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A099376 An inverse Chebyshev transform of x^3. 1
0, 1, 4, 14, 48, 165, 572, 2002, 7072, 25194, 90440, 326876, 1188640, 4345965, 15967980, 58929450, 218349120, 811985790, 3029594040, 11338026180, 42550029600, 160094486370, 603784920024, 2282138106804, 8643460269248 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The sequence is 0,0,0,1,0,4,0,14,0,...with zeros restored. Second binomial transform of (-1)^n*A003518(n). Second binomial transform of expansion of x^3c(-x)^8, where c(x) is g.f. of A000108. The g.f. is transformed to x^3 under the Chebyshev transformation A(x)->(1/(1+x^2))A(x/(1+x^2)). For a sequence b(n), this corresponds to taking sum{k=0..floor(n/2), C(n-k,k)(-1)^k*b(n-2k)}, or sum{k=0..n, C((n+k)/2,k)b(k)(-1)^((n-k)/2)(1+(-1)^(n-k))/2}.

Let X_n be the set of all noncrossing set partitions of an n-element set which either do not contain {n-1,n} as a block, or which do not contain the block {n} whenever 1 and n-1 are in the same block. For n>0, (-1)^n*a(n) gives the value of the Möbius function of X_{n+2} ordered by dual refinement between the discrete and the full partition. For example, X_3 is a chain consisting of 3 elements and its Möbius function between least and greatest element therefore takes the value a(1)=0. - Henri Mühle, Jan 10 2017

LINKS

Table of n, a(n) for n=0..24.

H. Mühle, Two Posets of Noncrossing Partitions Coming From Undesired Parking Spaces, arXiv:1701.02109 [math.CO], 2017.

FORMULA

G.f.: (1-2x)^4(sqrt((1+2x)/(1-2x))-1)^8/(256x^5); a(n)=sum{k=0..n, (k+1)C(n, (n-k)/2)(-1)^k(C(3, k)-3C(2, k)+3C(1, k)-C(0, k))(1+(-1)^(n-k))/(n+k+2)}.

a(n)=A002057(n-1). - Michael Somos, Jul 31 2005

Given an ellipse with eccentricity e and major and minor axis a and b respectively, then ((a-b)/ (a+b))^2 = 1*(e/2)^4 +4*(e/2)^6 +14*(e/2)^8 +48*(e/2)^10 + ... - Michael Somos, Apr 11 2007

E.g.f.: exp(2x)(Bessel_I(1,2x)-Bessel_I(3,2x)). - Paul Barry, Jun 04 2007

Conjecture: (n+3)*(n-1)*a(n) -2*n*(2*n+1)*a(n-1)=0. - R. J. Mathar, Sep 26 2012

a(n) = A000108(n+1)-2*A000108(n) for n>0. - Henri Mühle, Jan 10 2017

MATHEMATICA

Table[CatalanNumber[n + 2] - 2 CatalanNumber[n + 1], {n, 0, 30}] (* or *)

Table[4 Binomial[2 # + 3, #]/(# + 4) &[n - 1], {n, 0, 30}] (* Michael De Vlieger, Jan 10 2017, latter after Harvey P. Dale at A002057 *)

PROG

(PARI) {a(n)= if(n<1, 0, n++; 2* binomial(2*n, n-2)/n)} /* Michael Somos, Apr 11 2007 */

CROSSREFS

Cf. A000108, A003518, A000245, A071718.

Sequence in context: A092489 A094827 A094667 * A002057 A047048 A071745

Adjacent sequences:  A099373 A099374 A099375 * A099377 A099378 A099379

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Oct 13 2004

STATUS

approved

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Last modified February 19 11:04 EST 2018. Contains 299330 sequences. (Running on oeis4.)