|
%I M2898 N1163 #692 Jul 24 2023 01:53:39
%S 1,1,3,11,45,197,903,4279,20793,103049,518859,2646723,13648869,
%T 71039373,372693519,1968801519,10463578353,55909013009,300159426963,
%U 1618362158587,8759309660445,47574827600981,259215937709463,1416461675464871
%N Schroeder's second problem (generalized parentheses); also called super-Catalan numbers or little Schroeder numbers.
%C If you are looking for the Schroeder numbers (a.k.a. large Schroder numbers, or big Schroeder numbers), see A006318.
%C Yang & Jiang (2021) call these the small 2-Schroeder numbers. - _N. J. A. Sloane_, Mar 28 2021
%C There are two schools of thought about the index for the first term. I prefer the indexing a(0) = a(1) = 1, a(2) = 3, a(3) = 11, etc.
%C a(n) is the number of ways to insert parentheses in a string of n+1 symbols. The parentheses must be balanced but there is no restriction on the number of pairs of parentheses. The number of letters inside a pair of parentheses must be at least 2. Parentheses enclosing the whole string are ignored.
%C Also length of list produced by a variant of the Catalan producing iteration: replace each integer k with the list 0,1,..,k,k+1,k,...,1,0 and get the length a(n) of the resulting (flattened) list after n iterations. - _Wouter Meeussen_, Nov 11 2001
%C Stanley gives several other interpretations for these numbers.
%C Number of Schroeder paths of semilength n (i.e., lattice paths from (0,0) to (2n,0), with steps H=(2,0), U=(1,1) and D(1,-1) and not going below the x-axis) with no peaks at level 1. Example: a(2)=3 because among the six Schroeder paths of semilength two HH, UHD, UUDD, HUD, UDH and UDUD, only the first three have no peaks at level 1. - _Emeric Deutsch_, Dec 27 2003
%C a(n+1) is the number of Dyck n-paths in which the interior vertices of the ascents are colored white or black. - _David Callan_, Mar 14 2004
%C Number of possible schedules for n time slots in the first-come first-served (FCFS) printer policy.
%C Also row sums of A086810, A033282, A126216. - _Philippe Deléham_, May 09 2004
%C a(n+1) is the number of pairs (u,v) of same-length compositions of n, 0's allowed in u but not in v and u dominates v (meaning u_1 >= v_1, u_1 + u_2 >= v_1 + v_2 and so on). For example, with n=2, a(3) counts (2,2), (1+1,1+1), (2+0,1+1). - _David Callan_, Jul 20 2005
%C The big Schroeder number (A006318) is the number of Schroeder paths from (0,0) to (n,n) (subdiagonal paths with steps (1,0) (0,1) and (1,1)). These paths fall in two classes: those with steps on the main diagonal and those without. These two classes are equinumerous and the number of paths in either class is the little Schroeder number a(n) (half the big Schroeder number). - Marcelo Aguiar (maguiar(AT)math.tamu.edu), Oct 14 2005
%C With offset 0, a(n) = number of (colored) Motzkin (n-1)-paths with each upstep U getting one of 2 colors and each flatstep F getting one of 3 colors. Example. With their colors immediately following upsteps/flatsteps, a(2) = 3 counts F1, F2, F3 and a(3)=11 counts U1D, U2D, F1F1, F1F2, F1F3, F2F1, F2F2, F2F3, F3F1, F3F2, F3F3. - _David Callan_, Aug 16 2006
%C Shifts left when INVERT transform applied twice. - _Alois P. Heinz_, Apr 01 2009
