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 13 results. Next

A026908 a(n) = A026907(2*n, n).

Original entry on oeis.org

1, 44, 198, 758, 2824, 10530, 39516, 149274, 567120, 2164898, 8297206, 31907214, 123053938, 475753142, 1843357752, 7155936162, 27826486272, 108369397602, 422614721742, 1650118887102, 6450143419962, 25238550057222
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A000108, A026907.

Programs

  • Magma
    A026908:= func< n | (n+1)*Catalan(n) +3*(n+3)*Catalan(n+2) -18 >;
[A026908(n): n in [0..40]]; // G. C. Greubel, Aug 22 2025
  • Mathematica
    With[{b=Binomial}, Table[b[2*n,n] +3*b[2*n+4,n+2] -18, {n,0,40}]] (* G. C. Greubel, Aug 22 2025 *)
  • SageMath
    def A026908(n): return binomial(2*n,n) +3*binomial(2*n+4,n+2) -18
print([A026908(n) for n in range(41)]) # G. C. Greubel, Aug 22 2025

Formula

a(n) = 4^n*Gamma(n+1/2)*(38 + 99*n + 49*n^2)*Pi^(-1/2)/Gamma(n+3) - 18 (based on guessed recurrence) - Mark van Hoeij, Oct 30 2011
From G. C. Greubel, Aug 22 2025: (Start)
a(n) = binomial(2*n, n) + 3*binomial(2*n+4, n+2) - 18.
a(n) = (n+2)*A000108(n) + 3*(n+3)*A000108(n+2) - 18.
G.f.: 2*(7 + 45*x + 2*x^2 - (5 - 23*x)*sqrt(1-4*x))/((1-x)*sqrt(1-4*x)*(1 + sqrt(1 - 4*x))^2).
E.g.f.: (1/x)*exp(2*x)*( 25*x*BesselI(0, 2*x) - 6*(1-4*x)*BesselI(1, 2*x) ) - 18*exp(x). (End)

A026909 a(n) = (1/2)*A026907(2*n, n).

Original entry on oeis.org

22, 99, 379, 1412, 5265, 19758, 74637, 283560, 1082449, 4148603, 15953607, 61526969, 237876571, 921678876, 3577968081, 13913243136, 54184698801, 211307360871, 825059443551, 3225071709981, 12619275028611, 49423455006501, 193732625020419, 760001601263697, 2983614441691035, 11720995167614703, 46074369462135607
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A000108, A026907, A026908.

Programs

  • Magma
    A026909:= func< n | ((n+1)*Catalan(n) +3*(n+3)*Catalan(n+2))/2 -9 >;
[A026909(n): n in [1..40]]; // G. C. Greubel, Aug 22 2025
  • Mathematica
    With[{b=Binomial}, Table[(b[2*n,n] +3*b[2*n+4,n+2] -18)/2, {n,40}]] (* G. C. Greubel, Aug 22 2025 *)
  • SageMath
    def A026909(n): return (binomial(2*n,n) +3*binomial(2*n+4,n+2))//2 -9
print([A026909(n) for n in range(1,41)]) # G. C. Greubel, Aug 22 2025

Formula

From G. C. Greubel, Aug 22 2025: (Start)
a(n) = (binomial(2*n, n) + 3*binomial(2*n+4, n+2) - 18)/2.
a(n) = ((n+2)*A000108(n) + 3*(n+3)*A000108(n+2))/2 - 9.
G.f.: 2*(3 + 25*x - x^2 - (3 - 13*x + x^2)*sqrt(1-4*x))/((1-x)*sqrt(1-4*x)*(1 + sqrt(1 - 4*x))^2).
E.g.f.: (1/(2*x))*exp(2*x)*( 25*x*BesselI(0, 2*x) - 6*(1-4*x)*BesselI(1, 2*x) ) - 9*exp(x) - 1/2. (End)

Extensions

More terms added by G. C. Greubel, Aug 22 2025

A026910 a(n) = A026907(2*n, n-1).

