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 1-6 of 6 results.

A037956 a(n) = binomial(n, floor((n-4)/2)).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 6, 7, 28, 36, 120, 165, 495, 715, 2002, 3003, 8008, 12376, 31824, 50388, 125970, 203490, 497420, 817190, 1961256, 3268760, 7726160, 13037895, 30421755, 51895935, 119759850, 206253075
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A035951, A035952, A035953, A035954, A035955, A035957.
Cf. A089940, A101491.

Programs

  • Magma
    [Binomial(n, Floor((n-4)/2)): n in [0..40]]; // G. C. Greubel, Jun 20 2022
  • Maple
    seq(binomial(n,floor((n-4)/2)),n=0..50); # Robert Israel, Oct 28 2019
  • Mathematica
    Table[Binomial[n,Floor[(n-4)/2]],{n,0,40}] (* Harvey P. Dale, Mar 02 2015 *)
  • SageMath
    [binomial(n, (n-4)//2) for n in (0..40)] # G. C. Greubel, Jun 20 2022

Formula

E.g.f.: Bessel_I(4,2x) + Bessel_I(5,2x). - Paul Barry, Feb 28 2006
(n+5)*(n-4)*a(n) = -(n^2-3*n-20)*a(n-1) - (n^2-13*n-88)*a(n-2) + 2*(2*n+3)*(n-2)*a(n-3) +20*(n-2)*(n-3)*a(n-4). - R. J. Mathar, Nov 24 2012
From Robert Israel, Oct 28 2019: (Start)
G.f.: 16*x^4*(1+2*x+sqrt(1-4*x^2))/(sqrt(1-4*x^2)*(1+sqrt(1-4*x^2))^5).
Mathar's recurrence verified using the D.E. (4*x^4-x^2)*y'' + (16*x^3+2*x^2-2*x)*y' + (8*x^2+2*x+20)*y = 0 satisfied by the G.f. (End)

A037957 a(n) = binomial(n, floor((n-6)/2)).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 8, 9, 45, 55, 220, 286, 1001, 1365, 4368, 6188, 18564, 27132, 77520, 116280, 319770, 490314, 1307504, 2042975, 5311735, 8436285, 21474180, 34597290, 86493225, 141120525, 347373600
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A035951, A035952, A035953, A035954, A035955, A035956.
Cf. A089940, A101491.

Programs

  • Magma
    [Binomial(n, Floor((n-6)/2)): n in [0..40]]; // G. C. Greubel, Jun 20 2022
  • Mathematica
    Table[Binomial[n,Floor[(n-6)/2]],{n,0,40}] (* Harvey P. Dale, May 16 2017 *)
  • PARI
    a(n)=binomial(n,n\2-3) \\ Charles R Greathouse IV, Oct 23 2023
  • SageMath
    [binomial(n, (n-6)//2) for n in (0..40)] # G. C. Greubel, Jun 20 2022

Formula

(n+7)*(n-6)*a(n) = 2*n*a(n-1) + 4*n*(n-1)*a(n-2). - R. J. Mathar, Jul 26 2015
From G. C. Greubel, Jun 20 2022: (Start)
G.f.: ((1 + x - 7*x^2 - 6*x^3 + 14*x^4 + 9*x^5 - 7*x^6 - 2*x^7) - (1 + x - 5*x^2 - 4*x^3 + 6*x^4 + 3*x^5 - x^6)*sqrt(1-4*x^2))/(2*x^7*sqrt(1-4*x^2)).
E.g.f.: BesselI(6, 2*x) + BesselI(7, 2*x). (End)

A037951 a(n) = binomial(n, floor((n-3)/2)).

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 6, 21, 28, 84, 120, 330, 495, 1287, 2002, 5005, 8008, 19448, 31824, 75582, 125970, 293930, 497420, 1144066, 1961256, 4457400, 7726160, 17383860, 30421755, 67863915, 119759850, 265182525
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A035952, A035953, A035954, A035955, A035956, A035957.
Cf. A089940, A101491.

