A180874 Lassalle's sequence connected with Catalan numbers and Narayana polynomials.

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

Offset: 1

Author

Jonathan Vos Post, Sep 22 2010

Keywords

• nonn
• easy

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^(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)

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 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)

Links

• G. C. Greubel, Table of n, a(n) for n = 1..250
• Colin Defant, Descents in t-Sorted Permutations, arXiv:1904.02613 [math.CO], 2019.
• Colin Defant, Michael Engen, and Jordan A. Miller, Stack-sorting, 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 Stack-Sorting, arXiv:2004.11367 [math.CO], 2020. See p. 37.
• Matthieu Josuat-Verges, Cumulants of the q-semicircular 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], 2010-2012.
• Michel Lassalle, Two integer sequences related to Catalan numbers, Journal of Combinatorial Theory, Series A, Volume 119, Issue 4, May 2012, Pages 923-935.
• Hanna Mularczyk, Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations, arXiv:1908.04025 [math.CO], 2019.

Crossrefs

Cf. A000108, A001263, A188664, A115369.

Cf. A036040, A097610, A126120, A238390, A263916, A257490.

Programs

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*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

Mathematica

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 *)

Formula

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

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..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