A002026 -id:A002026 - cp's OEIS frontend

A052731 E.g.f. [1-x -sqrt(1-2x-3x^2)]/(2x) - [1+x-sqrt(1-2x-3x^2)]/2 .

0, 0, 0, 6, 48, 600, 8640, 151200, 3064320, 71124480, 1857945600, 54007430400, 1729195776000, 60483053030400, 2294881337548800, 93889711948032000, 4120492394962944000, 193100926276177920000, 9624765220305371136000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 687

spec := [S,{C=Prod(B,Z),S=Prod(B,C),B=Union(S,Z,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

D-finite with recurrence: a(1)=0; a(2)=0; a(3)=6; (-15*n+15*n^3+15*n^2+3*n^4-18)*a(n) +(-n^3-7*n^2-16*n-12)*a(n+1) +(-3*n^2-16*n-21)*a(n+2) +(n+4)*a(n+3)=0; a(4)=48; a(5)=600.
Conjecture: a(n) = n!*A002026(n-2). - R. J. Mathar, Oct 16 2013
a(n) ~ sqrt(2) * 3^(n - 1/2) * n^(n-1) / exp(n). - Vaclav Kotesovec, Nov 19 2021

A106489 Triangle read by rows: T(n,k) is the number of short bushes with n edges and having the leftmost leaf at height k (a short bush is an ordered tree with no nodes of outdegree 1).

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 9, 5, 1, 21, 12, 3, 51, 30, 9, 1, 127, 76, 25, 4, 323, 196, 69, 14, 1, 835, 512, 189, 44, 5, 2188, 1353, 518, 133, 20, 1, 5798, 3610, 1422, 392, 70, 6, 15511, 9713, 3915, 1140, 230, 27, 1, 41835, 26324, 10813, 3288, 726, 104, 7, 113634, 71799, 29964

Offset: 2

Emeric Deutsch, May 29 2005

Basically, the mirror image of A020474. Row n has floor(n/2) terms (first row is row 2). Row sums yield the Riordan numbers (A005043). Column 1 yields the Motzkin numbers (A001006); column 2 yields A002026; column 3 yields A005322; column 4 yields A005323; column 4 yields A005324; column 5 yields A005325; column 6 yields A005326.

T(n,k) is the number of Riordan paths (Motzkin paths with no flatsteps on the x-axis) with k returns to the x-axis. For example, T(6,2) = 5 counts UDUFFD, UDUUDD, UFDUFD, UFFDUD, UUDDUD where U = (1,1) is an upstep, F = (1,0) is a flatstep, and D = (1,-1) is a downstep. - David Callan, Dec 12 2021

			Column 1 yields the Motzkin numbers: indeed, if from each short bush, having leftmost leaf at height 1, we drop the leftmost edge, then we obtain the so-called bushes, known to be counted by the Motzkin numbers.
Triangle begins:
   1;
   1;
   2,  1;
   4,  2;
   9,  5,  1;
  21, 12,  3;
  51, 30,  9,  1.

F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.
Sen-Peng Eu, Shu-Chung Liu, and Yeong-Nan Yeh, Taylor expansions for Catalan and Motzkin numbers, Advances in Applied Mathematics 29, Issue 3 (2002), 345-357 (see p. 352).

Cf. A020474, A005043, A001006, A002026, A005322, A005323, A005324, A005325, A005326, A064189.

S:=1/2/(z+z^2)*(1+z-sqrt(1-2*z-3*z^2)): G:=simplify(t*z^2*S/(1-z*S-t*z^2*S)): Gserz:=simplify(series(G,z=0,19)): for n from 2 to 17 do P[n]:=sort(coeff(Gserz,z^n)) od: for n from 2 to 17 do seq(coeff(P[n],t^k),k=1..floor(n/2)) od; # yields sequence in triangular form

(* To generate the sequence *)
CoefficientList[CoefficientList[Series[(1-t-2xt^2-Sqrt[1-2t-3t^2])/(2t^2(1-x+xt+x^2t^2)), {t,0,10}], t], x] // Flatten
(* To generate the triangle *)
CoefficientList[Series[(1-t-2xt^2-Sqrt[1-2t-3t^2])/(2t^2(1-x+xt+x^2t^2)), {t, 0, 10}], {t, x}] // MatrixForm
Table[If[n < 2 k, 0, GegenbauerC[n-2k,-n+k-1,-1/2](k+1)/(n-k+1)], {n,0,10}, {k,0,5}] // MatrixForm
(* Emanuele Munarini, Feb 10 2018 *)

