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 1-7 of 7 results.

A134761 a(n) = (1/2)*( (1 + (-1)^n)*A134760(n/2) + (1 - (-1)^n) ).

Original entry on oeis.org

1, 1, 3, 1, 11, 1, 39, 1, 139, 1, 503, 1, 1847, 1, 6863, 1, 25739, 1, 97239, 1, 369511, 1, 1410863, 1, 5408311, 1, 20801199, 1, 80233199, 1, 310235039, 1, 1202160779, 1, 4667212439, 1, 18150270599, 1, 70690527599, 1, 275693057639, 1, 1076515748879, 1, 4208197927439, 1
Offset: 0

Views

Author

Gary W. Adamson, Nov 09 2007

Keywords

Comments

Second inverse binomial transform of A134760.
A134760 interpolated with 1's.
Former name: A007318^(-2) * A134760. - G. C. Greubel, May 27 2024

Examples

			The first few terms are (1, 1, 3, 1, 11, 1, 39, ...), since A134760 = (1, 3, 11, 39, 139, 503, ...).

Links

Crossrefs

Cf. A000984, A134760.

Programs

  • Magma
    A134761:= func< n | (n mod 2 eq 0) select 2*Binomial(2*Floor(n/2), Floor(n/2)) - 1 else 1 >;
[A134761(n): n in [0..70]]; // G. C. Greubel, May 27 2024
  • Mathematica
    Table[If[EvenQ[n], 2*(1+Floor[n/2])*CatalanNumber[Floor[n/2]]-1, 1], {n,0,70}] (* G. C. Greubel, May 27 2024 *)
  • SageMath
    def A134761(n): return 1 if (n%2==0) else 2*binomial(2*(n//2), (n//2)) -1
[A134761(n) for n in range(71)] # G. C. Greubel, May 27 2024

Formula

From G. C. Greubel, May 27 2024: (Start)
a(n) = (1/2)*( (1 + (-1)^n)*A134760(n/2) + (1 - (-1)^n) ).
G.f.: 2/sqrt(1 - 4*x^2) - 1/(1 + x).
E.g.f.: 2*BesselI(0, 2*x) - exp(-x).
a(n) = (-(n-1)*(3*n-4)*a(n-1) + 4*(3*n^2 -10*n +7)*a(n-2) + 4*(n-2)*(3*n-4)*a(n-3))/(n*(3*n-7)), with a(0) = a(1) = 1, a(2) = 3. (End)

Extensions

New name and terms a(14) onward added by G. C. Greubel, May 27 2024

A109128 Triangle read by rows: T(n,k) = T(n-1,k-1) + T(n-1,k) + 1 for 0

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 11, 7, 1, 1, 9, 19, 19, 9, 1, 1, 11, 29, 39, 29, 11, 1, 1, 13, 41, 69, 69, 41, 13, 1, 1, 15, 55, 111, 139, 111, 55, 15, 1, 1, 17, 71, 167, 251, 251, 167, 71, 17, 1, 1, 19, 89, 239, 419, 503, 419, 239, 89, 19, 1, 1, 21, 109, 329, 659, 923, 923, 659, 329, 109, 21, 1
Offset: 0

Views

Author

Reinhard Zumkeller, Jun 20 2005

Keywords

Comments

Eigensequence of the triangle = A001861. - Gary W. Adamson, Apr 17 2009

Examples

			Triangle begins as:
  1;
  1   1;
  1   3   1;
  1   5   5   1;
  1   7  11   7   1;
  1   9  19  19   9   1;
  1  11  29  39  29  11   1;
  1  13  41  69  69  41  13   1;
  1  15  55 111 139 111  55  15   1;
  1  17  71 167 251 251 167  71  17   1;
  1  19  89 239 419 503 419 239  89  19   1;

Links

Crossrefs

Cf. A000035, A000295, A001861, A005408, A014473, A028387, A030662.
Cf. A134760, A281362, A327767.
Cf. A000325 (row sums).
Sequence m*binomial(n,k) - (m-1): A007318 (m=1), this sequence (m=2), A131060 (m=3), A131061 (m=4), A131063 (m=5), A131065 (m=6), A131067 (m=7), A168625 (m=8).