Original entry on oeis.org

28, 154, 627, 2414, 9201, 35094, 134259, 515302, 1983678, 7656374, 29619728, 114822606, 445925707, 1734610302, 6757306947, 26358212502, 102937963242, 402444721782, 1574952822852, 6169131608622, 24184904949252, 94885715007390, 372536351222832, 1463611239163614, 5753766284232606, 22632379126906174, 89072478723401106
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A000108, A026907.

Programs

  • Magma
    A026910:= func< n | n*Catalan(n) +3*(n+2)*Catalan(n+2) -18 >;
[A026910(n): n in [1..40]]; // G. C. Greubel, Aug 22 2025
  • Mathematica
    Table[n*CatalanNumber[n] +3*(n+2)*CatalanNumber[n+2] -18, {n,40}] (* G. C. Greubel, Aug 22 2025 *)
  • SageMath
    def A026910(n): return binomial(2*n,n-1) +3*binomial(2*n+4,n+1) -18
print([A026910(n) for n in range(1,41)]) # G. C. Greubel, Aug 22 2025

Formula

From G. C. Greubel, Aug 22 2025: (Start)
a(n) = n*A000108(n) + 3*(n+2)*A000108(n+2) - 18.
a(n) = binomial(2*n, n-1) + 3*binomial(2*n+4, n+1) - 18.
G.f.: (3 - 9*x + 7*x^2 - 3*x^3 + 2*x^4 - (3 - 3*x + 7*x^2 + 17*x^3 + 12*x^4)*sqrt(1-4*x))/( 2*(1-x)*x^3*sqrt(1-4*x) ).
E.g.f.: 6 - 18*exp(x) - (1/x^2)*exp(2*x)*( 6*x(1-4*x)*BesselI(0, 2*x) - (6 - 12*x + 25*x^2)*BesselI(1, 2*x) ). (End)

Extensions

More terms added by G. C. Greubel, Aug 22 2025

A026911 a(n) = A026907(2*n, n-2).

Original entry on oeis.org

67, 348, 1495, 6108, 24501, 97456, 385900, 1524066, 6009720, 23675882, 93226503, 367005692, 1444728537, 5687662392, 22395051912, 88199397642, 347448657492, 1369107075762, 5396498311992, 21277355051610, 83918011194996, 331073286677058, 1306540603377930, 5157617675058838, 20365730134359298, 80440031466243942
Offset: 2

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A026907.

Programs

  • Magma
    A026911:= func< n | Binomial(2*n,n-2) +3*Binomial(2*n+4,n) -18 >;
[A026911(n): n in [2..45]]; // G. C. Greubel, Aug 23 2025
  • Mathematica
    Table[Binomial[2*n,n-2] +3*Binomial[2*n+4,n] -18, {n,2,45}] (* G. C. Greubel, Aug 23 2025 *)
  • SageMath
    def A026911(n): return binomial(2*n,n-2) +3*binomial(2*n+4,n) -18
print([A026911(n) for n in range(2,46)]) # G. C. Greubel, Aug 23 2025

Formula

From G. C. Greubel, Aug 23 2025: (Start)
a(n) = binomial(2*n, n-2) + 3*binomial(2*n+4, n) - 18.
G.f.: (3 - 15*x + 19*x^2 - 11*x^3 + 6*x^4 - 2*x^5 - (3 - 9*x + 7*x^2 - 3*x^3 + 8*x^4 + 30*x^5)*sqrt(1-4*x))/(2*(1-x)*x^4*sqrt(1-4*x)).
E.g.f.: 15 - 18*exp(x) + (1/x^3)*exp(2*x)*(6*x*(3 - 4*x + 4*x^2)*BesselI(0, 2*x) - 6*(3 - 4*x + 5*x^2 - 4*x^3)*BesselI(1, 2*x) + x^3*BesselI(2, 2*x) ). (End)

Extensions

More terms added by G. C. Greubel, Aug 23 2025

A026914 T(n,[ n/2 ]), T given by A026907.

