|
%I M1659 #743 Jan 17 2024 01:24:16
%S 1,2,6,22,90,394,1806,8558,41586,206098,1037718,5293446,27297738,
%T 142078746,745387038,3937603038,20927156706,111818026018,600318853926,
%U 3236724317174,17518619320890,95149655201962,518431875418926,2832923350929742,15521467648875090
%N Large Schröder numbers (or large Schroeder numbers, or big Schroeder numbers).
%C For the little Schröder numbers (or little Schroeder numbers, or small Schroeder numbers) see A001003.
%C The number of perfect matchings in a triangular grid of n squares (n = 1, 4, 9, 16, 25, ...). - _Roberto E. Martinez II_, Nov 05 2001
%C a(n) is the number of subdiagonal paths from (0, 0) to (n, n) consisting of steps East (1, 0), North (0, 1) and Northeast (1, 1) (sometimes called royal paths). - _David Callan_, Mar 14 2004
%C Twice A001003 (except for the first term).
%C a(n) is the number of dissections of a regular (n+4)-gon by diagonals that do not touch the base. (A diagonal is a straight line joining two nonconsecutive vertices and dissection means the diagonals are noncrossing though they may share an endpoint. One side of the (n+4)-gon is designated the base.) Example: a(1)=2 because a pentagon has only 2 such dissections: the empty one and the one with a diagonal parallel to the base. - _David Callan_, Aug 02 2004
%C a(n) is the number of separable permutations, i.e., permutations avoiding 2413 and 3142 (see Shapiro and Stephens). - _Vincent Vatter_, Aug 16 2006
%C _Eric W. Weisstein_ comments that the Schröder numbers bear the same relationship to the Delannoy numbers (A001850) as the Catalan numbers (A000108) do to the binomial coefficients. - _Jonathan Vos Post_, Dec 23 2004
%C a(n) is the number of lattice paths from (0, 0) to (n+1, n+1) consisting of unit steps north N = (0, 1) and variable-length steps east E = (k, 0), with k a positive integer, that stay strictly below the line y = x except at the endpoints. For example, a(2) = 6 counts 111NNN, 21NNN, 3NNN, 12NNN, 11N1NN, 2N1NN (east steps indicated by their length). If the word "strictly" is replaced by "weakly", the counting sequence becomes the little Schröder numbers, A001003 (offset). - _David Callan_, Jun 07 2006
%C a(n) is the number of dissections of a regular (n+3)-gon with base AB that do not contain a triangle of the form ABP with BP a diagonal. Example: a(1) = 2 because the square D-C | | A-B has only 2 such dissections: the empty one and the one with the single diagonal AC (although this dissection contains the triangle ABC, BC is not a diagonal). - _David Callan_, Jul 14 2006
%C a(n) is the number of (colored) Motzkin n-paths with each upstep and each flatstep at ground level getting one of 2 colors and each flatstep not at ground level getting one of 3 colors. Example: With their colors immediately following upsteps/flatsteps, a(2) = 6 counts U1D, U2D, F1F1, F1F2, F2F1, F2F2. - _David Callan_, Aug 16 2006
%C The Hankel transform of this sequence is A006125(n+1) = [1, 2, 8, 64, 1024, 32768, ...]; example: Det([1, 2, 6, 22; 2, 6, 22, 90; 6, 22, 90, 394; 22, 90, 394, 1806]) = 64. - _Philippe Deléham_, Sep 03 2006
%C Triangle A144156 has row sums equal to A006318 with left border A001003. - _Gary W. Adamson_, Sep 12 2008
%C a(n) is also the number of order-preserving and order-decreasing partial transformations (of an n-chain). Equivalently, it is the order of the Schröder monoid, PC sub n. - _Abdullahi Umar_, Oct 02 2008
%C Sum_{n >= 0} a(n)/10^n - 1 = (9 - sqrt(41))/2. - _Mark Dols_, Jun 22 2010
%C 1/sqrt(41) = Sum_{n >= 0} Delannoy number(n)/10^n. - _Mark Dols_, Jun 22 2010
%C a(n) is also the dimension of the space Hoch(n) related to Hochschild two-cocycles. - Ph. Leroux (ph_ler_math(AT)yahoo.com), Aug 24 2010
%C Let W = (w(n, k)) denote the augmentation triangle (as at A193091) of A154325; then w(n, n) = A006318(n). - _Clark Kimberling_, Jul 30 2011
%C Conjecture: For each n > 2, the polynomial sum_{k = 0}^n a(k)*x^{n-k} is irreducible modulo some prime p < n*(n+1). - _Zhi-Wei Sun_, Apr 07 2013
%C From _Jon Perry_, May 24 2013: (Start)
%C Consider a Pascal triangle variant where T(n, k) = T(n, k-1) + T(n-1, k-1) + T(n-1, k), i.e., the order of performing the calculation must go from left to right (A033877). This sequence is the rightmost diagonal.
