A045720 -id:A045720 - cp's OEIS frontend

A035119 Related to A045720 and A035101.

0, 0, 1, 18, 285, 4680, 82845, 1595790, 33453945, 760970700, 18705542625, 494764058250, 14023390706325, 424278354099600, 13653335491921125, 465794724725079750, 16796514560465264625, 638448710154151396500

Offset: 1

Wolfdieter Lang

3rd column of triangular array A035342. a(n) = (2*n+1)*a(n-1) + A035101(n-1), n >= 3, a(2)=0.
a(n) gives the number of organically labeled forests (sets) with three rooted ordered trees with n non-root vertices. Organic labeling means that the vertex labels along the (unique) path from the root to any of the leaves (degree 1, non-root vertices) is increasing. W. Lang, Aug 07 2007.
a(n), n>=3, enumerates unordered n-vertex forests composed of three plane (ordered) ternary (3-ary) trees with increasing vertex labeling. See A001147 (number of increasing ternary trees) and a D. Callan comment there. For a picture of some ternary trees see a W. Lang link under A001764.

			a(4)=18 for the number of forests (sets) of three increasing labeled rooted trees with 4 non-root vertices and three root labels 0: [(0,4),{(0,1),(0,2)},(0,3)]; [(0,4),{(0,2),(0,1)},(0,3)]; [(0,4),{(0,1),(0,3)},(0,2)]; [(0,4),{(0,3),(0,1)},(0,2)]; [(0,4),{(0,2),(0,3)},(0,1)]; [(0,4),{(0,3),(0,2)},(0,1)]; [(0,4),(0,1,2),(0,3)]; [(0,4),(0,1,3),(0,2)]; [(0,4),(0,2,3),(0,1)]; [{(0,4),(0,1)},(0,2),(0,3)]; [{(0,1),(0,4)},(0,2),(0,3)]; [{(0,4),(0,2)},(0,1),(0,3)]; [{(0,2),(0,4)},(0,1),(0,3)]; [{(0,4),(0,3)},(0,1),(0,2)]; [{(0,3),(0,4)},(0,1),(0,2)]; [(0,1,4),(0,2),(0,3)]; [(0,2,4),(0,1),(0,3)]; [(0,3,4),(0,1),(0,2)].
a(4)=18 increasing ternary 3-forest with n=4 vertices: there are three 3-forests (two one vertex trees together with any of the three different 2-vertex trees) each with six increasing labelings. W. Lang, Sep 14 2007.

Cf. A000108, A045720, A035101, A035342.

a(n) = n!*((n+2)*binomial(2*n, n)/4-3*2^(2*n-3))/(3*2^(n-2)); a(n)= n!*A045720(n-3)/(3*2^(n-2)), n >= 3; E.g.f. (4/3)*(x*c(x/2)*(1-2*x)^(-1/2)/2)^3 = (2*x/3)*((1-x/2)*c(x/2)-1)/(1-2*x)^(3/2), where c(x) = g.f. for Catalan numbers A000108, a(0) := 0.

A035324 A convolution triangle of numbers, generalizing Pascal's triangle A007318.

Original entry on oeis.org

1, 3, 1, 10, 6, 1, 35, 29, 9, 1, 126, 130, 57, 12, 1, 462, 562, 312, 94, 15, 1, 1716, 2380, 1578, 608, 140, 18, 1, 6435, 9949, 7599, 3525, 1045, 195, 21, 1, 24310, 41226, 35401, 19044, 6835, 1650, 259, 24, 1, 92378, 169766, 161052, 97954, 40963, 12021, 2450

Offset: 1

Wolfdieter Lang

Replacing each '2' in the recurrence by '1' produces Pascal's triangle A007318(n-1,m-1). The columns appear as A001700, A008549, A045720, A045894, A035330, ...
Triangle T(n,k), 1 <= k <= n, given by (0, 3/1, 1/3, 5/3, 3/5, 7/5, 5/7, 9/7, 7/9, 11/9, 9/11, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 28 2012
Riordan array (1, c(x)/sqrt(1-4x)) where c(x) = g.f. for Catalan numbers A000108, first column (k = 0) omitted. - Philippe Deléham, Jan 28 2012

			Triangle begins:
    1;
    3,   1;
   10,   6,   1;
   35,  29,   9,   1;
  126, 130,  57,  12,   1;
  462, 562, 312,  94,  15,   1;
Triangle (0, 3, 1/3, 5/3, 3/5, ...) DELTA (1,0,0,0,0,0, ...) has an additional first column (1,0,0,...).

