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A180874
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Lassalle's sequence connected with Catalan numbers and Narayana polynomials.
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12
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1, 1, 5, 56, 1092, 32670, 1387815, 79389310, 5882844968, 548129834616, 62720089624920, 8646340208462880, 1413380381699497200, 270316008395632253340, 59800308109377016336155, 15151722444639718679892150, 4359147487054262623576455600
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OFFSET
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1,3
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COMMENTS
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Defined by the recurrence formula in Theorem 1, page 2 of Lasalle.
Let G(t) = Sum_{n>=0} t^(2n)/(n!(n+1)!) = exp(c.t) be the e.g.f. of the aerated Catalan numbers c_n of A126120.
R = x + H(D) = x + d/dD log[G(D)] = x + D - D^3/3! + 5 D^5/5! - 56 D^7/7! + ... = x + e^(r. D) generates a signed, aerated version of this entry's sequence a(n), (r.)^(2n+1) = r(2n+1) = (-1)^n a(n+1) for n>=0 and r(0) = a(0) = 0, and is, with D = d/dx, the raising operator for the Appell polynomials P(n,x) of A097610, where P(n,x) = (c. + x)^n = Sum{k=0 to n} binomial(n,k) c_k x^(n-k) with c_k = A126120(k), i.e., R P(n,x) = P(n+1,x).
d/dt log[G(t)] = e^(r.t) = e^(q.t) / e^(c.t) = Ev[c. e^(c.t)] / Ev[e^(c.t)] = e^(q.t) e^(d.t) = [Sum_{n>=0} 2n t^(2n-1)/(n!(n+1)!)] / [Sum_{n>=0} t^(2n)/(n!(n+1)!)] with Ev[..] denoting umbral evaluation, so q(n) = c(n+1) = A126120(n+1) and d(2n) = (-1)^n A238390(n) and vanishes otherwise. Then (r. + c.)^n = q(n) = Sum_{k=0..n} binomial(n,k) r(k) c(n-k) and (q. + d.)^n = r(n), relating A180874, A126120 (A000108), and A238390 through binomial convolutions.
The sequence can also be represented in terms of the Faber polynomials of A263916 as a(n) = |(2n-1)! F(2n,0,b(2),0,b(4),0,..)| = |h(2n)| where b(2n) = 1/(n!(n + 1)!) = A126120(2n)/(2n)! = A000108(n)/(2n)!, giving h(0) = 1, h(1) = 0, h(2) = 1, h(3) = 0, h(4) = -1, h(5) = 0, h(6) = 5, h(7) = 0, h(8) = -56, ..., implying, among other relations, that A000108(n/2)= A126120(n) = Bell(n,0,h(2),0,h(4),...), the Bell polynomials of A036040 which reduce to A257490 in this case.
(End)
a(n) is the number of pairs (rho,r), where rho is a matching on [2n] and r is an acyclic orientation of the crossing graph of rho in which the block containing 1 is the only source (see the Josuat-Verges paper or the Defant-Engen-Miller paper for definitions).
a(n) is the number of permutations of [2n-1] that have exactly 1 preimage under West's stack-sorting map.
a(n) is the number of valid hook configurations of permutations of [2n-1] that have n-1 hooks (see the paper by Defant, Engen, and Miller for definitions).
Say a binary tree is full if every vertex has either 0 or 2 children. If u is a left child in such a tree, then we can start at the sibling of u and travel down left edges until reaching a leaf v. Call v the leftmost nephew of u. A decreasing binary plane tree on [m] is a binary plane tree labeled with the elements of [m] in which every nonroot vertex has a label that is smaller than the label of its parent. a(n) is the number of full decreasing binary plane trees on [2n-1] in which every left child has a label that is larger than the label of its leftmost nephew.
(End)
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LINKS
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FORMULA
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a(n) = (-1)^(n-1) * (C(n)+Sum_{j=1..n-1} (-1)^j *binomial(2n-1,2j-1) * a(j) *C(n-j)), where C() = A000108(). - R. J. Mathar, Apr 17 2011, corrected by Vaclav Kotesovec, Feb 28 2014
a(n) ~ (n!)^2 / (sqrt(Pi) * n^(3/2) * r^n), where r = BesselJZero[1, 1]^2/16 = 0.917623165132743328576236110539381686855099186384686... - Vaclav Kotesovec, added Feb 28 2014, updated Mar 01 2014
Define E(m,n) by E(1,1) = 1, E(n,n) = 0 for n > 1, and E(m,n) = Sum_{j=1..m} Sum_{i=1..n-m-1} binomial(n-m-1,i-1) * F_j(i+j-1) * F_{m-j}(n-j-i) for 0 <= m < n, where F_m(n) = Sum_{j=m..n} E_j(n). Then a(n) = F_0(2n-1). - Colin Defant, Sep 06 2018
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MAPLE
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A000108 := proc(n) binomial(2*n, n)/(1+n) ; end proc:
A180874 := proc(n) option remember; if n = 1 then 1; else A000108(n)+add((-1)^j*binomial(2*n-1, 2*j-1)*procname(j)*A000108(n-j), j=1..n-1) ; %*(-1)^(n-1) ; end if; end proc: # R. J. Mathar, Apr 16 2011
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MATHEMATICA
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nmax=20; a = ConstantArray[0, nmax]; a[[1]]=1; Do[a[[n]] = (-1)^(n-1)*(Binomial[2*n, n]/(n+1) + Sum[(-1)^j*Binomial[2n-1, 2j-1]*a[[j]]* Binomial[2*(n-j), n-j]/(n-j+1), {j, 1, n-1}]), {n, 2, nmax}]; a (* Vaclav Kotesovec, Feb 28 2014 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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