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 1-5 of 5 results.

A099035 a(n) = (n+1)*2^(n-1) - 1.

Original entry on oeis.org

1, 5, 15, 39, 95, 223, 511, 1151, 2559, 5631, 12287, 26623, 57343, 122879, 262143, 557055, 1179647, 2490367, 5242879, 11010047, 23068671, 48234495, 100663295, 209715199, 436207615, 905969663, 1879048191, 3892314111, 8053063679
Offset: 1

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Author

Ralf Stephan, Sep 28 2004

Keywords

Comments

Row sums of triangle A135852. - Gary W. Adamson, Dec 01 2007
Binomial transform of [1, 4, 6, 8, 10, 12, 14, 16, ...]. Equals A128064 * A000225, (A000225 starting 1, 3, 7, 15, ...). - Gary W. Adamson, Dec 28 2007

Links

Crossrefs

First differences of A066524.
Cf. A135852, A128064, A000225.

Programs

Formula

a(n) = A057711(n+1) - 1 = A058966(n+3)/2 = (A087323(n)-1)/2 = (A074494(n+1)-2)/3 = (A003261(n+1)-3)/4 = A036289(n+1)/4 - 1, n>0.
a(n) = A131056(n+1) - 2. - Juri-Stepan Gerasimov, Oct 02 2011
From Colin Barker, Mar 23 2012: (Start)
a(n) = 5*a(n-1) - 8*a(n-2) + 4*a(n-3).
G.f.: x*(1-2*x^2)/((1-x)*(1-2*x)^2). (End)
E.g.f.: ((2*x+1)*exp(2*x) - 2*exp(x) + 1)/2. - G. C. Greubel, Dec 31 2017

A131055 1 followed by repeats of 2*k.

Original entry on oeis.org

1, 2, 2, 4, 4, 6, 6, 8, 8, 10, 10, 12, 12, 14, 14, 16, 16, 18, 18, 20, 20, 22, 22, 24, 24, 26, 26, 28, 28, 30, 30, 32, 32, 34, 34, 36, 36, 38, 38, 40, 40, 42, 42, 44, 44, 46, 46, 48, 48, 50, 50, 52, 52, 54, 54, 56, 56, 58, 58, 60, 60, 62, 62, 64, 64, 66, 66, 68, 68, 70, 70, 72
Offset: 1

Views

Author

Gary W. Adamson, Jun 12 2007

Keywords

Links

Crossrefs

Cf. A131056.

Programs

  • Maple
    seq(`if`(n=1, 1, n-(n mod 2)), n=1..72);
  • Mathematica
    Join[{1}, Table[2*Floor[i/2], {i, 2, 81}]] (* Stefan Steinerberger, Jun 13 2007 *)
With[{c=2*Range[40]},Join[{1},Riffle[c,c]]] (* Harvey P. Dale, Jul 25 2019 *)

Formula

Inverse binomial transform of A131056: (1, 3, 7, 17, 41, 97, 225, ...).
From Colin Barker, Oct 28 2012: (Start)
a(n) = ((-1)^n + 2*n-1)/2 for n > 1.
a(n) = a(n-1) + a(n-2) - a(n-3) for n > 1.
G.f.: x*(x^3-x^2+x+1)/((x-1)^2*(x+1)). (End)
E.g.f.: x*(exp(x) + 1) - sinh(x). - Stefano Spezia, Jan 28 2024

Extensions

More terms from Stefan Steinerberger, Jun 13 2007

A141222 Expansion of -1/(2*x) + (2*x-1)^2/(2*x*(1-4x)^(3/2)).

Original entry on oeis.org

1, 5, 22, 95, 406, 1722, 7260, 30459, 127270, 529958, 2200276, 9111830, 37650172, 155266100, 639191160, 2627302995, 10784089350, 44208873390, 181025067300, 740483276610, 3026059513620, 12355464845100
Offset: 0

Views

Author

Paul Barry, Jun 14 2008

Keywords

Comments

Apply Riordan array (1/sqrt(1-4x), xc(x)) to A131056, c(x) the g.f. of A000108.
Apply Riordan array (c(x)/sqrt(1-4*x), x*c(x)^2) to A131055.
Hankel transform appears to be (-1)^n*A085046(n).
Coefficients T(2*n+1,n) of triangle A103450. [Emanuele Munarini, Jun 01 2012, corrected by Werner Schulte, Nov 27 2021]

Links

Programs

  • Mathematica
    Table[((1+3*n+n^2)*Binomial[2*n, n])/(n+1),{n,0,20}] (* Vaclav Kotesovec, Feb 13 2014 *)
CoefficientList[Series[-1/(2*x)+(2*x-1)^2/(2*x*(1-4x)^(3/2)),{x,0,20}],x] (* Vaclav Kotesovec, Feb 13 2014 *)
a[n_] := (1 + 3 n + n^2) CatalanNumber[n];
Table[a[n], {n, 0, 21}] (* Peter Luschny, Nov 28 2021 *)
  • Maxima
    a(n):=sum(binomial(2*n,k)*binomial(n+1,2*n-k),k,0,n); makelist(a(n),n,0,40); /* Emanuele Munarini, Jun 01 2012 */

Formula

a(n) = Sum_{k=0..n} (1 + (k+1)*2^(k-1) - 0^k/2)*C(2n-k,n-k); a(n) = Sum_{k=0..n} C(2n,k)*C(n+1,2n-k).
Equals the Narayana transform (A001263) of integer squares. - Gary W. Adamson, Jul 29 2011
Conjecture: (n+1)*a(n) + 2*(-3*n-1)*a(n-1) + 4*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 24 2012
From Vaclav Kotesovec, Feb 13 2014: (Start)
G.f.: -1/(2*x) + (2*x-1)^2/(2*x*(1-4x)^(3/2)).
a(n) = (1 + 3*n + n^2) * C(2*n,n) / (n+1).
Recurrence: (n+1)*(n^2 + n - 1)*a(n) = 2*(2*n-1)*(n^2 + 3*n + 1)*a(n-1).
(End)

