A026022 -id:A026022 - cp's OEIS frontend

A026029 Number of (s(0), s(1), ..., s(2n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 3, s(2n) = 3. Also T(2n,n), where T is defined in A026022.

1, 2, 6, 20, 69, 242, 858, 3068, 11050, 40052, 145996, 534888, 1968685, 7276050, 26993490, 100490220, 375287550, 1405622460, 5278838100, 19873977240, 74994427170, 283595947284, 1074568266756, 4079184055640, 15511924233204, 59083160374952, 225384613313944

Offset: 0

Clark Kimberling

nonn

Hankel transform is A008619(n+1). - Paul Barry, May 11 2009

Michael De Vlieger, Table of n, a(n) for n = 0..1000
Andrei Asinowski and Cyril Banderier, From geometry to generating functions: rectangulations and permutations, arXiv:2401.05558 [cs.DM], 2024. See page 2.
Arturo Merino and Torsten Mütze, Combinatorial generation via permutation languages. III. Rectangulations, arXiv:2103.09333 [math.CO], 2021.

CoefficientList[Series[(1 - 2*x)*(-1 + Sqrt[1 - 4*x] + 2*x)^2 / (4*x^4), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 03 2019 *)

Expansion of (1+x^2*C^4)*C^2, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
a(n) = Sum_{k=0..n} C(n, k)*Sum_{i=0..k} C(k, 2i)*A000108(i+1). - Paul Barry, Jul 18 2003
a(n) = Sum_{k=0..3} A039599(n,k) = A000108(n) + A000245(n) + A000344(n) + A000588(n) = A026012(n) + A000588(n). - Philippe Deléham, Nov 12 2008
a(n) = C(2n,n) - C(2n,n-4). - Paul Barry, May 11 2009
Conjecture: (n+4)*a(n) + 6*(-n-2)*a(n-1) + 4*(2*n-1)*a(n-2) = 0. - R. J. Mathar, Nov 24 2012
a(n) ~ 4^(n+2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 03 2019
E.g.f.: exp(2*x)*(BesselI(0, 2*x) - BesselI(4, 2*x)). - Stefano Spezia, Jan 17 2024

A026023 a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n and s(0) = 3. Also a(n) = Sum{T(n,k), k = 0,1,...,[ (n+3)/2 ]}, where T is defined in A026022.

Original entry on oeis.org

1, 2, 4, 8, 15, 30, 56, 112, 210, 420, 792, 1584, 3003, 6006, 11440, 22880, 43758, 87516, 167960, 335920, 646646, 1293292, 2496144, 4992288, 9657700, 19315400, 37442160, 74884320, 145422675, 290845350, 565722720, 1131445440, 2203961430, 4407922860

Offset: 0

Clark Kimberling

nonn

a(n)/2^n is the probability that a random walker starting at x=4 and jumping +-1 with equal probability at each time step is not adsorbed at the boundary x=0 at time n. - Robert M. Ziff, Nov 10 2014

Paolo Xausa, Table of n, a(n) for n = 0..1000
Christian Krattenthaler, Daniel Yaqubi, Some determinants of path generating functions, II, arXiv:1802.05990 [math.CO], 2018, Adv. Appl. Math. 101 (2018), 232-265.

Cf. A001791, A162551 (bisections).

Module[{r=Range[0,20],b},Riffle[b=Binomial[2r+2,r],2b]] (* Paolo Xausa, Dec 14 2023 *)

a(2n) = C(2n+2, n), a(2n+1) = 2*a(2n).
E.g.f.: dif(Bessel_I(1,2x)+2*Bessel_I(2,2x)+Bessel_I(3,2x),x). - Paul Barry, Jun 09 2007
O.g.f.: -1/2*(-1+4*x^2+(1-8*x^2+20*x^4-16*x^6)^(1/2))/x^4/(2*x-1). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009
Conjecture: (n+4)*(n-1)*a(n) +(n-1)*(n+1)*a(n-1) -2*(n+1)*(2*n+1)*a(n-2) -4*(n-1)*(n+1)*a(n-3)=0. - R. J. Mathar, Sep 29 2012

Definition corrected by Herbert Kociemba, May 08 2004

A026026 a(n) = number of (s(0), s(1), ..., s(2n-1)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 3, s(2n-1) = 4. Also a(n) = T(2n-1,n-1), where T is defined in A026022.

