A026148 -id:A026148 - cp's OEIS frontend

A026123 a(n) = number of (s(0),s(1),...,s(n)) such that every s(i) is a nonnegative integer, s(0) = 1, s(n) = 2, |s(1) - s(0)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = T(n,n-1), where T is the array in A026120; a(n) = U(n,n+1), where U is the array in A026148.

1, 4, 10, 28, 76, 209, 575, 1589, 4405, 12253, 34189, 95679, 268503, 755457, 2130717, 6023235, 17063139, 48434514, 137741280, 392407134, 1119766942, 3200326627, 9160055809, 26254474379, 75348899051, 216515177336, 622887159274

Offset: 2

Clark Kimberling

nonn

First differences of A026134.

G.f.: z^2(-1+(1-z)^2M^3), with M the g.f. of the Motzkin numbers (A001006).

D-finite with recurrence: (n+5)*a(n) +5*(-n-3)*a(n-1) +(5*n+1)*a(n-2) +(5*n+3)*a(n-3) +6*(-n+3)*a(n-4)=0. - R. J. Mathar, Jun 23 2013

A026163 Sum{T(k,k-1)}, k = 1,2,...,n, where T is the array in A026148.

Original entry on oeis.org

1, 2, 6, 16, 45, 126, 356, 1008, 2862, 8140, 23188, 66144, 188916, 540216, 1546560, 4432512, 12717513, 36526626, 105016686, 302228080, 870613689, 2510249302, 7244285436, 20924179920, 60487084775, 174994990326, 506669921982

Offset: 1

Clark Kimberling

nonn

Cf. A026148.

Equals T(n, n-1), where T is the array in A026323.

Conjectures from Mark van Hoeij, Oct 30 2011: (Start)
a(n) = -4*(-3)^(1/2)*(-1)^n*((n^3+11*n^2+48*n+45)*hypergeom([1/2, n+2],[1],4/3)+(3*n^2+11*n+15)*hypergeom([1/2, n+3],[1],4/3))/((n+3)*(n+5)*(n+6)*(7+n))
G.f.: (2*x-1)*((x+1)^(1/2)*(1-3*x)^(1/2)*(x-1)*(x^2+2*x-1)+x^4-4*x^3-2*x^2+4*x-1)/(2*x^8). (End)
Conjecture: -(n+7)*(3*n-31)*a(n) +3*(-n^2-35*n-76)*a(n-1) +2*(32*n^2+27*n-459)*a(n-2) +(-47*n^2+286*n-204)*a(n-3) -3*(37*n-51)*(n-2)*a(n-4)=0. - R. J. Mathar, Jun 23 2013

A026165 Number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 2, |s(1) - s(0)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also sum of numbers in row n+1 of the array T in A026148.

Original entry on oeis.org

1, 2, 6, 17, 49, 141, 407, 1177, 3411, 9904, 28808, 83931, 244895, 715534, 2093262, 6130767, 17974779, 52751358, 154950378, 455524203, 1340182539, 3945723033, 11624603235, 34268836707, 101081770181, 298320243976, 880875609552

Offset: 0

Clark Kimberling

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

CoefficientList[Series[E^x/x^2*((2*x^2-2*x)*BesselI[0, 2*x]+(2-x+2*x^2)*BesselI[1, 2*x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec after Vladeta Jovovic, Feb 01 2014 *)

a(n) = Sum_{k=0..n} binomial(n, k)*binomial(k+1, floor(k/2)). - Vladeta Jovovic, Sep 18 2003
E.g.f.: exp(x)/x^2*((2*x^2-2*x)*BesselI(0, 2*x)+(2-x+2*x^2)*BesselI(1, 2*x)). - Vladeta Jovovic, Sep 23 2003
Conjecture: (n+3)*a(n) + (-5*n-9)*a(n-1) + (5*n+1)*a(n-2) + 5*(n-1)*a(n-3) + 6*(-n+3)*a(n-4) = 0. - R. J. Mathar, Jun 23 2013
Recurrence: (n+3)*(2*n^2 - n + 1)*a(n) = (4*n^3 + 10*n^2 + 7*n - 5)*a(n-1) + 3*(n-1)*(2*n^2 + 3*n + 2)*a(n-2). - Vaclav Kotesovec, Feb 01 2014
a(n) ~ 2 * 3^(n+1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 01 2014

A026151 a(n) = T(n,n), where T is the array in A026148.

Original entry on oeis.org

1, 1, 2, 4, 12, 32, 89, 246, 685, 1912, 5355, 15038, 42339, 119484, 337935, 957738, 2719539, 7736208, 22044444, 62916696, 179841270, 514793944, 1475586687, 4234989242, 12169352003, 35009302296, 100826681530, 290683486132, 838873595124

Offset: 0

Clark Kimberling

nonn

Cf. A026148

First differences of A026325 for n > 1.

a(0)-a(1) inserted and a(20) corrected by Sean A. Irvine, Sep 18 2019

A026152 a(n) = T(n,n-1), where T is the array in A026148.

Original entry on oeis.org

1, 4, 10, 29, 81, 230, 652, 1854, 5278, 15048, 42956, 122772, 351300, 1006344, 2885952, 8285001, 23809113, 68490060, 197211394, 568385609, 1639635613, 4734036134, 13679894484, 39562904855, 114507905551, 331674931656, 961408814434

Offset: 2

Clark Kimberling

nonn

First differences of A026163.

