salzer] - Sean A. Irvine, Dec 20 2016
Numerators of coefficients for central differences M_{4}^(2*n).
(Formerly M5035 N2173)
+10
10
1, 1, 1, 17, 31, 1, 5461, 257, 73, 1271, 60787, 241, 22369621, 617093, 49981, 16843009, 5726623061, 7957, 91625968981, 61681, 231927781, 50991843607, 499069107643, 4043309297, 1100586419201, 5664905191661, 1672180312771
COMMENTS
Numerators in the expansion of (2*sinh(x/2))^4 = x^4 + (1/6)*x^6 + (1/80)*x^8 + (17/30240)*x^10 + ....
Let f(x) be a polynomial in x. The expansion of (2*sinh(x/2))^4 leads to a formula for the fourth central differences: f(x+2) - 4*f(x+1) + 6*f(x) - 4*f(x-1) + f(x-2) = (2*sinh(D/2))^4(f(x)) = D^4(f(x)) + (1/6)*D^6(f(x)) + (1/80)* D^8(f(x)) + (17/30240)*D^10(f(x)) + ..., where D denotes the differential operator d/dx. (End)
REFERENCES
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).
MAPLE
gf := (sinh(2*sqrt(x)) - 2*sinh(sqrt(x)))*sqrt(x):
ser := series(gf, x, 40): seq(numer(coeff(ser, x, n)), n=2..28); # Peter Luschny, Oct 05 2019
Denominators of coefficients for central differences M_{3}'^(2*n+1).
(Formerly M3676 N1499)
+10
10
1, 4, 40, 12096, 604800, 760320, 217945728000, 697426329600, 16937496576000, 30964207376793600, 187333454629601280000, 111407096483020800000, 1814811575069725360128000000, 10162944820390462016716800000
REFERENCES
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).
FORMULA
a(n) are the denominators in the expansion of (1/2)*(d/dx)(2*sinh(sqrt(x)/2))^4 =
x + (1/4)*x^2 + (1/40)*x^3 + (17/12096)*x^4 + (31/604800)*x^5 + ...
The a(n) also appear as denominators in the difference formula: (1/2)*f(x+2) - f(x+1) + f(x-1) - (1/2)*f(x-2) = D^3(f(x)) + (1/4)*D^5(f(x)) + (1/40)*D^7(f(x)) + (17/12096)*D^9(f(x)) + ..., where D denotes the differential operator d/dx.
(End)
MAPLE
gf := (sinh(2*sqrt(x)) - 2*sinh(sqrt(x)))/sqrt(x): ser := series(gf, x, 20):
seq(denom(coeff(ser, x, n)), n=1..14); # Peter Luschny, Oct 05 2019
Numerators of central difference coefficients M_{3}^(2n+1).
(Formerly M4894 N2097)
+10
9
1, 1, 13, 41, 671, 73, 597871, 7913, 28009, 792451, 170549237, 19397633, 317733228541, 9860686403, 75397891, 170314355593, 2084647712458321, 29327731093, 168856464709124011, 3063310184201, 499338236699611, 535201577273701757, 23571643935246013553
COMMENTS
Numerators in the expansion of (2*sinh(x/2))^3 = x^3 + (1/8)*x^5 + (13/1920)*x^7 + (41/193536)*x^9 + ....
Let f(x) be a polynomial in x. The expansion of (2*sinh(x/2))^3 leads to a formula for the third central differences: f(x+3/2) - 3*f(x+1/2) + 3*f(x-1/2) - f(x-3/2) = (2*sinh(D/2))^3(f(x)) = D^3(f(x)) + (1/8)*D^5(f(x)) + (13/1920)* D^7(f(x)) + ..., where D denotes the differential operator d/dx. (End)
REFERENCES
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).
T(n, k) = Sum_{j=0..k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n), triangle read by rows, n >= 0 and 0 <= k <= n.
+10
8
1, 0, 1, 0, 1, 12, 0, 1, 60, 360, 0, 1, 252, 5040, 20160, 0, 1, 1020, 52920, 604800, 1814400, 0, 1, 4092, 506880, 12640320, 99792000, 239500800, 0, 1, 16380, 4684680, 230630400, 3632428800, 21794572800, 43589145600, 0, 1, 65532, 42653520, 3952428480
EXAMPLE
Triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 1, 12;
[3] 0, 1, 60, 360;
[4] 0, 1, 252, 5040, 20160;
[5] 0, 1, 1020, 52920, 604800, 1814400;
[6] 0, 1, 4092, 506880, 12640320, 99792000, 239500800;
[7] 0, 1, 16380, 4684680, 230630400, 3632428800, 21794572800, 43589145600;
MAPLE
T := (n, k) -> add((-1)^j*binomial(2*k, j)*(k-j)^(2*n), j=0..k):
for n from 0 to 8 do seq(T(n, k), k=0..n) od;
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