%C Number of increasing tableaux of shape (n,n). An increasing tableau is a semistandard tableaux with strictly increasing rows and columns, and set of entries an initial segment of the positive integers. Example: a(2) = 3 because of the three tableaux (12)(34), (13)(24), (12)(23). - _Oliver Pechenik_, Apr 22 2014
%C Number of ordered trees with no vertex of outdegree 1 and having n+1 leaves (called sometimes Schröder trees). - _Emeric Deutsch_, Dec 13 2014
%C Number of dissections of a convex (n+2)-gon by nonintersecting diagonals. Example: a(2)=3 because for a square ABCD we have (i) no diagonal, (ii) dissection with diagonal AC, and (iii) dissection with diagonal BD. - _Emeric Deutsch_, Dec 13 2014
%C The little Schroeder numbers are the moments of the Marchenko-Pastur law for the case c=2 (although the moment m0 is 1/2 instead of 1): 1/2, 1, 3, 11, 45, 197, 903, ... - _Jose-Javier Martinez_, Apr 07 2015
%C Number of generalized Motzkin paths with no level steps at height 0, from (0,0) to (2n,0), and consisting of steps U=(1,1), D=(1,-1) and H2=(2,0). For example, for n=3, we have the 11 paths: UDUDUD, UUDDUD, UDUUDD, UH2DUD, UDUH2D, UH2H2D, UUDUDD, UUUDDD, UUH2DD, UUDH2D, UH2UDD. - _José Luis Ramírez Ramírez_, Apr 20 2015
%C REVERT transform of A225883. - _Vladimir Reshetnikov_, Oct 25 2015
%C Total number of (nonempty) faces of all dimensions in the associahedron K_{n+1} of dimension n-1. For example, K_4 (a pentagon) includes 5 vertices and 5 edges together with K_4 itself (5 + 5 + 1 = 11), while K_5 includes 14 vertices, 21 edges and 9 faces together with K_5 itself (14 + 21 + 9 + 1 = 45). - _Noam Zeilberger_, Sep 17 2018
%C a(n) is the number of interval posets of permutations with n minimal elements that have exactly two realizers, up to a shift by 1 in a(4). See M. Bouvel, L. Cioni, B. Izart, Theorem 17 page 13. - _Mathilde Bouvel_, Oct 21 2021
%C a(n) is the number of sequences of nonnegative integers (u_1, u_2, ..., u_n) such that (i) u_1 = 1, (ii) u_i <= i for all i, (iii) the nonzero u_i are weakly increasing. For example, a(2) = 3 counts 10, 11, 12, and a(3) = 11 counts 100, 101, 102, 103, 110, 111, 112, 113, 120, 122, 123. See link below. - _David Callan_, Dec 19 2021
%C a(n) is the number of parking functions of size n avoiding the patterns 132 and 213. - _Lara Pudwell_, Apr 10 2023
%C a(n+1) is the number of Schroeder paths from (0,0) to (2n,0) in which level steps at height 0 come in 2 colors. - _Alexander Burstein_, Jul 23 2023
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%H Olivier Gérard, <a href="/A001003/a001003a.pdf">Illustration of initial terms (a)</a>
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%H <a href="/index/Par#parens">Index entries for sequences related to parenthesizing</a>
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%F D-finite with recurrence: (n+1) * a(n) = (6*n-3) * a(n-1) - (n-2) * a(n-2) if n>1. a(0) = a(1) = 1.
%F a(n) = 3*a(n-1) + 2*A065096(n-2) (n>2). If g(x) = 1 + 3x + 11x^2 + 45x^3 + ... + a(n)*x^n + ..., then g(x) = 1 + 3(x*g(x)) + 2(x*g(x))^2, g(x)^2 = 1 + 6x + 31x^2 + 156x^3 + ... + A065096(n)*x^n + ... - _Paul D. Hanna_, Jun 10 2002
%F a(n+1) = -a(n) + 2*Sum_{k=1..n} a(k)*a(n+1-k). - _Philippe Deléham_, Jan 27 2004
%F a(n-2) = (1/(n-1))*Sum_{k=0..n-3} binomial(n-1,k+1)*binomial(n-3,k)*2^(n-3-k) for n >= 3 [G. Polya, Elemente de Math., 12 (1957), p. 115.] - _N. J. A. Sloane_, Jun 13 2015
%F G.f.: (1 + x - sqrt(1 - 6*x + x^2) )/(4*x) = 2/(1 + x + sqrt(1 - 6*x + x^2)).