Original entry on oeis.org

1, 13, 44, 90, 198, 370, 758, 1403, 2824, 5256, 10530, 19749, 39516, 74628, 149274, 283551, 567120, 1082440, 2164898, 4148594, 8297206, 15953598, 31907214, 61526960, 123053938, 237876562, 475753142, 921678867, 1843357752
Offset: 0

Views

Author

Clark Kimberling

Keywords

A026915 a(n) = T(n,0) + T(n,1) + ... + T(n,n), T given by A026907.

Original entry on oeis.org

1, 26, 100, 272, 640, 1400, 2944, 6056, 12304, 24824, 49888, 100040, 200368, 401048, 802432, 1605224, 3210832, 6422072, 12844576, 25689608, 51379696, 102759896, 205520320, 411041192, 822082960, 1644166520, 3288333664, 6576667976, 13153336624, 26306673944
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • PARI
    Vec((1+22*x+x^2)/((1-x)^2*(1-2*x)) + O(x^40)) \\ Colin Barker, Feb 18 2016

Formula

a(n) = 49*2^n - 24*n - 48.
a(n) = 4*a(n-1)-5*a(n-2)+2*a(n-3) for n>2. G.f.: (1+22*x+x^2) / ((1-x)^2*(1-2*x)). - Colin Barker, Feb 18 2016

A026917 a(n) = Sum{T(i,j)}, 0<=j<=i, 0<=i<=n, T given by A026907.

Original entry on oeis.org

1, 27, 127, 399, 1039, 2439, 5383, 11439, 23743, 48567, 98455, 198495, 398863, 799911, 1602343, 3207567, 6418399, 12840471, 25685047, 51374655, 102754351, 205514247, 411034567, 822075759, 1644158719, 3288325239, 6576658903, 13153326879, 26306663503
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Programs

  • Mathematica
    LinearRecurrence[{5,-9,7,-2},{1,27,127,399},30] (* Harvey P. Dale, Jul 03 2017 *)
  • PARI
    Vec((1+22*x+x^2)/((1-x)^3*(1-2*x)) + O(x^40)) \\ Colin Barker, Feb 18 2016

Formula

a(n) = 98*2^n - 12*n^2 - 60*n - 97.
From Colin Barker, Feb 18 2016: (Start)
a(n) = 5*a(n-1)-9*a(n-2)+7*a(n-3)-2*a(n-4) for n>3.
G.f.: (1+22*x+x^2) / ((1-x)^3*(1-2*x)).
(End)

A026912 T(2n-1,n-1), T given by A026907.

Original entry on oeis.org

13, 90, 370, 1403, 5256, 19749, 74628, 283551, 1082440, 4148594, 15953598, 61526960, 237876562, 921678867, 3577968072, 13913243127, 54184698792, 211307360862, 825059443542, 3225071709972, 12619275028602, 49423455006492
Offset: 1

Views

Author

Clark Kimberling

Keywords

A026913 T(2n-1,n-2), T given by A026907.

Original entry on oeis.org

46, 239, 993, 3927, 15327, 59613, 231733, 901220, 3507762, 13666112, 53295628, 208049127, 812931417, 3179338857, 12444969357, 48753264432, 191137360902, 749893379292, 2944059898632, 11565629920632, 45462260000880, 178803726202404
Offset: 2

Views

Author

Clark Kimberling

Keywords

A026916 a(n) = T(n,0) + T(n,1) + ... + T(n,[ n/2 ]), T given by A026907.

Original entry on oeis.org

1, 13, 72, 136, 419, 700, 1851, 3028, 7564, 12412, 30209, 50020, 119942, 200524, 475853, 802612, 1888976, 3211036, 7504737, 12844804, 29838451, 51379948, 118713767, 205520596, 472568449, 822083260, 1882043403
Offset: 0

Views

Author

Clark Kimberling

Keywords

Showing 1-10 of 13 results. Next

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

Content is available under The OEIS End-User License Agreement.