Original entry on oeis.org

1, 3, 10, 35, 125, 451, 1638, 5980, 21930, 80750, 298452, 1106921, 4118725, 15371475, 57528750, 215867880, 811985790, 3061229850, 11565545100, 43782423750, 166051490514, 630877833102, 2400830868860, 9150602070760, 34927775872500, 133502608167292

Offset: 2

Clark Kimberling

nonn

Expansion of (1+x^2*C^4)*C^3, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.

Conjecture: -(n+3)*(11*n-38)*a(n) +2*(35*n^2-86*n-102)*a(n-1) -4*(13*n-30)*(2*n-5)*a(n-2)=0. - R. J. Mathar, Jun 22 2013

Definition corrected by Herbert Kociemba, May 02 2004

A026027 a(n) = number of (s(0), s(1), ..., s(2n-1)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 3, s(2n-1) = 6. Also a(n) = T(2n-1,n-2), where T is defined in A026022.

Original entry on oeis.org

1, 5, 21, 84, 329, 1274, 4900, 18768, 71706, 273581, 1043119, 3976700, 15163785, 57848910, 220830360, 843621600, 3225477150, 12342901410, 47274639594, 181230395304, 695384769002, 2670587146260, 10265249568536, 39491671919840

Offset: 2

Clark Kimberling

nonn

C(2n-1, n-2) - C(2n-1, n-6). G.f.: (1+x^2C^4)*C^5, where C=(1-sqrt(1-4x))/(2x). - Ralf Stephan, Jan 09 2005

Conjecture: (13*n+118)*(n+5)*a(n) +(13*n^2-969*n-3490)*a(n-1) +2*(-203*n^2+1080*n+758)*a(n-2) +4*(73*n-182)*(2*n-5)*a(n-3)=0. - R. J. Mathar, Jun 22 2013

A026030 a(n) = T(2n,n-1), where T is defined in A026022.

Original entry on oeis.org

1, 4, 15, 56, 209, 780, 2912, 10880, 40698, 152456, 572033, 2150040, 8095425, 30535260, 115377660, 436698240, 1655607390, 6286707000, 23908446510, 91057063344, 347281885818, 1326262602104, 5071418015120, 19415851639296, 74419447792340

Offset: 1

Clark Kimberling

nonn

a(n) = number of (s(0), s(1), ..., s(2n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 3, s(2n) = 5.

			x + 4*x^2 + 15*x^3 + 56*x^4 + 209*x^5 + 780*x^6 + 2912*x^7 + 10880*x^8 + ...

Cf. A001075.

{a(n) = binomial( 2*n, n-1) - binomial( 2*n, n-5)} /* Michael Somos, Jan 08 2012 */

a(n) = C(2n, n-1) - C(2n, n-5). G.f.: (1+x^2C^4)*C^4, where C=(1-sqrt(1-4x))/(2x). - Ralf Stephan, Jan 09 2005
G.f.: 2*x*(1-2*x) / ((1-2*x)*(1-4*x+x^2) + (1-x)*(1-3*x)*sqrt(1-4*x)). - Michael Somos, Jan 08 2012
Conjecture: (n+5)*a(n) -2*(5*n+16)*a(n-1) +(35*n+47)*a(n-2) +2*(-25*n+14)*a(n-3) +12*(2*n-5)*a(n-4)=0. - R. J. Mathar, Jun 15 2014

A026031 a(n) = number of (s(0), s(1), ..., s(2n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 3, s(2n) = 7. Also a(n) = T(2n,n-2), where T is defined in A026022.