Extensions

Name of the sequence corrected by Vaclav Kotesovec, Feb 13 2014

A046688 Antidiagonals of square array in which k-th row (k>0) is an arithmetic progression of difference 2^(k-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 5, 1, 1, 5, 7, 9, 9, 1, 1, 6, 9, 13, 17, 17, 1, 1, 7, 11, 17, 25, 33, 33, 1, 1, 8, 13, 21, 33, 49, 65, 65, 1, 1, 9, 15, 25, 41, 65, 97, 129, 129, 1, 1, 10, 17, 29, 49, 81, 129, 193, 257, 257, 1, 1, 11, 19, 33, 57, 97, 161, 257, 385, 513, 513, 1
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			From _Gus Wiseman_, May 08 2021: (Start):
Array A(m,n) = 1 + n*2^(m-1) begins:
       n=0: n=1: n=2: n=3: n=4: n=5: n=6: n=7: n=8: n=9:
  m=0:   1    1    1    1    1    1    1    1    1    1
  m=1:   1    2    3    5    9   17   33   65  129  257
  m=2:   1    3    5    9   17   33   65  129  257  513
  m=3:   1    4    7   13   25   49   97  193  385  769
  m=4:   1    5    9   17   33   65  129  257  513 1025
  m=5:   1    6   11   21   41   81  161  321  641 1281
  m=6:   1    7   13   25   49   97  193  385  769 1537
  m=7:   1    8   15   29   57  113  225  449  897 1793
  m=8:   1    9   17   33   65  129  257  513 1025 2049
  m=9:   1   10   19   37   73  145  289  577 1153 2305
Triangle T(n,k) = 1 + (n-k)*2^(k-1) begins:
   1
   1   1
   1   2   1
   1   3   3   1
   1   4   5   5   1
   1   5   7   9   9   1
   1   6   9  13  17  17   1
   1   7  11  17  25  33  33   1
   1   8  13  21  33  49  65  65   1
   1   9  15  25  41  65  97 129 129   1
   1  10  17  29  49  81 129 193 257 257   1
   1  11  19  33  57  97 161 257 385 513 513   1
(End)

References

  • G. H. Hardy, A Theorem Concerning the Infinite Cardinal Numbers, Quart. J. Math., 35 (1904), p. 90 = Collected Papers, Vol. VII, p. 430.

Links

Crossrefs

Row sums are A000079.
Diagonal n = m + 1 of the array is A002064.
Diagonal n = m of the array is A005183.
Column m = 1 of the array is A094373.
Diagonal n = m - 1 of the array is A131056.
A002109 gives hyperfactorials (sigma: A260146, omega: A303281).
A009998(k,n) = n^k.
A009999(n,k) = n^k.
A057156 = (2^n)^(2^n).
A062319 counts divisors of n^n.
Cf. A000169, A000272, A000312, A036289, A343656, A343658.

Programs

  • Mathematica
    Table[If[k==0,1,n*2^(k-1)+1],{n,0,9},{k,0,9}] (* ARRAY, Gus Wiseman, May 08 2021 *)
Table[If[k==0,1,1+(n-k)*2^(k-1)],{n,0,10},{k,0,n}] (* TRIANGLE, Gus Wiseman, May 08 2021 *)
  • PARI
    A(m,n)={if(m>0, 1+n*2^(m-1), 1)}
{ for(m=0, 10, for(n=0, 10, print1(A(m,n), ", ")); print) } \\ Andrew Howroyd, Mar 07 2020

Formula

A(m,n) = 1 + n*2^(m-1) for m > 1. - Andrew Howroyd, Mar 07 2020
As a triangle, T(n,k) = A(k,n-k) = 1 + (n-k)*2^(k-1). - Gus Wiseman, May 08 2021

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Apr 06 2000

A166335 Exponential Riordan array [1+x*sinh(x), x].

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 6, 0, 1, 4, 0, 12, 0, 1, 0, 20, 0, 20, 0, 1, 6, 0, 60, 0, 30, 0, 1, 0, 42, 0, 140, 0, 42, 0, 1, 8, 0, 168, 0, 280, 0, 56, 0, 1, 0, 72, 0, 504, 0, 504, 0, 72, 0, 1, 10, 0, 360, 0, 1260, 0, 840, 0, 90, 0, 1
Offset: 0

Views

Author

Paul Barry, Oct 12 2009

Keywords

Examples

			Triangle begins
1,
0, 1,
2, 0, 1,
0, 6, 0, 1,
4, 0, 12, 0, 1,
0, 20, 0, 20, 0, 1,
6, 0, 60, 0, 30, 0, 1,
0, 42, 0, 140, 0, 42, 0, 1,
8, 0, 168, 0, 280, 0, 56, 0, 1,
0, 72, 0, 504, 0, 504, 0, 72, 0, 1,
10, 0, 360, 0, 1260, 0, 840, 0, 90, 0, 1

Crossrefs

Row sums are A131056. Diagonal sums are A166336. Central coefficients are A166337.

Programs

  • Mathematica
    (* The function RiordanArray is defined in A256893. *)
RiordanArray[1 + # Sinh[#]&, #&, 11, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

Formula

Number triangle T(n,k)=[k<=n]*C(n,k)*((n-k)+0^((n-k)/2))(1+(-1)^(n-k))/2.
Showing 1-5 of 5 results.

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

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