cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 18 results. Next

A026010 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) = 2. Also a(n) = sum of numbers in row n+1 of array T defined in A026009.

Original entry on oeis.org

1, 2, 4, 7, 14, 25, 50, 91, 182, 336, 672, 1254, 2508, 4719, 9438, 17875, 35750, 68068, 136136, 260338, 520676, 999362, 1998724, 3848222, 7696444, 14858000, 29716000, 57500460, 115000920, 222981435, 445962870, 866262915, 1732525830, 3370764540
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

Conjecture: a(n) is the number of integer compositions of n + 2 in which the even parts appear as often at even positions as at odd positions (confirmed up to n = 19). - Gus Wiseman, Mar 17 2018

Examples

			The a(3) = 7 compositions of 5 in which the even parts appear as often at even positions as at odd positions are (5), (311), (131), (113), (221), (122), (11111). Missing are (41), (14), (32), (23), (212), (2111), (1211), (1121), (1112). - _Gus Wiseman_, Mar 17 2018

Links

Crossrefs

First differences of A050168. Pairwise sums of A037952.
Cf. A000712, A001405, A005774, A045931, A063886, A097613, A130780, A171966, A239241, A299926, A300061, A300787, A300788, A300789.

Programs

  • Magma
    [(&+[Binomial(Floor((n+k)/2), Floor(k/2)): k in [0..n]]): n in [0..40]]; // G. C. Greubel, Nov 08 2018
  • Mathematica
    Array[Sum[Binomial[Floor[(# + k)/2], Floor[k/2]], {k, 0, #}] &, 34, 0] (* Michael De Vlieger, May 16 2018 *)
Table[2^(-1 + n)*(((2 + 3*#)*Gamma[(1 + #)/2])/(Sqrt[Pi]*Gamma[2 + #/2]) &[n + Mod[n, 2]]), {n,0,40}] (* Peter Pein, Nov 08 2018 *)
Table[(1/2)^((5 - (-1)^n)/2)*(6*n + 7 - 3*(-1)^n)*CatalanNumber[(2*n + 1 - (-1)^n)/4], {n, 0, 40}] (* G. C. Greubel, Nov 08 2018 *)
  • PARI
    vector(40, n, n--; sum(k=0,n, binomial(floor((n+k)/2), floor(k/2)))) \\ G. C. Greubel, Nov 08 2018

Formula

a(2*n) = ((3*n + 1)/(2*n + 1))*C(2*n + 1, n)= A051924(1+n), n>=0, a(2*n-1) = a(2*n)/2 = A097613(1+n), n >= 1. - Herbert Kociemba, May 08 2004
a(n) = Sum_{k=0..n} binomial(floor((n+k)/2), floor(k/2)). - Paul Barry, Jul 15 2004
Inverse binomial transform of A005774: (1, 3, 9, 26, 75, 216, ...). - Gary W. Adamson, Oct 22 2007
Conjecture: (n+3)*a(n) - 2*a(n-1) + (-5*n-3)*a(n-2) + 2*a(n-3) + 4*(n-3)*a(n-4) = 0. - R. J. Mathar, Jun 20 2013
a(n) = (1/2)^((5 - (-1)^n)/2)*(6*n + 7 - 3*(-1)^n)*Catalan((2*n + 1 - (-1)^n)/4), where Catalan is the Catalan number = A000108. - G. C. Greubel, Nov 08 2018

A026013 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) = 2, s(2n) = 4. Also a(n) = T(2n,n-1), where T is the array defined in A026009.

Original entry on oeis.org

1, 4, 15, 55, 200, 726, 2639, 9620, 35190, 129200, 476102, 1760673, 6533150, 24319050, 90795375, 339929880, 1275977670, 4801199400, 18106714050, 68430306750, 259129680264, 983085703116, 3736124441990, 14222020085880, 54221213973500
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A000108, A026009.

Programs

  • Magma
    [Binomial(2*n, n-1) - Binomial(2*n, n-4): n in [1..30]]; // G. C. Greubel, Mar 18 2021
  • Mathematica
    Rest[CoefficientList[Series[(-1 + Sqrt[1 - 4*x])^3 * (1 - x) * (-1 + Sqrt[1 - 4*x] + 2*x) / (16*x^4), {x, 0, 20}], x]] (* Vaclav Kotesovec, Sep 03 2019 *)
With[{f = CatalanNumber}, Table[f[n+3] -4*f[n+2] +3*f[n+1] +Sum[f[j+1]*f[n-j-1], {j, 0, n-1}], {n,30}]] (* G. C. Greubel, Mar 18 2021 *)
  • Sage
    [binomial(2*n, n-1) - binomial(2*n, n-4) for n in (1..30)] # G. C. Greubel, Mar 18 2021