Programs

  • Haskell
    a109128 n k = a109128_tabl !! n !! k
a109128_row n = a109128_tabl !! n
a109128_tabl = iterate (\row -> zipWith (+)
   ([0] ++ row) (1 : (map (+ 1) $ tail row) ++ [0])) [1]
-- Reinhard Zumkeller, Apr 10 2012
  • Magma
    [2*Binomial(n,k) -1: k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 12 2020
  • Maple
    A109128 := proc(n,k)
    2*binomial(n,k)-1 ;
end proc: # R. J. Mathar, Jul 12 2016
  • Mathematica
    Table[2*Binomial[n,k] -1, {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)
  • Sage
    [[2*binomial(n,k) -1 for k in (0..n)] for n in (0..12)] # G. C. Greubel, Mar 12 2020

Formula

T(n,k) = T(n-1,k-1) + T(n-1,k) + 1 with T(n,0) = T(n,n) = 1.
Sum_{k=0..n} T(n, k) = A000325(n+1) (row sums).
T(n, k) = 2*binomial(n,k) - 1. - David W. Cantrell (DWCantrell(AT)sigmaxi.net), Sep 30 2007
T(n, 1) = 2*n - 1 = A005408(n+1) for n>0.
T(n, 2) = n^2 + n - 1 = A028387(n-2) for n>1.
T(n, k) = Sum_{j=0..n-k} C(n-k,j)*C(k,j)*(2 - 0^j) for k <= n. - Paul Barry, Apr 27 2006
T(n,k) = A014473(n,k) + A007318(n,k), 0 <= k <= n. - Reinhard Zumkeller, Apr 10 2012
From G. C. Greubel, Apr 06 2024: (Start)
T(n, n-k) = T(n, k).
T(2*n, n) = A134760(n).
T(2*n-1, n) = A030662(n), for n >= 1.
Sum_{k=0..n-1} T(n, k) = A000295(n+1), for n >= 1.
Sum_{k=0..n} (-1)^k*T(n, k) = 2*[n=0] - A000035(n+1).
Sum_{k=0..n-1} (-1)^k*T(n, k) = A327767(n), for n >= 1.
Sum_{k=0..floor(n/2)} T(n-k, k) = A281362(n).
Sum_{k=0..floor((n-1)/2)} T(n-k, k) = A281362(n-1) - (1+(-1)^n)/2 for n >= 1.
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = b(n), where b(n) is the repeating pattern {1,1,0,-2,-3,-1,2,2,-1,-3,-2,0} with b(n) = b(n-12). (End)

Extensions

Offset corrected by Reinhard Zumkeller, Apr 10 2012

A134758 a(n) = A000984(n) + n.

Original entry on oeis.org

1, 3, 8, 23, 74, 257, 930, 3439, 12878, 48629, 184766, 705443, 2704168, 10400613, 40116614, 155117535, 601080406, 2333606237, 9075135318, 35345263819, 137846528840, 538257874461, 2104098963742, 8233430727623, 32247603683124, 126410606437777, 495918532948130
Offset: 0

Views

Author

Gary W. Adamson, Nov 09 2007

Keywords

Links

Crossrefs

Cf. A000984, A134759, A134760, A134761, A134762, A134763, A134770, A134771.

Programs

  • Magma
    [n+(n+1)*Catalan(n): n in [0..40]]; // G. C. Greubel, May 28 2024
  • Mathematica
    Table[Binomial[2n,n]+n,{n,0,40}] (* Harvey P. Dale, Dec 10 2011 *)
  • SageMath
    [n+binomial(2*n,n) for n in range(41)] # G. C. Greubel, May 28 2024

Formula

G.f.: ((1-x)^2 + x*sqrt(1-4*x))/((1-x)^2*sqrt(1-4*x)). - Harvey P. Dale, Dec 10 2011
From G. C. Greubel, May 28 2024: (Start)
E.g.f.: x*exp(x) + exp(2*x)*BesselI(0, 2*x).
a(n) = (2*(2*n-1)*a(n-1) - (3*n^2 - 6*n + 2))/n. (End)

Extensions

More terms from Harvey P. Dale, Dec 10 2011

A134759 a(n) = 2*A000984(n) - (n+1).