%C Triangle begins:
%C 1;
%C 1, 2;
%C 1, 4, 6;
%C 1, 6, 16, 22;
%C 1, 8, 30, 68, 90;
%C ... (End)
%C a(n) is the number of permutations avoiding 2143, 3142 and one of the patterns among 246135, 254613, 263514, 524361, 546132. - _Alexander Burstein_, Oct 05 2014
%C a(n) is the number of semi-standard Young tableaux of shape n x 2 with consecutive entries. That is, j in P and 1 <= i<= j imply i in P. - _Graham H. Hawkes_, Feb 15 2015
%C a(n) is the number of unary-rooted size n unary-binary trees (each node has either 1 or 2 degree out). - _John Bodeen_, May 29 2017
%C Conjecturally, a(n) is the number of permutations pi of length n such that s(pi) avoids the patterns 231 and 321, where s denotes West's stack-sorting map. - _Colin Defant_, Sep 17 2018
%C a(n) is the number of n X n permutation matrices which percolate under the 2-neighbor bootstrap percolation rule (see Shapiro and Stephens). The number of general n X n matrices of weight n which percolate is given in A146971. - _Jonathan Noel_, Oct 05 2018
%C a(n) is the number of permutations of length n+1 which avoid 3142 and 3241. The permutations are precisely the permutations that are sortable by a decreasing stack followed by an increasing stack in series. - _Rebecca Smith_, Jun 06 2019
%C a(n) is the number of permutations of length n+1 avoiding the partially ordered pattern (POP) {3>1, 4>1, 1>2} of length 4. That is, the number of length n+1 permutations having no subsequences of length 4 in which the second element is the smallest, and the first element is smaller than the third and fourth elements. - _Sergey Kitaev_, Dec 10 2020
%C Named after the German mathematician Ernst Schröder (1841-1902). - _Amiram Eldar_, Apr 15 2021
%C a(n) is the number of sequences of nonnegative integers (u_1, u_2, ..., u_n) such that (i) u_i <= i for all i, and (ii) the nonzero u_i are weakly increasing. For example, a(2) = 6 counts 00, 01, 02, 10, 11, 12. See link "Some bijections for lattice paths" at A001003. - _David Callan_, Dec 18 2021
%C a(n) is the number of separable elements of the Weyl group of type B_n/C_n (see Gaetz and Gao). - _Fern Gossow_, Jul 31 2023
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%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%F G.f.: (1 - x - (1 - 6*x + x^2)^(1/2))/(2*x).
%F a(n) = 2*hypergeom([-n+1, n+2], [2], -1). - _Vladeta Jovovic_, Apr 24 2003
%F For n > 0, a(n) = (1/n)*Sum_{k = 0..n} 2^k*C(n, k)*C(n, k-1). - _Benoit Cloitre_, May 10 2003
%F The g.f. satisfies (1 - x)*A(x) - x*A(x)^2 = 1. - _Ralf Stephan_, Jun 30 2003
%F For the asymptotic behavior, see A001003 (remembering that A006318 = 2*A001003). - _N. J. A. Sloane_, Apr 10 2011
%F From _Philippe Deléham_, Nov 28 2003: (Start)
%F Row sums of A088617 and A060693.
