A024996 -id:A024996 - cp's OEIS frontend

A024997 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0 = s(n), |s(i) - s(i-1)| = 1 for i = 1,2; |s(i) - s(i-1)| <= 1 for i >= 3. Also a(n) = T(n,n), where T is the array defined in A024996.

2, 8, 20, 58, 162, 462, 1318, 3782, 10886, 31436, 91016, 264134, 768094, 2237640, 6529284, 19079574, 55826166, 163538472, 479588844, 1407813438, 4136307798, 12163015662, 35793391662, 105407889930, 310620540202, 915913267652, 2702265079208

Offset: 3

Clark Kimberling

nonn

Second differences of the central trinomial coefficients A002426. - T. D. Noe, Mar 16 2005

G. C. Greubel, Table of n, a(n) for n = 3..1000

Cf. A025179.

Rest[Differences[CoefficientList[Series[1/Sqrt[(1 + x) (1 - 3 x)], {x, 0, 30}], x], 2]] (* Harvey P. Dale, May 11 2013 *)
Table[2 Sum[Binomial[n, 2 k] Binomial[2 k + 1, k + 1], {k, 0, Floor[n/2]}],  {n, 1, 25}] (* G. C. Greubel, Mar 01 2017 *)
Rest[Rest[CoefficientList[Series[((1 - x)^2 - (1 - x) Sqrt[1 - 2 x - 3 x^2])/(x Sqrt[1 - 2 x - 3 x^2]), {x, 0, 15}], x]]] (* G. C. Greubel, Mar 02 2017 *)

x='x +O('x^50); Vec(((1-x)^2-(1-x +2*x^2)*sqrt(1-2*x-3*x^2)) /(x*sqrt(1 - 2*x -3*x^2))) \\ G. C. Greubel, Mar 01 2017

a(n) = 2*A025179(n-1).
From G. C. Greubel, Mar 01 2017: (Start)
a(n) = 2*Sum_{k=0..floor(n/2)} binomial(n, 2*k)*binomial(2*k+1, k+1), for n>=1.
O.g.f.: ((1-x)^2-(1-x+2*x^2)*sqrt(1-2*x-3*x^2)) / sqrt(1-2*x-3*x^2) [corrected by Charles R Greathouse IV, Mar 05 2017]
E.g.f.: 2*exp(x)*(BesselI(0, 2*x) + BesselI(2, 2*x)). (End)

A024998 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, |s(i) - s(i-1)| = 1 for i = 1,2; |s(i) - s(i-1)| <= 1 for i >= 3, s(n) = 1. Also a(n) = T(n,n-1), where T is the array defined in A024996.

Original entry on oeis.org

1, 0, 3, 6, 19, 52, 150, 428, 1232, 3552, 10275, 29790, 86559, 251980, 734773, 2145822, 6275145, 18373296, 53856153, 158025186, 464112297, 1364247180, 4013353932, 11815188000, 34807249134, 102606325136, 302646363725, 893175905778

Offset: 1

Clark Kimberling

nonn

Comments from David Callan, Jul 15 2004: "a(n+1) is the number of Motzkin (2n)-paths whose longest level segment has length n. (A level segment is a maximal sequence of contiguous flatsteps.) Example: with n=3, the paths counted by a(4) are FFFUDF, FFFUFD, FUDFFF, FUFFFD, UFDFFF, UFFFDF. Here is a bijection to the sequences (s(i)) above.
"Given such a Motzkin (2n)-path, delete the (unique) longest level segment to split the path into A,B. Form the path BUA (U can be recovered as the up step immediately following the rightmost of the lowest points on this path). This path will not start or end with F.
"Transfer the level segment (if any) following the first step to the end. Code the resulting path with 1 for U, 0 for F and -1 for D. Then take partial sums (including the empty sum) to get a sequence (s(i)). Example: UF^9UFFUDDDF -> U,UFFUDDDF -> UFFUDDDFUU -> UUDDDFUUFF -> 1,1,-1,-1,-1,0,1,1,0,0 -> (0,1,2,1,0,-1,-1,0,1,1,1)."