Conjecture: -(n+7)*(31*n-87)*a(n) +(164*n^2+313*n-2115)*a(n-1) +(-182*n^2+229*n+891)*a(n-2) +(-164*n^2+59*n+519)*a(n-3) +3*(71*n-104)*(n-3)*a(n-4)=0. - R. J. Mathar, Jun 23 2013

A026153 T(n,n-2), where T is the array in A026148.

Original entry on oeis.org

1, 2, 7, 20, 60, 176, 517, 1512, 4415, 12870, 37477, 109044, 317109, 921870, 2679510, 7787904, 22636503, 65804638, 191332945, 556456060, 1618813834, 4710869108, 13713658368, 39935698400, 116340344575, 339050396646, 988474306017

Offset: 2

Clark Kimberling

nonn

First differences of A026327.

Conjecture: -(n+8)*(n-118)*a(n) +(-n^2-561*n-2878)*a(n-1) +2*(38*n^2+203*n-1070)*a(n-2) +2*(-116*n^2+781*n+2493)*a(n-3) +(13*n^2-1012*n+3888)*a(n-4) +3*(107*n-423)*(n-4)*a(n-5)=0. - R. J. Mathar, Jun 23 2013

A026154 a(n) = T(n,n-3), where T is the array in A026148.

Original entry on oeis.org

1, 3, 11, 35, 111, 343, 1049, 3177, 9559, 28611, 85293, 253461, 751296, 2222442, 6563598, 19359022, 57038247, 167911721, 493972165, 1452419661, 4268753126, 12542145548, 36841741320, 108202146520, 317748977715, 933052724721, 2739805981773

Offset: 3

Clark Kimberling

nonn

First differences of A026328.

Conjecture: (n+9)*a(n) +(-7*n-47)*a(n-1) +(13*n+51)*a(n-2) +(5*n+37)*a(n-3) +2*(-13*n-6)*a(n-4) +2*(n-13)*a(n-5) +12*(n-4)*a(n-6)=0. - R. J. Mathar, Jun 23 2013

A026155 T(n,n-4), where T is the array in A026148.

Original entry on oeis.org

1, 4, 16, 56, 189, 616, 1967, 6182, 19205, 59124, 180726, 549276, 1661646, 5007520, 15042722, 45068836, 134727499, 401991436, 1197519631, 3562523314, 10585937404, 31424706800, 93206486620, 276253350360, 818278951035, 2422518642516

Offset: 4

Clark Kimberling

nonn

First differences of A026329.

Conjectures from Chai Wah Wu, Mar 23 2020: (Start)
a(n) = 12*a(n-1) - 53*a(n-2) + 90*a(n-3) + 26*a(n-4) - 256*a(n-5) + 160*a(n-6) + 224*a(n-7) - 203*a(n-8) - 84*a(n-9) + 77*a(n-10) + 14*a(n-11) - 7*a(n-12) for n > 15.
G.f.: x^4*(x - 1)^2*(x^6 - 6*x^5 - 13*x^4 + 8*x^3 + 8*x^2 - 6*x + 1)/((7*x^6 + 7*x^5 - 21*x^4 + 14*x^2 - 7*x + 1)*(x^6 - 3*x^5 - 5*x^4 + 8*x^3 + 4*x^2 - 5*x + 1)). (End)

A026156 a(n) = T(2n-1,n), where T is the array in A026148.

Original entry on oeis.org

1, 4, 20, 111, 616, 3464, 19656, 112370, 646272, 3735324, 21678696, 126256361, 737513400, 4319243760, 25352692620, 149108262264, 878502339712, 5184007871160, 30633757767144, 181253849830625, 1073680925191128, 6366778606697544, 37790436148353900, 224505805610383128

Offset: 1

Clark Kimberling

nonn

Offset corrected and more terms from Sean A. Irvine, Sep 18 2019

A026157 a(n) = T(2n,n), where T is the array in A026148.

Original entry on oeis.org

1, 1, 7, 35, 189, 1038, 5796, 32734, 186494, 1069793, 6170554, 35752906, 207940493, 1213271620, 7098581115, 41631465591, 244669507994, 1440588095878, 8495970243502, 50179297862166, 296763850899633

Offset: 0

Clark Kimberling

nonn

cp's OEIS Frontend

A026123 a(n) = number of (s(0),s(1),...,s(n)) such that every s(i) is a nonnegative integer, s(0) = 1, s(n) = 2, |s(1) - s(0)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = T(n,n-1), where T is the array in A026120; a(n) = U(n,n+1), where U is the array in A026148.

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A026163 Sum{T(k,k-1)}, k = 1,2,...,n, where T is the array in A026148.

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A026165 Number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 2, |s(1) - s(0)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also sum of numbers in row n+1 of the array T in A026148.

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A026151 a(n) = T(n,n), where T is the array in A026148.

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A026152 a(n) = T(n,n-1), where T is the array in A026148.

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A026153 T(n,n-2), where T is the array in A026148.

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A026154 a(n) = T(n,n-3), where T is the array in A026148.

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A026155 T(n,n-4), where T is the array in A026148.

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A026156 a(n) = T(2n-1,n), where T is the array in A026148.

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Extensions

A026157 a(n) = T(2n,n), where T is the array in A026148.

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