Formula

G.f.: (x + x^2*C^3)*C^3 where C = g.f. for Catalan numbers A000108.
E.g.f.: exp(2*x)*(Bessel_I(1,2*x)-Bessel_I(4,2*x)). - Paul Barry, Jun 04 2007
Conjecture: (n+4)*(n+1)*a(n) -4*(n^2+4*n+6)*a(n-1) +4*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 13 2012
a(n) ~ 15 * 4^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 03 2019
From G. C. Greubel, Mar 18 2021: (Start)
a(n) = C(n+3) - 4*C(n+2) + 3*C(n+1) + Sum_{j=0..n-1} C(j+1)*C(n-j-1), where C(n) are the Catalan numbers (A000108).
a(n) = binomial(2*n, n-1) - binomial(2*n, n-4) = A026009(2*n, n-1). (End)

Extensions

Corrected by Franklin T. Adams-Watters, Oct 25 2006

A026014 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) = 2, s(2n) = 6. Also a(n) = T(2n,n-2), where T is the array defined in A026009.

Original entry on oeis.org

1, 6, 28, 119, 483, 1911, 7448, 28764, 110466, 422807, 1615152, 6163885, 23514855, 89714835, 342411120, 1307613480, 4997082510, 19111589280, 73154916744, 280265589198, 1074685552094, 4124573481446, 15843809385168, 60914041121640
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

First differences of A000588.
Cf. A000108, A026009.

Programs

  • Magma
    [Binomial(2*n, n-2) - Binomial(2*n, n-5): n in [2..30]]; // G. C. Greubel, Mar 19 2021
  • Mathematica
    Table[Binomial[2*n, n-2] - Binomial[2*n, n-5], {n, 2, 30}] (* G. C. Greubel, Mar 19 2021 *)
  • Sage
    [binomial(2*n, n-2) - binomial(2*n, n-5) for n in (2..30)] # G. C. Greubel, Mar 19 2021

Formula

-(n-2)*(n+5)*(n+23)*a(n) +(-n^3+127*n^2+188*n-432)*a(n-1) +2*(n-1)*(2*n-3)*(5*n-24)*a(n-2) = 0. - R. J. Mathar, Jun 20 2013
From G. C. Greubel, Mar 19 2021: (Start)
G.f.: (1-x)*(1 -7*x +14*x^2 -7*x^3 -(1 -5*x +6*x^2 -x^3)*sqrt(1-4*x))/(2*x^5).
G.f.: (1-x)*x^2*C(x)^7, where C(x) is the g.f. of the Catalan numbers (A000108).
E.g.f.: exp(2*x)*(BesselI(2, 2*x) - BesselI(5, 2*x)).
a(n) = binomial(2*n, n-2) - binomial(2*n, n-5) = A026009(2*n, n-2).
a(n) = 1 if n = 2 else f(n) - f(n-1), where f(n) = Sum_{j=0..n-2} C(n-j-2)*(C(j+5) -4*C(j+4) +3*C(j+3)) and C(n) are the Catalan numbers. (End)
From G. C. Greubel, Mar 22 2021: (Start)
a(n) = C(n+4) -6*C(n+3) +11*C(n+2) -7*C(n+1) +C(n).
a(n) = 21*(n*(n-1)*(n^2+n+4)/((n+2)*(n+3)*(n+4)*(n+5)))*C(n), where C(n) are the Catalan numbers. (End)

A026015 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) = 2, s(2n) = 8. Also a(n) = T(2n,n-3), where T is the array defined in A026009.

Original entry on oeis.org

1, 8, 45, 219, 987, 4248, 17748, 72675, 293436, 1172908, 4653935, 18366075, 72186075, 282861360, 1105877880, 4316224860, 16825024134, 65525448960, 255024693434, 992116674142, 3858537980286, 15004402265424, 58343871881400
Offset: 3

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

First differences of A001392.
Cf. A000108, A026009.

Programs

  • Magma
    [Binomial(2*n, n-3) - Binomial(2*n, n-6): n in [3..30]]; // G. C. Greubel, Mar 19 2021
  • Mathematica
    Table[Binomial[2*n, n-3] - Binomial[2*n, n-6], {n, 3, 30}] (* G. C. Greubel, Mar 19 2021 *)
  • Sage
    [binomial(2*n, n-3) - binomial(2*n, n-6) for n in (3..30)] # G. C. Greubel, Mar 19 2021