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Wolfdieter Lang, First 10 rows.

Cf. A000108, A007318, A039598.
Row sums: A049027(n), n >= 1.
Alternating row sums give A000108 (Catalan numbers).
If offset 0 (n >= m >= 0): convolution triangle based on A001700 (central binomial coeffs. of odd order).

a035324 n k = a035324_tabl !! (n-1) !! (k-1)
a035324_row n = a035324_tabl !! (n-1)
a035324_tabl = map snd $ iterate f (1, [1]) where
   f (i, xs)  = (i + 1, map (`div` (i + 1)) $
      zipWith (+) ((map (* 2) $ zipWith (*) [2 * i + 1 ..] xs) ++ [0])
                  ([0] ++ zipWith (*) [2 ..] xs))
-- Reinhard Zumkeller, Jun 30 2013

a[n_, m_] /; n >= m >= 1 := a[n, m] = 2*(2*(n-1) + m)*(a[n-1, m]/n) + m*(a[n-1, m-1]/n); a[n_, m_] /; n < m = 0; a[n_, 0] = 0; a[1, 1] = 1; Flatten[ Table[ a[n, m], {n, 1, 10}, {m, 1, n}]] (* Jean-François Alcover, Feb 21 2012, from first formula *)

@cached_function
def T(n, k):
    if n == 0: return n^k
    return sum(binomial(2*i-1, i)*T(n-1, k-i) for i in (1..k-n+1))
A035324 = lambda n,k: T(k, n)
for n in (1..8): print([A035324(n, k) for k in (1..n)]) # Peter Luschny, Aug 16 2016

a(n+1, m) = 2*(2*n+m)*a(n, m)/(n+1) + m*a(n, m-1)/(n+1), n >= m >= 1; a(n, m) := 0, n

G.f. for column m: ((x*c(x)/sqrt(1-4*x))^m)/x, where c(x) = g.f. for Catalan numbers A000108.

a(n, m) =: s2(3; n, m).

With offset 0 (0 <= k <= n), T(n,k) = Sum_{j>=0} A039598(n,j)*binomial(j,k). - Philippe Deléham, Mar 30 2007

T(n+1,n) = 3*n = A008585(n).

T(n,k) = T(n-1,k-1) + 3*T(n-1,k) + Sum_{i>=0} T(n-1,k+1+i)*(-1)^i. - Philippe Deléham, Feb 23 2012

T(n,m) = Sum_{k=m..n} k*binomial(k-1,k-m)*2^(k-m)*binomial(2*n-k-1,n-k)/n. - Vladimir Kruchinin, Aug 07 2013

A045894 4-fold convolution of A001700(n), n >= 0.

Original entry on oeis.org

1, 12, 94, 608, 3525, 19044, 97954, 486000, 2345930, 11081880, 51447036, 235454848, 1064832173, 4767347796, 21160397050, 93223960784, 408037319262, 1775744775592, 7688699122724, 33140226601920, 142262721338146

Offset: 0

Wolfdieter Lang

Indranil Ghosh, Table of n, a(n) for n = 0..1500
José Agapito, Ângela Mestre, Maria M. Torres, and Pasquale Petrullo, On One-Parameter Catalan Arrays, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.1.
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.

Cf. A001700, A000108, A045720.

Table[(n + 11)*4^(n + 2) - (n + 5) Binomial[2 (n + 4), n + 4]/2, {n, 0, 20}] (* Michael De Vlieger, Feb 18 2017 *)

import math
def C(n,r):
    f=math.factorial
    return f(n)/f(r)/f(n-r)
def A045894(n):
    return (n+11)*4**(n+2)-(n+5)*C(2*(n+4),(n+4))/2 # Indranil Ghosh, Feb 18 2017

a(n) = (n+11)*4^(n+2) - (n+5)*binomial(2*(n+4), n+4)/2;
G.f.: c(x)^4/(1-4*x)^2, where c(x) = g.f. for Catalan numbers A000108;
recursion: a(n)= (2*(2*n+10)/(n+4))*a(n-1) + (4/(n+4))*A045720(n), a(0)=1.