Original entry on oeis.org

1, 2, 9, 36, 135, 498, 1841, 6856, 25731, 97230, 369501, 1410852, 5408299, 20801186, 80233185, 310235024, 1202160763, 4667212422, 18150270581, 70690527580, 275693057619, 1076515748858, 4208197927417, 16466861455176, 64495207366175, 252821212875478
Offset: 0

Views

Author

Gary W. Adamson, Nov 09 2007

Keywords

Links

Crossrefs

Cf. A000108, A000984, A134758, A134760, A134761, A134762, A134763, A134770, A134771.

Programs

  • Magma
    [(n+1)*(2*Catalan(n)-1): n in [0..40]]; // G. C. Greubel, May 28 2024
  • Mathematica
    Table[2 Binomial[2n,n]-n-1,{n,0,30}] (* Harvey P. Dale, Aug 07 2023 *)
  • SageMath
    [2*binomial(2*n,n) -(n+1) for n in range(41)] # G. C. Greubel, May 28 2024

Formula

From G. C. Greubel, May 28 2024: (Start)
a(n) = (n+1)*(2*A000108(n) - 1).
a(n) = (2*(2*n-1)*a(n-1) + 3*n*(n-1))/n.
G.f.: 2/sqrt(1-4*x) - 1/(1-x)^2.
E.g.f.: 2*exp(2*x)*BesselI(0, 2*x) - (1+x)*exp(x). (End)

Extensions

More terms from Harvey P. Dale, Aug 07 2023

A134762 a(n) = 3*A000984(n) - 2.

Original entry on oeis.org

1, 4, 16, 58, 208, 754, 2770, 10294, 38608, 145858, 554266, 2116294, 8112466, 31201798, 120349798, 465352558, 1803241168, 7000818658, 27225405898, 106035791398, 413539586458, 1614773623318, 6312296891158, 24700292182798, 96742811049298, 379231819313254
Offset: 0

Views

Author

Gary W. Adamson, Nov 09 2007

Keywords

Comments

Second inverse binomial transform of the sequence = A134763, (same as a(n) but with interpolated two's).

Links

Crossrefs

Cf. A000108, A000984, A134758, A134759, A134760, A134761, A134763, A134770, A134771.

Programs

  • Magma
    [3*(n+1)*Catalan(n)-2: n in [0..40]]; // G. C. Greubel, May 28 2024
  • Mathematica
    Table[3*Binomial[2*n,n]-2, {n,0,40}] (* G. C. Greubel, May 28 2024 *)
  • PARI
    a(n) = 3*binomial(2*n, n) - 2; \\ Michel Marcus, Nov 22 2013
  • SageMath
    [3*binomial(2*n,n) -2 for n in range(41)] # G. C. Greubel, May 28 2024

Formula

G.f.: 3/sqrt(1-4*x) - 2/(1-x). - Sergei N. Gladkovskii, Nov 21 2013
From G. C. Greubel, May 28 2024: (Start)
a(n) = 3*(n+1)*A000108(n) - 2.
a(n) = (2*(2*n-1)*a(n-1) + 2*(3*n-2))/n.
E.g.f.: 3*exp(2*x)*BesselI(0, 2*x) - 2*exp(x). (End)

Extensions

More terms from Michel Marcus, Nov 22 2013

A134763 a(n) = (1/2)*( (1+(-1)^n)*A134762(n/2) + 2*(1-(-1)^n) ).

Original entry on oeis.org

1, 2, 4, 2, 16, 2, 58, 2, 208, 2, 754, 2, 2770, 2, 10294, 2, 38608, 2, 145858, 2, 554266, 2, 2116294, 2, 8112466, 2, 31201798, 2, 120349798, 2, 465352558, 2, 1803241168, 2, 7000818658, 2, 27225405898, 2, 106035791398, 2, 413539586458, 2, 1614773623318, 2, 6312296891158, 2
Offset: 0

Views

Author

Gary W. Adamson, Nov 09 2007

Keywords

Comments

Second inverse binomial transform of A134762.
A134762 interpolated with two's.
Former name: A000718^(-2) * A134762. - G. C. Greubel, May 28 2024

Examples

			First few terms of the sequence are: (1, 2, 4, 2, 16, 2, 58, ...), interpolating two's in the sequence A134762: (1, 4, 16, 58, ...).