Formula

-(n+6)*(n-3)*a(n) +2*(3*n^2+3*n-20)*a(n-1) +(-9*n^2+15*n+20)*a(n-2) +2*(n-2)*(2*n-5)*a(n-3) = 0. - R. J. Mathar, Jun 20 2013
From G. C. Greubel, Mar 19 2021: (Start)
G.f.: (1-x)*((1-3*x)*(1 -6*x +9*x^2 -3*x^3) -(1-x)*(1 -6*x +9*x^2 -x^3)*sqrt(1-4*x))/(2*x^6).
G.f.: (1-x)*x^3*C(x)^9, where C(x) is the g.f. of the Catalan numbers (A000108).
E.g.f.: exp(2*x)*(BesselI(3, 2*x) - BesselI(6, 2*x)).
a(n) = binomial(2*n, n-3) - binomial(2*n, n-6) = A026009(2*n, n-3).
a(n) = f(n) - f(n-1), where f(n) = Sum_{j=0..n-3} C(n-j-3)*(C(j+7) -6*C(j+6) +10*C(j+5) -4*C(j+4)) and C(n) are the Catalan numbers. (End)
From G. C. Greubel, Mar 22 2021: (Start)
a(n) = C(n+5) -8*C(n+4) +22*C(n+3) -25*C(n+2) +11*C(n+1) -C(n).
a(n) = (9/20)*(binomial(n,3)/binomial(n+6,5))*(3*n^2 +3*n +20)*C(n). (End)

A026017 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) = 2, s(2n-1) = 5. Also a(n) = T(2n-1,n-2), where T is the array defined in A026009.

Original entry on oeis.org

1, 5, 21, 83, 319, 1209, 4550, 17068, 63954, 239666, 898909, 3375825, 12697035, 47833905, 180510210, 682341000, 2583591150, 9798281910, 37218303330, 141585223494, 539395269462, 2057771255210, 7860697923436, 30065829471048, 115135255095140, 441410428339972
Offset: 2

Views

Author

Clark Kimberling

Keywords

Crossrefs

First differences of A003517.

Formula

Expansion of (1+x^1*C^3)*C^4, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
Conjecture: (n+4)*a(n) +(-8*n-17)*a(n-1) +(19*n+1)*a(n-2) +6*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Jun 20 2013

A026018 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) = 2, s(2n-1) = 7. Also a(n) = T(2n-1,n-3), where T is the array defined in A026009.

Original entry on oeis.org

1, 7, 36, 164, 702, 2898, 11696, 46512, 183141, 716243, 2788060, 10817820, 41880930, 161900910, 625272480, 2413491360, 9313307370, 35936613414, 138680365704, 535290282632, 2066802226236, 7983111461732, 30848211650592
Offset: 3

Views

Author

Clark Kimberling

Keywords

Crossrefs

First differences if A003518.

Formula

Conjecture: -(n+5)*(3*n-37)*a(n) +3*(-n^2-84*n-173)*a(n-1) +2*(32*n^2+295*n+254)*a(n-2) -8*(n+25)*(2*n-5)*a(n-3)=0. - R. J. Mathar, Jun 20 2013

A026021 T(n,[ n/2 ]), where T is the array defined in A026009.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 19, 34, 62, 117, 207, 407, 704, 1430, 2431, 5070, 8502, 18122, 30056, 65246, 107236, 236436, 385662, 861764, 1396652, 3157325, 5088865, 11622015, 18642420, 42961470, 68624295, 159419670, 253706790, 593636670, 941630580
Offset: 0

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A026009.

Formula

Conjecture: -(n+7)*(5*n-34)*a(n) +8*(-n-14)*a(n-1) +(27*n^2-83*n-428)*a(n-2) +8*(4*n+5)*a(n-3) 44*(n-3)*(7*n-27)*a(n-4)=0. - R. J. Mathar, Jun 20 2013

A027287 Self-convolution of array T given by A026009.

Original entry on oeis.org

1, 2, 6, 15, 54, 118, 471, 973, 4128, 8262, 36481, 71534, 324876, 627926, 2911923, 5568348, 26241376, 49767432, 237550401, 447567059, 2158650078, 4045351674, 19679880649, 36716551814, 179920446576, 334415846660
Offset: 0

Views

Author

Clark Kimberling

Keywords

A027288 a(n) = Sum_{k=0..floor(n/2)} T(n,k) * T(n,k+1), with T given by A026009.

Original entry on oeis.org

1, 4, 12, 46, 145, 552, 1820, 6918, 23562, 89645, 312664, 1191885, 4232345, 16170728, 58229100, 222997812, 811985790, 3116622123, 11451785640, 44047806725, 163072988442, 628465576101, 2341484487832, 9040083756276, 33863310322100, 130957152428622
Offset: 0

Views

Author

Clark Kimberling

Keywords

Extensions

More terms from Sean A. Irvine, Oct 26 2019

A027289 a(n) = Sum_{k=0..floor(n/2)-1} T(n,k) * T(n,k+2), with T given by A026009.

Original entry on oeis.org

1, 3, 18, 55, 264, 847, 3744, 12465, 52868, 180895, 749220, 2617173, 10680656, 37915317, 153240864, 551019963, 2212417314, 8038946075, 32129224712, 117758845827, 469105048654, 1731917060158, 6882935960496, 25569740611270, 101442831429264, 378875492015643
Offset: 2

Views

Author

Clark Kimberling

Keywords

Extensions

More terms from Sean A. Irvine, Oct 26 2019
Showing 1-10 of 18 results. Next

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