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Rest[CoefficientList[Series[(1-x)^2*((1-x)*(1-2*x-3*x^2)^(-1/2)-1)/(2*x), {x, 0, 20}], x]] (* Vaclav Kotesovec, Feb 13 2014 *)

a(n+1)=sum(j=1..n/2, C[n-1, 2j-1] C[2j+1, j]). This sum counts the Motzkin (2n)-paths above by number j of up steps and the sequences (s(i)) by number j of indices i for which s(i) - s(i-1) = -1. GF: (1-x)^2 ( (1-x)(1-2x-3x^2)^(-1/2) - 1 )/(2x). - David Callan, Jul 15 2004
Conjecture: (n+1)*a(n) +(-3*n+2)*a(n-1) +(-n-7)*a(n-2) +3*(n-4)*a(n-3)=0. - R. J. Mathar, Jun 23 2013
a(n) ~ 2 * 3^(n+1/2) / (9 * sqrt(Pi*n)). - Vaclav Kotesovec, Feb 13 2014

A026069 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, |s(i) - s(i-1)| = 1 for i = 1,2; |s(i) - s(i-1)| <= 1 for i >= 3, s(n) = 2. Also a(n) = T(n,n-2), where T is the array defined in A024996.

Original entry on oeis.org

1, 5, 13, 40, 116, 342, 1002, 2941, 8629, 25333, 74405, 218659, 642955, 1891683, 5568867, 16403283, 48342867, 142548639, 420546039, 1241293314, 3665526270, 10829045472, 32005684340, 94632148659, 279909001851, 828235716571, 2451561077995

Offset: 3

Clark Kimberling

nonn

First differences of A025180. Second differences of A014531.

a(29) corrected by Sean A. Irvine, Sep 14 2019

A026070 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, |s(i) - s(i-1)| = 1 for i = 1,2; |s(i) - s(i-1)| <= 1 for i >= 3, s(n) = 3. Also a(n) = T(n,n-3), where T is the array defined in A024996.

Original entry on oeis.org

1, 2, 8, 24, 76, 232, 707, 2136, 6429, 19282, 57695, 172316, 513955, 1531362, 4559271, 13566288, 40349619, 119972214, 356634978, 1059985776, 3150165270, 9361450868, 27819215185, 82670528056, 245680350995, 730149455646, 2170105711452

Offset: 3

Clark Kimberling

nonn

First differences of A025181. Second differences of A014532.

Conjecture: (n+3)*a(n) +(-5*n-6)*a(n-1) +(3*n-11)*a(n-2) +(11*n-8)*a(n-3) +2*(-2*n+17)*a(n-4) +6*(-n+6)*a(n-5)=0. - R. J. Mathar, Jun 23 2013

A026071 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, |s(i) - s(i-1)| = 1 for i = 1,2; |s(i) - s(i-1)| <= 1 for i >= 3, s(n) = 4. Also a(n) = T(n,n-4), where T is the array defined in A024996.

Original entry on oeis.org

1, 3, 12, 40, 133, 427, 1352, 4224, 13080, 40216, 122980, 374452, 1136226, 3438150, 10380048, 31279728, 94114125, 282804759, 848886180, 2545759328, 7628718845, 22845628531, 68377674280, 204560102800, 611720539235, 1828673918721

Offset: 4

Clark Kimberling

nonn

First differences of A025182.

Conjecture: -(n-4)*(n+4)*a(n) +(4*n+7)*(n-4)*a(n-1) +(-2*n^2+23*n-12)*a(n-2) -(4*n+3)*(n-4)*a(n-3) +3*(n-4)*(n-5)*a(n-4)=0. - R. J. Mathar, Jun 22 2013

A026078 a(n) = T(n,[ n/2 ]), where T is the array defined in A024996.

Original entry on oeis.org

1, 1, 2, 1, 5, 8, 24, 40, 133, 218, 736, 1203, 4135, 6743, 23452, 38194, 134043, 218115, 770864, 1253614, 4455462, 7242629, 25859380, 42022984, 150615223, 244700887, 879876040, 1429252775, 5153445895, 8369948745, 30251941860, 49127900370

Offset: 0

Clark Kimberling

nonn

A026079 a(n) = Sum_{k = 1..n} T(k,k-1), where T is the array defined in A024996.