%F a(n) = Sum_{k = 0..n} C(n+k, n)*C(n, k)/(k+1). (End)
%F With offset 1: a(1) = 1, a(n) = a(n-1) + Sum_{i = 1..n-1} a(i)*a(n-i). - _Benoit Cloitre_, Mar 16 2004
%F a(n) = Sum_{k = 0..n} A000108(k)*binomial(n+k, n-k). - _Benoit Cloitre_, May 09 2004
%F a(n) = Sum_{k = 0..n} A011117(n, k). - _Philippe Deléham_, Jul 10 2004
%F a(n) = (CentralDelannoy(n+1) - 3 * CentralDelannoy(n))/(2*n) = (-CentralDelannoy(n+1) + 6 * CentralDelannoy(n) - CentralDelannoy(n-1))/2 for n >= 1, where CentralDelannoy is A001850. - _David Callan_, Aug 16 2006
%F From _Abdullahi Umar_, Oct 11 2008: (Start)
%F A123164(n+1) - A123164(n) = (2*n+1)*a(n) (n >= 0).
%F and 2*A123164(n) = (n+1)*a(n) - (n-1)*a(n-1) (n > 0). (End)
%F Define the general Delannoy numbers d(i, j) as in A001850. Then a(k) = d(2*k, k) - d(2*k, k-1) and a(0) = 1, Sum_{j=0..n} ((-1)^j * (d(n, j) + d(n-1, j-1)) * a(n-j)) = 0. - _Peter E John_, Oct 19 2006
%F Given an integer t >= 1 and initial values u = [a_0, a_1, ..., a_{t-1}], we may define an infinite sequence Phi(u) by setting a_n = a_{n-1} + a_0*a_{n-1} + a_1*a_{n-2} + ... + a_{n-2}*a_1 for n >= t. For example, Phi([1]) is the Catalan numbers A000108. The present sequence is (essentially) Phi([2]). - _Gary W. Adamson_, Oct 27 2008
%F G.f.: 1/(1-2x/(1-x/(1-2x/(1-x/(1-2x/(1-x/(1-2x/(1-x/(1-2x/(1-x.... (continued fraction). - _Paul Barry_, Dec 08 2008
%F G.f.: 1/(1 - x - x/(1 - x - x/(1 - x - x/(1 - x - x/(1 - x - x/(1 - ... (continued fraction). - _Paul Barry_, Jan 29 2009
%F a(n) ~ ((3 + 2*sqrt(2))^n)/(n*sqrt(2*Pi*n)*sqrt(3*sqrt(2) - 4))*(1-(9*sqrt(2) + 24)/(32*n) + ...). - G. Nemes (nemesgery(AT)gmail.com), Jan 25 2009
%F Logarithmic derivative yields A002003. - _Paul D. Hanna_, Oct 25 2010
%F a(n) = the upper left term in M^(n+1), M = the production matrix:
%F 1, 1, 0, 0, 0, 0, ...
%F 1, 1, 1, 0, 0, 0, ...
%F 2, 2, 1, 1, 0, 0, ...
%F 4, 4, 2, 1, 1, 0, ...
%F 8, 8, 8, 2, 1, 1, ...
%F ... - _Gary W. Adamson_, Jul 08 2011
%F a(n) is the sum of top row terms in Q^n, Q = an infinite square production matrix as follows:
%F 1, 1, 0, 0, 0, 0, ...
%F 1, 1, 2, 0, 0, 0, ...
%F 1, 1, 1, 2, 0, 0, ...
%F 1, 1, 1, 1, 2, 0, ...