%F a(n) ~ W*(3+sqrt(8))^n*n^(-3/2) where W = (1/4)*sqrt((sqrt(18)-4)/Pi) [See Knuth I, p. 534, or Perez. Note that the formula on line 3, page 475 of Flajolet and Sedgewick seems to be wrong - it has to be multiplied by 2^(1/4).] - _N. J. A. Sloane_, Apr 10 2011
%F The correct asymptotic for A001003 is a(n) ~ W*(3+sqrt(8))^n*n^(-3/2), where W = (1+sqrt(2))/(2*2^(3/4)*sqrt(Pi)) = 0.404947065905750651736243... Result in book by D. Knuth (p. 539 of 3rd edition, exercise 12) is for sequence b(n), but a(n) = b(n+1)/2. Therefore is asymptotic a(n) ~ b(n) * (3+sqrt(8))/2. - _Vaclav Kotesovec_, Sep 09 2012
%F The Hankel transform of this sequence gives A006125 = 1, 1, 2, 8, 64, 1024, ...; example: det([1, 1, 3, 11; 1, 3, 11, 45; 3, 11, 45, 197; 11, 45, 197, 903]) = 2^6 = 64. - _Philippe Deléham_, Mar 02 2004
%F a(n+1) = Sum_{k=0..floor((n-1)/2)} 2^k * 3^(n-1-2k) * binomial(n-1, 2k) * Catalan(k). This formula counts colored Dyck paths by the same parameter by which Touchard's identity counts ordinary Dyck paths: number of DDUs (U=up step, D=down step). See also Gouyou-Beauchamps reference. - _David Callan_, Mar 14 2004
%F a(n) = (1/(n+1))*Sum_{k=0..n} C(n+1, k)*C(2n-k, n)*(-1)^k*2^(n-k) [with offset 0]; a(n) = (1/(n+1))*Sum_{k=0..n} C(n+1, k+1)*C(n+k, k)*(-1)^(n-k)*2^k [with offset 0]; a(n) = Sum_{k=0..n} (1/(k+1))*C(n, k)*C(n+k, k)*(-1)^(n-k)*2^k [with offset 0]; a(n) = Sum_{k=0..n} A088617(n, k)*(-1)^(n-k)*2^k [with offset 0]. - _Paul Barry_, May 24 2005
%F E.g.f. of a(n+1) is exp(3*x)*BesselI(1, 2*sqrt(2)*x)/(sqrt(2)*x). - _Vladeta Jovovic_, Mar 31 2004
%F Reversion of (x-2*x^2)/(1-x) is g.f. offset 1.
%F For n>=1, a(n) = Sum_{k=0..n} 2^k*N(n, k) where N(n, k) = (1/n)*C(n, k)*C(n, k+1) are the Narayana numbers (A001263). - _Benoit Cloitre_, May 10 2003 [This formula counts colored Dyck paths by number of peaks, which is easy to see because the Narayana numbers count Dyck paths by number of peaks and the number of peaks determines the number of interior ascent vertices.]
%F a(n) = Sum_{k=0..n} A088617(n, k)*2^k*(-1)^(n-k). - _Philippe Deléham_, Jan 21 2004
%F For n > 0, a(n) = (1/(n+1)) * Sum_{k = 0 .. n-1} binomial(2*n-k, n)*binomial(n-1, k). This formula counts colored Dyck paths (as above) by number of white vertices. - _David Callan_, Mar 14 2004
%F a(n-1) = (d^(n-1)/dx^(n-1))((1-x)/(1-2*x))^n/n!|_{x=0}. (For a proof see the comment on the unsigned row sums of triangle A111785.)
%F a(n) = (1/n)*Sum_{k=1..n} binomial(n, k)*binomial(n+k, k-1) = hypergeom([1-n, n+2], [2], -1), n>=1. - _Wolfdieter Lang_, Sep 12 2005
%F a(n) = hypergeom([1-n, -n], [2], 2) for n>=0. - _Peter Luschny_, Sep 22 2014
%F a(m+n+1) = Sum_{k>=0} A110440(m, k)*A110440(n, k)*2^k = A110440(m+n, 0). - _Philippe Deléham_, Sep 14 2005
%F Sum over partitions formula (reference Schroeder paper p. 362, eq. (1) II). Number the partitions of n according to Abramowitz-Stegun pp. 831-832 (see reference under A105805) with k=1..p(n)= A000041(n). For n>=1: a(n-1) = Sum_{k=2..p(n)} A048996(n,k)*a(1)^e(k, 1)*a(1)^e(k, 2)*...*a(n-2)^e(k, n-1) if the k-th partition of n in the mentioned order is written as (1^e(k, 1), 2^e(k, 2), ..., (n-1)e(k, n-1)). Note that the first (k=1) partition (n^1) has to be omitted. - _Wolfdieter Lang_, Aug 23 2005
%F Starting (1, 3, 11, 45, ...), = row sums of triangle A126216 = A001263 * [1, 2, 4, 8, 16, ...]. - _Gary W. Adamson_, Nov 30 2007
%F From _Paul Barry_, May 15 2009: (Start)
%F G.f.: 1/(1+x-2x/(1+x-2x/(1+x-2x/(1+x-2x/(1-.... (continued fraction).
%F G.f.: 1/(1-x/(1-x-x/(1-x-x/(1-x-x/(1-... (continued fraction).
%F G.f.: 1/(1-x-2x^2/(1-3x-2x^2/(1-3x-2x^2/(1-... (continued fraction). (End)
%F G.f.: 1 / (1 - x / (1 - 2*x / (1 - x / (1 - 2*x / ... )))). - _Michael Somos_, May 19 2013
%F a(n) = (LegendreP(n+1,3)-3*LegendreP(n,3))/(4*n) for n>0. - _Mark van Hoeij_, Jul 12 2010 [This formula is mentioned in S.-J. Kettle's 1982 letter - see link. _N. J. A. Sloane_, Jun 13 2015]
%F From _Gary W. Adamson_, Jul 08 2011: (Start)
%F a(n) = upper left term in M^n, where M is the production matrix:
%F 1, 1, 0, 0, 0, 0, ...