Original entry on oeis.org

1, 3, 6, 12, 31, 83, 233, 661, 1893, 5445, 15720, 45510, 132069, 384049, 1118822, 3264644, 9539789, 27913085, 81769238, 239794424, 703906721, 2068153901, 6081507833, 17896695833, 52703944967, 155310270103, 457956633828, 1351132539606, 3988457429607

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A025179.

Join[{1, 3}, Table[Sum[ Binomial[n, 2*k]*Binomial[2*k + 1, k + 1], {k, 0, Floor[n/2]}] + 2, {n,2,50}]] (* G. C. Greubel, May 22 2017 *)

concat([1,3], for(n=2,50, print1(2 + sum(k=0,floor(n/2), binomial(n, 2*k)*binomial(2*k+1, k+1)), ", "))) \\ G. C. Greubel, May 22 2017

a(n) = A025179(n) + 2, n>2.

A026080 Sum{T(n,k)}, k = 0,1,...,n, where T is the array defined in A024996.

Original entry on oeis.org

1, 2, 4, 7, 22, 64, 191, 567, 1689, 5033, 15013, 44809, 133816, 399802, 1194949, 3572693, 10684758, 31962456, 95633229, 286193409, 856610214, 2564317356, 7677475521, 22988860305, 68843627049, 206183053485, 617563017927, 1849887488987

Offset: 0

Clark Kimberling

nonn

First differences of A027914.

A026072 a(n) = T(2n-1,n), where T is the array defined in A024996.

Original entry on oeis.org

1, 3, 13, 76, 427, 2427, 13871, 79729, 460473, 2670394, 15540822, 90717305, 530946697, 3114680815, 18308774295, 107817336510, 635942014275, 3756423495198, 22217666030882, 131563775350068

Offset: 1

Clark Kimberling

nonn

A026073 T(2n,n), where T is the array defined in A024996.

Original entry on oeis.org

1, 2, 5, 24, 133, 736, 4135, 23452, 134043, 770864, 4455462, 25859380, 150615223, 879876040, 5153445895, 30251941860, 177938418225, 1048452751872, 6187400646130, 36566048896896, 216370786646054

Offset: 0

Clark Kimberling

nonn

Bisection of A026078.

cp's OEIS Frontend

A024997 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0 = s(n), |s(i) - s(i-1)| = 1 for i = 1,2; |s(i) - s(i-1)| <= 1 for i >= 3. Also a(n) = T(n,n), where T is the array defined in A024996.

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A024998 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, |s(i) - s(i-1)| = 1 for i = 1,2; |s(i) - s(i-1)| <= 1 for i >= 3, s(n) = 1. Also a(n) = T(n,n-1), where T is the array defined in A024996.

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A026069 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, |s(i) - s(i-1)| = 1 for i = 1,2; |s(i) - s(i-1)| <= 1 for i >= 3, s(n) = 2. Also a(n) = T(n,n-2), where T is the array defined in A024996.

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A026070 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, |s(i) - s(i-1)| = 1 for i = 1,2; |s(i) - s(i-1)| <= 1 for i >= 3, s(n) = 3. Also a(n) = T(n,n-3), where T is the array defined in A024996.

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A026071 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, |s(i) - s(i-1)| = 1 for i = 1,2; |s(i) - s(i-1)| <= 1 for i >= 3, s(n) = 4. Also a(n) = T(n,n-4), where T is the array defined in A024996.

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A026078 a(n) = T(n,[ n/2 ]), where T is the array defined in A024996.

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A026079 a(n) = Sum_{k = 1..n} T(k,k-1), where T is the array defined in A024996.

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A026080 Sum{T(n,k)}, k = 0,1,...,n, where T is the array defined in A024996.

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A026072 a(n) = T(2n-1,n), where T is the array defined in A024996.

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A026073 T(2n,n), where T is the array defined in A024996.

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