A210064 Total number of 231 patterns in the set of permutations avoiding 123.

Original entry on oeis.org

0, 0, 1, 11, 81, 500, 2794, 14649, 73489, 356960, 1691790, 7864950, 36000186, 162697176, 727505972, 3223913365, 14176874193, 61926666824, 268931341414, 1161913686618, 4997204887550, 21404922261112, 91351116184716, 388581750349946, 1647982988377786

Offset: 1

Cheyne Homberger, Mar 16 2012

nonn

a(n) is the total number of 231 (and also 312) patterns in the set of all 123 avoiding n-permutations. Also the number of 231 (or 213, or 312) patterns in the set of all 132 avoiding n-permutations.

			a(3) = 1 since there is only one 231 pattern in the set {132,213,231,312,321}.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Cheyne Homberger, Expected patterns in permutation classes, Electronic Journal of Combinatorics, 19(3) (2012), P43.

Cf. A045720.

Rest[CoefficientList[Series[x/(2*(1-4*x)^2) + (x-1)/(2*(1-4*x)^(3/2)) + 1/(2 - 8*x), {x, 0, 20}], x]] (* Vaclav Kotesovec, Mar 15 2014 *)

x='x+O('x^50); concat([0,0], Vec(x/(2*(1-4*x)^2) + (x-1)/(2*(1-4*x)^(3/2)) + 1/(2 - 8*x))) \\ G. C. Greubel, May 31 2017

G.f.: x/(2*(1-4*x)^2) + (x-1)/(2*(1-4*x)^(3/2)) + 1/(2 - 8*x).
a(n) ~ n * 2^(2*n-3) * (1 - 6/sqrt(Pi*n)). - Vaclav Kotesovec, Mar 15 2014
Conjecture: n*(n-3)*a(n) +2*(-4*n^2+11*n-2)*a(n-1) +8*(n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Oct 08 2016

A143019 Infinite square array read by antidiagonals: a(q,n) is the coefficient of z^n in the series expansion of C(z)^q/(1-4z)^(3/2), where C(z) = (1-sqrt(1-4z))/(2z) is the Catalan function (q,n = 0,1,2,...).

Original entry on oeis.org

1, 1, 6, 1, 7, 30, 1, 8, 38, 140, 1, 9, 47, 187, 630, 1, 10, 57, 244, 874, 2772, 1, 11, 68, 312, 1186, 3958, 12012, 1, 12, 80, 392, 1578, 5536, 17548, 51480, 1, 13, 93, 485, 2063, 7599, 25147, 76627, 218790, 1, 14, 107, 592, 2655, 10254, 35401, 112028, 330818

Offset: 0

Emeric Deutsch, Jul 24 2008

a(q,n) = a(q-1,n) + a(q+1,n-1).

Row 0 is A002457; row 1 is A000531; row 2 is A029760; row 3 is A045720.

			Array starts:
  1, 6, 30, 140,  630, ...
  1, 7, 38, 187,  874, ...
  1, 8, 47, 244, 1186, ...
  1, 9, 57, 312, 1578, ...
  ...

Cf. A002457, A000531, A029760, A045720.

a:=proc(q,n) options operator, arrow: sum(4^i*binomial(2*n-2*i+q, n-i), i= 0.. n) end proc: aa:=proc(q,n) options operator, arrow: a(q-1,n-1) end proc: matrix(10,10,aa); # yields sequence in matrix form

a(q,n) = Sum_{i=0..n} 4^i*binomial(2n-2i+q, n-i).

Showing 1-5 of 5 results.

cp's OEIS Frontend

A035119 Related to A045720 and A035101.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A035324 A convolution triangle of numbers, generalizing Pascal's triangle A007318.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Sage

Formula

A045894 4-fold convolution of A001700(n), n >= 0.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Formula

A210064 Total number of 231 patterns in the set of permutations avoiding 123.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A143019 Infinite square array read by antidiagonals: a(q,n) is the coefficient of z^n in the series expansion of C(z)^q/(1-4z)^(3/2), where C(z) = (1-sqrt(1-4z))/(2z) is the Catalan function (q,n = 0,1,2,...).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula