

A180874


Lassalle's sequence connected with Catalan numbers and Narayana polynomials.


12



1, 1, 5, 56, 1092, 32670, 1387815, 79389310, 5882844968, 548129834616, 62720089624920, 8646340208462880, 1413380381699497200, 270316008395632253340, 59800308109377016336155, 15151722444639718679892150, 4359147487054262623576455600
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OFFSET

1,3


COMMENTS

Defined by the recurrence formula in Theorem 1, page 2 of Lasalle.
From Tom Copeland, Jan 26 2016: (Start)
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^(nk) 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^(2n1)/(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(nk) 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) = (2n1)! 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)
From Colin Defant, Sep 06 2018: (Start)
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 JosuatVerges paper or the DefantEngenMiller paper for definitions).
a(n) is the number of permutations of [2n1] that have exactly 1 preimage under West's stacksorting map.
a(n) is the number of valid hook configurations of permutations of [2n1] that have n1 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 [2n1] in which every left child has a label that is larger than the label of its leftmost nephew.
(End)


LINKS

G. C. Greubel, Table of n, a(n) for n = 1..250
Colin Defant, Descents in tSorted Permutations, arXiv:1904.02613 [math.CO], 2019.
Colin Defant, Michael Engen, and Jordan A. Miller, Stacksorting, set partitions, and Lassalle's sequence, arXiv:1809.01340 [math.CO], 2018.
Colin Defant, Catalan Intervals and Uniquely Sorted Permutations, arXiv:1904.02627 [math.CO], 2019.
Colin Defant, Troupes, Cumulants, and StackSorting, arXiv:2004.11367 [math.CO], 2020. See p. 37.
Matthieu JosuatVerges, Cumulants of the qsemicircular law, Tutte polynomials, and heaps, arXiv:1203.3157 [math.CO], 2012.
Michel Lassalle, Catalan numbers and a new integer sequence, arXiv:1009.4225 [math.CO], 20102012.
Michel Lassalle, Two integer sequences related to Catalan numbers, Journal of Combinatorial Theory, Series A, Volume 119, Issue 4, May 2012, Pages 923935.
Hanna Mularczyk, Lattice Paths and PatternAvoiding Uniquely Sorted Permutations, arXiv:1908.04025 [math.CO], 2019.


FORMULA

a(n) = (1)^(n1) * (C(n)+Sum_{j=1..n1} (1)^j *binomial(2n1,2j1) * a(j) *C(nj)), where C() = A000108().  R. J. Mathar, Apr 17 2011, corrected by Vaclav Kotesovec, Feb 28 2014
E.g.f.: Sum_{k>=0} a(k)*x^(2*k+2)/(2*k+2)! = log(x/BesselJ(1,2*x)).  Sergei N. Gladkovskii, Dec 28 2011
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..nm1} binomial(nm1,i1) * F_j(i+j1) * F_{mj}(nji) for 0 <= m < n, where F_m(n) = Sum_{j=m..n} E_j(n). Then a(n) = F_0(2n1).  Colin Defant, Sep 06 2018


MAPLE

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*n1, 2*j1)*procname(j)*A000108(nj), j=1..n1) ; %*(1)^(n1) ; end if; end proc: # R. J. Mathar, Apr 16 2011


MATHEMATICA

nmax=20; a = ConstantArray[0, nmax]; a[[1]]=1; Do[a[[n]] = (1)^(n1)*(Binomial[2*n, n]/(n+1) + Sum[(1)^j*Binomial[2n1, 2j1]*a[[j]]* Binomial[2*(nj), nj]/(nj+1), {j, 1, n1}]), {n, 2, nmax}]; a (* Vaclav Kotesovec, Feb 28 2014 *)


CROSSREFS

Cf. A000108, A001263, A188664, A115369.
Cf. A036040, A097610, A126120, A238390, A263916, A257490.
Sequence in context: A192562 A060080 A203522 * A336243 A163793 A226425
Adjacent sequences: A180871 A180872 A180873 * A180875 A180876 A180877


KEYWORD

nonn,easy


AUTHOR

Jonathan Vos Post, Sep 22 2010


STATUS

approved