%F 2, 2, 2, 0, 0, 0, ...
%F 1, 1, 1, 1, 0, 0, ...
%F 2, 2, 2, 2, 2, 0, ...
%F 1, 1, 1, 1, 1, 1, ...
%F ... (End)
%F From _Gary W. Adamson_, Aug 23 2011: (Start)
%F a(n) is the sum of top row terms of Q^(n-1), where Q is the infinite square production matrix:
%F 1, 2, 0, 0, 0, ...
%F 1, 1, 2, 0, 0, ...
%F 1, 1, 1, 2, 0, ...
%F 1, 1, 1, 1, 2, ...
%F ... (End)
%F Let h(t) = (1-t)^2/(2*(1-t)^2-1) = 1/(1-(2*t+3*t^2+4*t^3+...)), an o.g.f. for A003480, then for A001003 a(n) = (1/n!)*((h(t)*d/dt)^n) t, evaluated at t=0, with initial n=1. (Cf. A086810.) - _Tom Copeland_, Sep 06 2011
%F A006318(n) = 2*a(n) if n>0. - _Michael Somos_, Mar 31 2007
%F BINOMIAL transform is A118376 with offset 0. REVERT transform is A153881. INVERT transform is A006318. INVERT transform of A114710. HANKEL transform is A139685. PSUM transform is A104858. - _Michael Somos_, May 19 2013
%F G.f.: 1 + x/(Q(0) - x) where Q(k) = 1 + k*(1-x) - x - x*(k+1)*(k+2)/Q(k+1) ; (continued fraction). - _Sergei N. Gladkovskii_, Mar 14 2013
%F a(n) = A144944(n,n) = A186826(n,0). - _Reinhard Zumkeller_, May 11 2013
%F a(n)=(-1)^n*LegendreP(n,-1,-3)/sqrt(2), n > 0, LegendreP(n,a,b) is the Legendre function. - _Karol A. Penson_, Jul 06 2013
%F Integral representation as n-th moment of a positive weight function W(x) = W_a(x) + W_c(x), where W_a(x) = Dirac(x)/2, is the discrete (atomic) part, and W_c(x) = sqrt(8-(x-3)^2)/(4*Pi*x) is the continuous part of W(x) defined on (3 sqrt(8),3+sqrt(8)): a(n) = int( x^n*W_a(x), x=-eps..eps ) + int( x^n*W_c(x), x = 3-sqrt(8)..3+sqrt(8) ), for any eps>0, n>=0. W_c(x) is unimodal, of bounded variation and W_c(3-sqrt(8)) = W_c(3+sqrt(8)) = 0. Note that the position of the Dirac peak (x=0) lies outside support of W_c(x). - _Karol A. Penson_ and Wojciech Mlotkowski, Aug 05 2013
%F G.f.: 1 + x/G(x) with G(x) = 1 - 3*x - 2*x^2/G(x) (continued fraction). - _Nikolaos Pantelidis_, Dec 17 2022
%e G.f. = 1 + x + 3*x^2 + 11*x^3 + 45*x^4 + 197*x^5 + 903*x^6 + 4279*x^7 + ...
%e a(2) = 3: abc, a(bc), (ab)c; a(3) = 11: abcd, (ab)cd, a(bc)d, ab(cd), (ab)(cd), a(bcd), a(b(cd)), a((bc)d), (abc)d, (a(bc))d, ((ab)c)d.
%e Sum over partitions formula: a(3) = 2*a(0)*a(2) + 1*a(1)^2 + 3*(a(0)^2)*a(1) + 1*a(0)^4 = 6 + 1 + 3 + 1 = 11.
%e a(4) = 45 since the top row of Q^3 = (11, 14, 12, 8, 0, 0, 0, ...); (11 + 14 + 12 + 8) = 45.