%F 1, 1, 1, 1, 1, 2, ...
%F ... - _Gary W. Adamson_, Aug 23 2011
%F From _Tom Copeland_, Sep 21 2011: (Start)
%F With F(x) = (1 - 3*x - sqrt(1 - 6*x + x^2))/(2*x) an o.g.f. (nulling the n = 0 term) for A006318, G(x) = x/(2 + 3*x + x^2) is the compositional inverse.
%F Consequently, with H(x) = 1/ (dG(x)/dx) = (2 + 3*x + x^2)^2 / (2 - x^2),
%F a(n) = (1/n!)*[(H(x)*d/dx)^n] x evaluated at x = 0, i.e.,
%F F(x) = exp[x*H(u)*d/du] u, evaluated at u = 0. Also, dF(x)/dx = H(F(x)). (End)
%F a(n-1) = number of ordered complete binary trees with n leaves having k internal vertices colored black, the remaining n - 1 - k internal vertices colored white, and such that each vertex and its rightmost child have different colors ([Drake, Example 1.6.7]). For a refinement of this sequence see A175124. - _Peter Bala_, Sep 29 2011
%F D-finite with recurrence: (n-2)*a(n-2) - 3*(2*n-1)*a(n-1) + (n+1)*a(n) = 0. - _Vaclav Kotesovec_, Oct 05 2012
%F G.f.: A(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) = (1 - G(0))/x; G(k) = 1 + x - 2*x/G(k+1); (continued fraction, 1-step). - _Sergei N. Gladkovskii_, Jan 04 2012
%F G.f.: A(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) = (G(0) - 1)/x; G(k) = 1 - x/(1 - 2/G(k+1)); (continued fraction, 2-step). - _Sergei N. Gladkovskii_, Jan 04 2012
%F a(n+1) = a(n) + Sum_{k=0..n} a(k)*(n-k). - _Reinhard Zumkeller_, Nov 13 2012
%F G.f.: 1/Q(0) 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(-1-n) = a(n). - _Michael Somos_, Apr 03 2013
%F G.f.: 1/x - 1 - U(0)/x, where U(k) = 1 - x - x/U(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Jul 16 2013
%F G.f.: (2 - 2*x - G(0))/(4*x), where G(k) = 1 + 1/( 1 - x*(6 - x)*(2*k - 1)/(x*(6 - x)*(2*k - 1) + 2*(k + 1)/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 16 2013
%F a(n) = 1/(n + 1) (Sum_{j=0..n} C(n+j, j)*C(n+j+1, j+1)*(Sum_{k=0..n-j} (-1)^k*C(n+j+k, k))). - _Graham H. Hawkes_, Feb 15 2015
%F a(n) = hypergeom([-n, n+1], [2], -1). - _Peter Luschny_, Mar 23 2015
%F a(n) = sqrt(2) * LegendreP(n, -1, 3) where LegendreP is the associated Legendre function of the first kind (in Maple's notation). - _Robert Israel_, Mar 23 2015
%F G.f. A(x) satisfies: A(x) = Sum_{j>=0} x^j * Sum_{k=0..j} binomial(j,k)*A(x)^k. - _Ilya Gutkovskiy_, Apr 11 2019
%e a(3) = 22 since the top row of Q^n = (6, 6, 6, 4, 0, 0, 0, ...); where 22 = (6 + 6 + 6 + 4).
%e G.f. = 1 + 2*x + 6*x^2 + 22*x^3 + 90*x^4 + 394*x^5 + 1806*x^6 + 8858*x^7 + 41586*x^8 + ...