G.f.: tz^2*S/(1 - zS - tz^2*S), where S = S(z) = (1 + z - sqrt(1 - 2z - 3z^2))/(2z(1+z)) is the g.f. of the short bushes (the Riordan numbers; A005043).
a(n,k) = T(n-k+1, n-2*k)*(k+1)/(n-k+1), for n >= 2k, where T(n,k) = A027907(n,k) are the trinomial coefficients. - Emanuele Munarini, Feb 10 2018
The rows are the antidiagonals of the Motzkin triangle A064189. - Peter Luschny, Feb 01 2025

A123261 Multiplicative encoding of Motzkin triangle (A026300).

Original entry on oeis.org

2, 6, 450, 405168750, 10326560651880195445980468750, 17149769349660883198128523550890723880659651223306378240865271303752564539222570800781250

Offset: 1

Jonathan Vos Post, Nov 06 2006

This is to A026300 "Motzkin triangle, T, read by rows; T(0,0) = T(1,0) = T(1,1) = 1; for n >= 2, T(n,0) = 1, T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k) for k = 1,2,...,n-1 and T(n,n) = T(n-1,n-2) + T(n-1,n-1)" as A007188 "Multiplicative encoding of Pascal triangle: Product p(i+1)^C(n,i)" is to A007318 "Pascal's triangle read by rows."

			a(1) = p(1)^T(1,1) = 2^1 = 2.
a(2) = p(1)^T(2,1) * p(2)^T(2,2) = 2^1 * 3^1 = 6.
a(3) = p(1)^T(3,1) * p(2)^T(3,2) * p(3)^T(3,3) = 2^1 * 3^2 * 5^2 = 450.
a(4) = 2^1 * 3^3 * 5^5 * 7^4 = 405168750.
a(5) = 2^1 * 3^4 * 5^9 * 7^12 * 11^9 = 10326560651880195445980468750.
a(6) = 2^1 * 3^5 * 5^14 * 7^25 * 11^30 * 13^21.
a(7) = 2^1 * 3^6 * 5^20 * 7^44 * 11^69 * 13^76 * 17^51.

Cf. A000040, A007188, A007318, A009766, A124061, Motzkin numbers (A001006) are T(n, n), other columns of T include A002026, A005322, A005323.

a(n) = Product_{i=1..n} p(i+1)^T(n,i), where T(n,i), are as in Motzkin triangle (A026300), T(0,0) = T(1,0) = T(1,1) = 1; for n >= 2, T(n,0) = 1, T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k) for k = 1,2,...,n-1 and T(n,n) = T(n-1,n-2) + T(n-1,n-1).

A337526 Number of length n permutations that are sorted to the identity by a consecutive-213-avoiding stack followed by a classical-21-avoiding stack.

Original entry on oeis.org

1, 2, 5, 15, 50, 180, 686, 2731

Offset: 1

Kai Zheng, Aug 30 2020

			For n = 3, the permutations 123, 213, 231, 312, and 321 are correctly sorted to the identity while the permutation 132 is not.

C. Defant and K. Zheng, Stack-Sorting with Consecutive-Pattern-Avoiding Stacks, arXiv:2008.12297 [math.CO], 2020.

Cf. A000108, A001006, A002026, A011782, A202062, A294790.

cp's OEIS Frontend

A052731 E.g.f. [1-x -sqrt(1-2x-3x^2)]/(2x) - [1+x-sqrt(1-2x-3x^2)]/2 .

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A106489 Triangle read by rows: T(n,k) is the number of short bushes with n edges and having the leftmost leaf at height k (a short bush is an ordered tree with no nodes of outdegree 1).

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A123261 Multiplicative encoding of Motzkin triangle (A026300).

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A337526 Number of length n permutations that are sorted to the identity by a consecutive-213-avoiding stack followed by a classical-21-avoiding stack.

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