%p t1 := (1/(4*x))*(1+x-sqrt(1-6*x+x^2)); series(t1,x,40);
%p invtr:= proc(p) local b; b:= proc(n) option remember; local i; `if`(n<1, 1, add(b(n-i) *p(i-1), i=1..n+1)) end end: a:='a': f:= (invtr@@2)(a): a:= proc(n) if n<0 then 1 else f(n-1) fi end: seq(a(n), n=0..30); # _Alois P. Heinz_, Apr 01 2009
%p # Computes n -> [a[0],a[1],..,a[n]]
%p A001003_list := proc(n) local j,a,w; a := array(0..n); a[0] := 1;
%p for w from 1 to n do a[w] := a[w-1]+2*add(a[j]*a[w-j-1],j=1..w-1) od;
%p convert(a,list) end: A001003_list(100); # _Peter Luschny_, May 17 2011
%t Table[Length[Flatten[Nest[ #/.a_Integer:> Join[Range[0, a + 1], Range[a, 0, -1]] &, {0}, n]]], {n, 0, 10}]; Sch[ 0 ] = Sch[ 1 ] = 1; Sch[ n_Integer ] := Sch[ n ] = (3(2n - 1)Sch[ n - 1 ] - (n - 2)Sch[ n - 2 ])/(n + 1); Array[ Sch, 24, 0]
%t (* Second program: *)
%t a[n_] := Hypergeometric2F1[-n + 1, n + 2, 2, -1]; a[0] = 1; Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Nov 09 2011, after _Vladeta Jovovic_ *)
%t a[ n_] := SeriesCoefficient[ (1 + x - Sqrt[1 - 6 x + x^2]) / (4 x), {x, 0, n}]; (* _Michael Somos_, Aug 26 2015 *)
%t Table[(KroneckerDelta[n] - GegenbauerC[n+1, -1/2, 3])/4, {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 25 2015 *)
%t a[n_] := -LegendreP[n, -1, 2, 3] I / Sqrt[2]; a[0] = 1;
%t Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Feb 16 2019 *)
%o (PARI) {a(n) = if( n<1, n==0, sum( k=0, n, 2^k * binomial(n, k) * binomial(n, k-1) ) / (2*n))}; /* _Michael Somos_, Mar 31 2007 */
%o (PARI) {a(n) = my(A); if( n<1, n==0, n--; A = x * O(x^n); n! * simplify( polcoef( exp(3*x + A) * besseli(1, 2*x * quadgen(8) + A), n)))}; /* _Michael Somos_, Mar 31 2007 */
%o (PARI) {a(n) = if( n<0, 0, n++; polcoef( serreverse( (x - 2*x^2) / (1 - x) + x * O(x^n)), n))}; /* _Michael Somos_, Mar 31 2007 */
%o (PARI) N=30; x='x+O('x^N); Vec( (1+x-(1-6*x+x^2)^(1/2))/(4*x) ) \\ _Hugo Pfoertner_, Nov 19 2018
%o (Python) # The objective of this implementation is efficiency.
%o # n -> [a(0), a(1), ..., a(n)]
%o def A001003_list(n):
%o a = [0 for i in range(n)]
%o a[0] = 1
%o for w in range(1, n):
%o s = 0
%o for j in range(1, w):
%o s += a[j]*a[w-j-1]
%o a[w] = a[w-1]+2*s
%o return a
%o # _Peter Luschny_, May 17 2011
%o (Sage) # Generalized algorithm of L. Seidel
%o def A001003_list(n) :
%o D = [0]*(n+1); D[1] = 1/2
%o b = True; h = 2; R = [1]
%o for i in range(2*n-2) :
%o if b :
%o for k in range(h,0,-1) : D[k] += D[k-1]
%o h += 1;
%o else :
%o for k in range(1,h, 1) : D[k] += D[k-1]
%o R.append(D[h-1]);
%o b = not b
%o return R
%o A001003_list(24) # _Peter Luschny_, Jun 02 2012
%o (Haskell)
%o a001003 = last . a144944_row -- _Reinhard Zumkeller_, May 11 2013
%o (Python)
%o from gmpy2 import divexact
%o A001003 = [1, 1]
%o for n in range(3,10**3):
%o A001003.append(divexact(A001003[-1]*(6*n-9)-(n-3)*A001003[-2],n))
%o # _Chai Wah Wu_, Sep 01 2014
%Y See A000108, A001190, A001699, A000081 for other ways to count parentheses.
%Y Cf. A000311, A006318 (Schroeder numbers), A010683, A065096.
%Y Row sums of A033282, A033877.
%Y Cf. A054726, A059435, A025240, A080243, A085403, A086456, A086616, A035011.
%Y Right-hand column 1 of triangle A011117.
%Y See also convolution triangle A011117.
%Y The sequences listed in Yang-Jiang's Table 1 appear to be A006318, A001003, A027307, A034015, A144097, A243675, A260332, A243676. - _N. J. A. Sloane_, Mar 28 2021
%Y Cf. A144156.
%K nonn,easy,nice,core
%O 0,3
%A _N. J. A. Sloane_
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