Original entry on oeis.org

1, 6, 28, 120, 494, 1988, 7888, 31008, 121125, 471086, 1826660, 7068360, 27313650, 105452700, 406923360, 1569869760, 6056194410, 23366193084, 90173331960, 348102883184, 1344324544156, 5193831553416, 20075820280544, 77637309982400

Offset: 2

Clark Kimberling

nonn

Table[Binomial[2n,n-2]-Binomial[2n,n-6],{n,2,30}] (* Harvey P. Dale, May 28 2017 *)

C(2n, n-2) - C(2n, n-6). G.f.: (1+x^2C^4)*C^6, where C=(1-sqrt(1-4x))/(2x). - Ralf Stephan, Jan 09 2005

Conjecture: (n+6)*a(n) +10*(-n-4)*a(n-1) +2*(17*n+32)*a(n-2) +4*(-11*n+4)*a(n-3) +8*(2*n-5)*a(n-4)=0. - R. J. Mathar, Jun 22 2013

A026034 T(n,[ n/2 ]), where T is defined in A026022.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 20, 35, 69, 125, 242, 451, 858, 1638, 3068, 5980, 11050, 21930, 40052, 80750, 145996, 298452, 534888, 1106921, 1968685, 4118725, 7276050, 15371475, 26993490, 57528750, 100490220, 215867880, 375287550, 811985790, 1405622460

Offset: 0

Clark Kimberling

nonn

a(2n) = A026029(n), a(2n+1) = A026026(n+2).

A027294 Self-convolution of row n of array T given by A026022.

Original entry on oeis.org

1, 2, 6, 20, 56, 208, 493, 1974, 4336, 18258, 38380, 167464, 342056, 1532376, 3067156, 14022168, 27646080, 128438893, 250291902, 1178090212, 2274524200, 10822168448, 20736353110, 99564168884, 189575748320, 917316990660

Offset: 0

Clark Kimberling

nonn

Cf. A026022.

A027295 a(n) = Sum_{k=0..m-1} T(n,k) * T(n,k+1), where m=n for n=0,1 and m=floor((n+3)/2) for n >= 2, and T given by A026022.

Original entry on oeis.org

1, 4, 15, 52, 195, 676, 2555, 8976, 34098, 121448, 463067, 1669340, 6384287, 23251620, 89157557, 327523680, 1258803182, 4658150520, 17940605163, 66803546472, 257777882894, 965023761064, 3730202020650, 14029655954272, 54315578806820, 205120107930192

Offset: 1

Clark Kimberling

nonn

More terms from Sean A. Irvine, Oct 26 2019

A027296 a(n) = Sum_{k=0..floor((n-1)/2)} T(n,k) * T(n,k+2), with T given by A026022.

Original entry on oeis.org

1, 6, 22, 100, 345, 1470, 5096, 21120, 74175, 302698, 1076922, 4352140, 15660827, 62876490, 228468272, 913093664, 3345466023, 13326904266, 49176362780, 195434713432, 725586231766, 2878605745972, 10744342665072, 42571577779200, 159638882959595, 631933882236306

Offset: 2

Clark Kimberling

nonn

Cf. A026022.

More terms from Sean A. Irvine, Oct 26 2019

cp's OEIS Frontend

A026029 Number of (s(0), s(1), ..., s(2n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 3, s(2n) = 3. Also T(2n,n), where T is defined in A026022.

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A026023 a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n and s(0) = 3. Also a(n) = Sum{T(n,k), k = 0,1,...,[ (n+3)/2 ]}, where T is defined in A026022.

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A026026 a(n) = number of (s(0), s(1), ..., s(2n-1)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 3, s(2n-1) = 4. Also a(n) = T(2n-1,n-1), where T is defined in A026022.

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A026027 a(n) = number of (s(0), s(1), ..., s(2n-1)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 3, s(2n-1) = 6. Also a(n) = T(2n-1,n-2), where T is defined in A026022.

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A026030 a(n) = T(2n,n-1), where T is defined in A026022.

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A026031 a(n) = number of (s(0), s(1), ..., s(2n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 3, s(2n) = 7. Also a(n) = T(2n,n-2), where T is defined in A026022.

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A026034 T(n,[ n/2 ]), where T is defined in A026022.

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A027294 Self-convolution of row n of array T given by A026022.

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A027295 a(n) = Sum_{k=0..m-1} T(n,k) * T(n,k+1), where m=n for n=0,1 and m=floor((n+3)/2) for n >= 2, and T given by A026022.

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A027296 a(n) = Sum_{k=0..floor((n-1)/2)} T(n,k) * T(n,k+2), with T given by A026022.

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