Programs

  • Magma
    [Binomial(n, Floor((n-3)/2)): n in [0..40]]; // G. C. Greubel, Jun 21 2022
  • Mathematica
    Table[Binomial[n,Floor[(n-3)/2]],{n,0,40}] (* Harvey P. Dale, Jul 09 2017 *)
  • SageMath
    [binomial(n, (n-3)//2) for n in (0..40)] # G. C. Greubel, Jun 21 2022

Formula

E.g.f.: Bessel_I(3, 2*x) + Bessel_I(4, 2*x) - Paul Barry, Feb 28 2006
(n+4)*(n-3)*a(n) = (-n^2+3*n+12)*a(n-1) + 2*(2*n+1)*(n-1)*a(n-2) + 4*(n-1)*(n-2)*a(n-3). - R. J. Mathar, Nov 24 2012
G.f.: ((1 +x -4*x^2 -3*x^3 +2*x^4) - (1 +x -2*x^2 -x^3)*sqrt(1-4*x^2))/(2*x^4*sqrt(1-4*x^2)). - G. C. Greubel, Jun 21 2022

A037953 a(n) = binomial(n, floor((n-5)/2)).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 7, 8, 36, 45, 165, 220, 715, 1001, 3003, 4368, 12376, 18564, 50388, 77520, 203490, 319770, 817190, 1307504, 3268760, 5311735, 13037895, 21474180, 51895935, 86493225, 206253075
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A035951, A035952, A035954, A035955, A035956, A035957.
Cf. A004526, A089940, A101491.

Programs

Formula

a(n) = binomial(n, A004526(n-5)). - Wesley Ivan Hurt, Nov 28 2013
(n+6)*(n-5)*a(n) = 2*(n)*a(n-1) + 4*(n-1)*n*a(n-2). - R. J. Mathar, Jul 26 2015
From G. C. Greubel, Jun 21 2022: (Start)
G.f.: ((1 +x -6*x^2 -5*x^3 +9*x^4 +5*x^5 -2*x^6) - (1 +x -4*x^2 -3*x^3 +3*x^4 +x^5)*sqrt(1-4*x^2))/(2*x^6*sqrt(1-4*x^2)).
E.g.f.: BesselI(5, 2*x) + BesselI(6, 2*x). (End)

A086905 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(k,floor(k/2)).

Original entry on oeis.org

1, 0, 2, 1, 5, 5, 15, 20, 50, 76, 176, 286, 638, 1078, 2354, 4081, 8789, 15521, 33099, 59279, 125477, 227239, 478193, 873885, 1830271, 3370029, 7030571, 13027729, 27088871, 50469889, 104647631, 195892564, 405187826, 761615284, 1571990936
Offset: 0

Views

Author

Vladeta Jovovic, Sep 19 2003

Keywords

Comments

Knödel walks starting and ending at 0, with n steps.

Links

Crossrefs

Cf. A036256, A001405.
First column of triangle A101491.

Programs

  • Mathematica
    Table[Sum[(-1)^(n-k)*Binomial[k,Floor[k/2]],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Mar 02 2014 *)
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*binomial(k,k\2)); \\ Michel Marcus, Dec 04 2016

Formula

G.f.: (sqrt((1+2*x)/(1-2*x))-1)/2/x/(1+x).
a(n) ~ 2^(n+3/2) / (3*sqrt(Pi*n)) * (1 - 2/(3*n)+ 3*(-1)^n/(4*n)). - Vaclav Kotesovec, Mar 02 2014

A037954 a(n) = binomial(n, floor((n-7)/2)).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 9, 10, 55, 66, 286, 364, 1365, 1820, 6188, 8568, 27132, 38760, 116280, 170544, 490314, 735471, 2042975, 3124550, 8436285, 13123110, 34597290, 54627300, 141120525, 225792840
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A035951, A035952, A035953, A035955, A035956, A035957.
Cf. A089940, A101491.

Programs

  • Magma
    [Binomial(n, Floor((n-7)/2)): n in [0..40]]; // G. C. Greubel, Jun 21 2022
  • Mathematica
    Table[Binomial[n,Floor[(n-7)/2]],{n,0,40}] (* Harvey P. Dale, Apr 15 2020 *)
  • SageMath
    [binomial(n, (n-7)//2) for n in (0..40)] # G. C. Greubel, Jun 21 2022

Formula

(n+8)*(n-7)*a(n) = 2*(n)*a(n-1) + 4*(n-1)*n*a(n-2). - R. J. Mathar, Jul 26 2015
From G. C. Greubel, Jun 21 2022: (Start)
G.f.: ((1 +x -8*x^2 -7*x^3 +20*x^4 +14*x^5 -16*x^6 -7*x^7 +2*x^8) - (1 +x -6*x^2 - 5*x^3 +10*x^4 +6*x^5 -4*x^6 -x^7)*sqrt(1-4*x^2))/(2*x^8*sqrt(1-4*x^2)).
E.g.f.: BesselI(7, 2*x) + BesselI(8, 2*x). (End)
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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