A026568 -id:A026568 - cp's OEIS frontend

A026569 a(n) = T(n,n), T given by A026568. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=0.

1, 1, 3, 5, 13, 27, 67, 153, 375, 893, 2189, 5319, 13089, 32155, 79479, 196573, 487833, 1212135, 3018355, 7525585, 18792303, 46980373, 117589689, 294613155, 738844719, 1854484305, 4658460165, 11710592711, 29458662005, 74151824271

Offset: 0

Clark Kimberling

nonn

Number of grand Motzkin n-paths avoiding UF. - David Scambler, Jun 20 2013

			For a(3) = 5 the five grand Motzkin paths are FDU, DFU, FUD, UDF and FFF. The paths containing UF, namely UFD and DUF, are avoided. - _David Scambler_, Jun 20 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Rui Duarte and António Guedes de Oliveira, Generating functions of lattice paths, Univ. do Porto (Portugal 2023).
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

Cf. A026568.

List([0..30], n-> Sum([0..Int(n/2)], k-> Binomial(2*k,k)*Binomial( n-k, k) )); # G. C. Greubel, Aug 03 2019

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/Sqrt((1-x)*(1-x-4*x^2)) )); // G. C. Greubel, Aug 03 2019

CoefficientList[Series[Sqrt[1/((1-x)(1-x-4x^2))],{x,0,30}],x] (* Harvey P. Dale, Oct 06 2011 *)

my(x='x+O('x^30)); Vec( 1/sqrt((1-x)*(1-x-4*x^2)) ) \\ G. C. Greubel, Aug 03 2019

(1/sqrt((1-x)*(1-x-4*x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 03 2019

a(n) = Sum_{k=0..floor(n/2)} binomial(2*k, k)*binomial(n-k, k). - Paul Barry, Sep 09 2004
G.f.: sqrt(1/((1-x)*(1-x-4*x^2))). - Ralf Stephan, Jan 08 2004
D-finite with recurrence: a(n) = 1/n*((2*n-1)*a(n-1) + (3*n-3)*a(n-2) - (4*n-6)*a(n-3)). - Vladeta Jovovic, Mar 12 2005
a(n) = Sum_{k=0..n} C(k, n-k)*C(2*(n-k), n-k). - Paul Barry, Jul 30 2005
G.f.: 1/(1-x-2*x^2/(1-0*x-x^2/(1-x-x^2/(1-0*x-2*x^2/(1-x-x^2/.... (continued fraction). Paul Barry, Dec 07 2008
a(n) ~ sqrt((5+13/sqrt(17))/8) * ((1+sqrt(17))/2)^n/sqrt(Pi*n). - Vaclav Kotesovec, Aug 10 2013

A026583 a(n) = Sum{T(i,j)}, 0<=j<=i, 0<=i<=2n, T given by A026568.

Original entry on oeis.org

1, 4, 11, 30, 77, 200, 511, 1314, 3361, 8620, 22067, 56550, 144821, 371024, 950311, 2434410, 6235657, 15973300, 40915931, 104809134, 268472861, 687709400, 1761600847, 4512438450, 11558841841, 29608595644, 75843963011

Offset: 0

Clark Kimberling

Amir Sapir, The Tower of Hanoi with Forbidden Moves, The Computer J. 47 (1) (2004) 20, case cyclic++, sequence d(n) (offset 1).
Index entries for linear recurrences with constant coefficients, signature (2,3,-4).

Cf. A006131, A026581 (first differences).

LinearRecurrence[{2,3,-4},{1,4,11},40] (* Harvey P. Dale, Apr 03 2024 *)

G.f.: (1+2x)/[(1-x)(1-x-4x^2)]. - Ralf Stephan, Feb 04 2004 (follows from first comment in A026581)

A026571 a(n) = T(n,n-2), T given by A026568. Also a(n) = number of integer strings s(0), ..., s(n) counted by T, such that s(n) = 2.

Original entry on oeis.org

1, 2, 7, 16, 44, 106, 273, 672, 1696, 4214, 10573, 26392, 66151, 165578, 415277, 1041480, 2615004, 6568450, 16512355, 41531360, 104526093, 263206638, 663143211, 1671581968, 4215574482, 10635988422, 26846320149, 67790042264

Offset: 2

Clark Kimberling

nonn

From Ricardo Gómez Aíza, Feb 26 2024: (Start)
The sequence corresponds to the cumulative distribution function of the number of petals in a rooted plane tree with nonempty flowers everywhere but on the root, with flowers made out of petals of size one.
Examples:
a(2)=1 because there is only one element of size 2, and it consists of the root with one descendant with a flower with a single petal attached to it;
a(3)=2 because again there is only one element of size 3 that consists of the root with one descendant with a flower with two petals attached to it;
a(4)=7 because there is one tree with the root and two descendants, each with a flower with one petal only (two petals in total), then there is one tree with the root and one descendant that also has a descendant, and both descendants with a flower with one petal only (two petals in total), and finally there is the tree with the root and one descendant with a flower with three petals. (End)

Michael De Vlieger, Table of n, a(n) for n = 2..2450
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See pp. 18-19.

CoefficientList[Series[(2*x^2 + x - 1 + (1 - x)*#)/(2*(x^3 - x^2)*#) &[Sqrt[(1 - x - 4*x^2)/(1 - x)]], {x, 0, 29}], x] (* Michael De Vlieger, Mar 03 2024 *)

Conjecture: (n+2)*a(n) + 3*(-n-1)*a(n-1) - 3*n*a(n-2) + 11*(n-1)*a(n-3) + 2*(n-6)*a(n-4) + 4*(-2*n+7)*a(n-5) = 0. - R. J. Mathar, Jun 23 2013
From Ricardo Gómez Aíza, Feb 26 2024: (Start)
G.f.: (2*x^2+x-1+(1-x)*p(x))/(2*(x^3-x^2)*p(x)) with p(x) = sqrt((1-x-4*x^2)/(1-x)).
a(n) ~ 16*sqrt((9-s)/(s*(s-1)^5*Pi*n))*(8/(s-1))^n where s=sqrt(17). (End)

A026572 a(n) = T(n,n-3), T given by A026568. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=3.