%p Order := 24: solve(series((y-y^2)/(1+y),y)=x,y); # then A(x)=y(x)/x
%p BB:=(-1-z-sqrt(1-6*z+z^2))/2: BBser:=series(BB, z=0, 24): seq(coeff(BBser, z, n), n=1..23); # _Zerinvary Lajos_, Apr 10 2007
%p A006318_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
%p for w from 1 to n do a[w] := 2*a[w-1]+add(a[j]*a[w-j-1], j=1..w-1) od; convert(a,list)end: A006318_list(22); # _Peter Luschny_, May 19 2011
%p A006318 := n-> add(binomial(n+k, n-k) * binomial(2*k, k)/(k+1), k=0..n): seq(A006318(n), n=0..22); # _Johannes W. Meijer_, Jul 14 2013
%p seq(simplify(hypergeom([-n,n+1],[2],-1)), n=0..100); # _Robert Israel_, Mar 23 2015
%t a[0] = 1; a[n_Integer] := a[n] = a[n - 1] + Sum[a[k]*a[n - 1 - k], {k, 0, n - 1}]; Array[a[#] &, 30]
%t InverseSeries[Series[(y - y^2)/(1 + y), {y, 0, 24}], x] (* then A(x) = y(x)/x *) (* _Len Smiley_, Apr 11 2000 *)
%t CoefficientList[Series[(1 - x - (1 - 6x + x^2)^(1/2))/(2x), {x, 0, 30}], x] (* _Harvey P. Dale_, May 01 2011 *)
%t a[ n_] := 2 Hypergeometric2F1[ -n + 1, n + 2, 2, -1]; (* _Michael Somos_, Apr 03 2013 *)
%t a[ n_] := With[{m = If[ n < 0, -1 - n, n]}, SeriesCoefficient[(1 - x - Sqrt[ 1 - 6 x + x^2])/(2 x), {x, 0, m}]]; (* _Michael Somos_, Jun 10 2015 *)
%t Table[-(GegenbauerC[n+1, -1/2, 3] + KroneckerDelta[n])/2, {n, 0, 30}] (* _Vladimir Reshetnikov_, Nov 12 2016 *)
%o (PARI) {a(n) = if( n<0, n = -1-n); polcoeff( (1 - x - sqrt( 1 - 6*x + x^2 + x^2 * O(x^n))) / 2, n+1)}; /* _Michael Somos_, Apr 03 2013 */
%o (PARI) {a(n) = if( n<1, 1, sum( k=0, n, 2^k * binomial( n, k) * binomial( n, k-1)) / n)};
%o (Sage) # Generalized algorithm of L. Seidel
%o def A006318_list(n) :
%o D = [0]*(n+1); D[1] = 1
%o b = True; h = 1; R = []
%o for i in range(2*n) :
%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 A006318_list(23) # _Peter Luschny_, Jun 02 2012
%o (Haskell)
%o a006318 n = a004148_list !! n
%o a006318_list = 1 : f [1] where
%o f xs = y : f (y : xs) where
%o y = head xs + sum (zipWith (*) xs $ reverse xs)
%o -- _Reinhard Zumkeller_, Nov 13 2012
%o (Python)
%o from gmpy2 import divexact
%o A006318 = [1, 2]
%o for n in range(3,10**3):
%o ....A006318.append(divexact(A006318[-1]*(6*n-9)-(n-3)*A006318[-2],n))
%o # _Chai Wah Wu_, Sep 01 2014
%o (GAP) Concatenation([1],List([1..25],n->(1/n)*Sum([0..n],k->2^k*Binomial(n,k)*Binomial(n,k-1)))); # _Muniru A Asiru_, Nov 29 2018
%Y Apart from leading term, twice A001003 (the small Schroeder numbers). Cf. A025240.
%Y Sequences A085403, A086456, A103137, A112478 are essentially the same sequence.
%Y Main diagonal of A033877.
%Y Cf. A002003, A004148, A088617, A060693, A144156.
%Y Row sums of A104219. Bisections give A138462, A138463.
%Y Row sums of A175124.
%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
%K nonn,easy,core,nice
%O 0,2
%A _N. J. A. Sloane_
%E Edited by _Charles R Greathouse IV_, Apr 20 2010
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