Links

Crossrefs

Cf. A000718, A000984, A001405, A134758, A134759, A134760, A134761, A134762, A134770, A134771.

Programs

  • Magma
    [3*((n+1) mod 2)*Binomial(n, Floor(n/2)) - 2*(-1)^n : n in [0..40]]; // G. C. Greubel, May 28 2024
  • Mathematica
    Table[(3/2)*(1+(-1)^n)*Binomial[n,n/2] -2*(-1)^n, {n,0,40}] (* G. C. Greubel, May 28 2024 *)
  • SageMath
    [3*((n+1)%2)*binomial(n, n//2) - 2*(-1)^n for n in range(41)] # G. C. Greubel, May 28 2024

Formula

From G. C. Greubel, May 28 2024: (Start)
a(n) = (1/2)*( (1+(-1)^n)*A134762(n/2) + 2*(1-(-1)^n) ).
a(n) = (3/2)*(1+(-1)^n)*A001405(n) - 2*(-1)^n.
G.f.: 3/sqrt(1-4*x^2) - 2/(1+x).
E.g.f.: 3*BesselI(0, 2*x) - 2*exp(-x). (End)

Extensions

Name change and terms a(14) onward added by G. C. Greubel, May 28 2024

A209245 Main diagonal of the triple recurrence x(i,j,k) = x(i-1,j,k) + x(i,j-1,k) + x(i,j,k-1) with x(i,j,k) = 1 if 0 in {i,j,k}.

Original entry on oeis.org

1, 3, 33, 543, 10497, 220503, 4870401, 111243135, 2602452993, 61985744967, 1497148260033, 36566829737727, 901314269530113, 22385640256615743, 559574590912019457, 14065064484334380543, 355222860485671141377, 9008982166319523972903, 229325469394627488082497
Offset: 0

Views

Author

Jon Perry, Jan 13 2013

Keywords

Comments

Level sums are defined as the sum of x(i,j,k) with i,j,k >= 0 and i+j+k = n. This gives 3*A164039(n-1) for n>0.
Slice x(1,j,k) with j,k >= 0 of the cube begins:
1, 1, 1, 1, 1, 1, 1, 1, ... A000012
1, 3, 5, 7, 9, 11, 13, 15, ... A005408
1, 5, 11, 19, 29, 41, 55, 71, ... A028387
1, 7, 19, 39, 69, 111, 167, 239, ... A108766(k+1)
1, 9, 29, 69, 139, 251, 419, 659, ...
1, 11, 41, 111, 251, 503, 923, 1583, ...
1, 13, 55, 167, 419, 923, 1847, 3431, ...
1, 15, 71, 239, 659, 1583, 3431, 6863, ...
The main diagonal of the slice is A134760.

Links

Crossrefs

Cf. A164039, A134760, A209288.
Column k=3 of A210472. - Alois P. Heinz, Jan 23 2013

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, 2*n+1,
      ((888-3020*n+3668*n^2-1912*n^3+364*n^4) *a(n-1)
       +3*(3*n-4)*(7*n-5)*(2*n-3)*(3*n-5) *a(n-2)) /
       ((2*n-1)*(7*n-12)*(n-1)^2))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jan 17 2013
  • Mathematica
    b[] = 0; b[args__] := b[args] = If[{args}[[1]] == 0, 1, Sum[b @@ Sort[ ReplacePart[{args}, i -> {args}[[i]] - 1]], {i, 1, Length[{args}]}]];
a[n_] := b @@ Table[n, 3];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 03 2018, from Alois P. Heinz's Maple code for A210472 *)

Formula

a(n) = x(n,n,n) with x(i,j,k) = 1 if 0 in {i,j,k} and x(i,j,k) = x(i-1,j,k) + x(i,j-1,k) + x(i,j,k-1) else.
a(n) ~ 3^(3*n+1/2) / (8*Pi*n). - Vaclav Kotesovec, Sep 07 2014
Showing 1-7 of 7 results.

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

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