Original entry on oeis.org

1, 2, 8, 19, 56, 140, 376, 953, 2474, 6286, 16097, 40880, 104069, 264052, 670414, 1699831, 4310546, 10924970, 27690075, 70168812, 177820791, 450618964, 1142004584, 2894347667, 7336297080, 18597140982, 47148420564

Offset: 3

Clark Kimberling

nonn

Also, a(n) = T(n,n-3), T given by A026584. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=3.

Cf. A026568, A026584.

Conjectured g.f.: (1/2)*((-3*x^2-x+1)*((x-1)/(4*x^2+x-1))^(1/2)-1+x+x^2)/x^3. - Mark van Hoeij, Oct 30 2011

Conjecture: (n+3)*a(n) +(-3*n-5)*a(n-1) +4*(-n-1)*a(n-2) +(13*n-9)*a(n-3) +(5*n-9)*a(n-4) +6*(-2*n+7)*a(n-5)=0. - R. J. Mathar, Jun 23 2013

A026570 a(n) = A026568(n,n-1), also a(n) = number of integer strings s(0),...,s(n) counted by A026568 such that s(n)=1.

Original entry on oeis.org

1, 1, 4, 7, 20, 43, 111, 259, 648, 1565, 3885, 9533, 23662, 58547, 145630, 362151, 903110, 2253615, 5633359, 14094035, 35304658, 88511733, 222115782, 557819793, 1401987930, 3526066273, 8874034647, 22346581133, 56304982154

Offset: 1

Clark Kimberling

nonn

Also a(n) = T'(n,n-1), T' given by A026584. Also a(n) = number of integer strings s(0),...,s(n) counted by T' such that s(n)=1.

Cf. A026568, A026569, A026584, A026585.

Conjecture: (n+1)*a(n) -2*n*a(n-1) +(-3*n-1)*a(n-2) +2*(2*n-3)*a(n-3)=0. - R. J. Mathar, Jun 23 2013

If recurrence is correct then a(n) = (A026569(n+1)-A026569(n))/2 = A026585(n+1)/2. - Mark van Hoeij, Nov 29 2024

A026573 a(n)=T(n,n-4), T given by A026568. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=4.

Original entry on oeis.org

1, 3, 12, 34, 103, 281, 778, 2072, 5524, 14488, 37918, 98474, 255137, 658351, 1695542, 4356520, 11177720, 28637452, 73294698, 187412326, 478861373, 1222765699, 3120722598, 7961152560, 20302100415, 51757554237

Offset: 4

Clark Kimberling

nonn

Conjecture: -2*(n-4)*(n+4)*a(n) +(5*n^2-5*n-46)*a(n-1) +(4*n^2+n+18)*a(n-2) +(-11*n^2+24*n-16)*a(n-3) +2*(n-2)*(2*n-5)*a(n-4)=0. - R. J. Mathar, Jun 23 2013

A026574 T(2n,n), T given by A026568.

Original entry on oeis.org

1, 1, 7, 19, 103, 341, 1809, 6336, 33899, 121483, 656199, 2381512, 12957809, 47419503, 259395664, 954961034, 5245001951, 19397198595, 106871608053

Offset: 0

Clark Kimberling

nonn

A026575 T(2n,n-1), T given by A026568.

Original entry on oeis.org

1, 2, 12, 38, 200, 701, 3682, 13294, 70665, 258430, 1388002, 5114572, 27682135, 102539204, 558172494, 2075658454, 11348521989, 42330184638

Offset: 1

Clark Kimberling

nonn

A026576 T(2n,n-2), T given by A026568.

Original entry on oeis.org

1, 3, 18, 65, 346, 1288, 6818, 25518, 136152, 510532, 2745347, 10309234, 55784794, 209766714, 1140670242, 4294635438, 23445204441

Offset: 2

Clark Kimberling

nonn

A026577 T(2n-1,n-1), T given by A026568.

Original entry on oeis.org

1, 2, 8, 34, 121, 606, 2155, 11306, 40717, 217666, 792351, 4280046, 15703156, 85410872, 315180458, 1722900242, 6385672193, 35040356022

Offset: 1

Clark Kimberling

nonn

cp's OEIS Frontend

A026569 a(n) = T(n,n), T given by A026568. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=0.

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A026583 a(n) = Sum{T(i,j)}, 0<=j<=i, 0<=i<=2n, T given by A026568.

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A026571 a(n) = T(n,n-2), T given by A026568. Also a(n) = number of integer strings s(0), ..., s(n) counted by T, such that s(n) = 2.

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Mathematica

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A026572 a(n) = T(n,n-3), T given by A026568. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=3.

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A026570 a(n) = A026568(n,n-1), also a(n) = number of integer strings s(0),...,s(n) counted by A026568 such that s(n)=1.

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A026573 a(n)=T(n,n-4), T given by A026568. Also a(n) = number of integer strings s(0),...,s(n) counted by T, such that s(n)=4.

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A026574 T(2n,n), T given by A026568.

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A026575 T(2n,n-1), T given by A026568.

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A026576 T(2n,n-2), T given by A026568.

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A026577 T(2n-1,n-1), T given by A026568.

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