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.

Previous Showing 11-17 of 17 results.

A026074 a(n) = T(3n,n), where T is the array defined in A024996.

Original entry on oeis.org

1, 1, 12, 91, 726, 5902, 48704, 406353, 3419020, 28958955, 246598716, 2109182557, 18106771400, 155929895685, 1346442959176, 11653675471305, 101071521560286, 878176794266261, 7642505705447340, 66606583068846975
Offset: 0

Views

Author

Clark Kimberling

Keywords

A026075 a(n) = T(4n,n), where T is the array defined in A024996.

Original entry on oeis.org

1, 2, 23, 230, 2395, 25518, 276298, 3026790, 33452225, 372278104, 4165976594, 46831604742, 528454140310, 5982318103300, 67908970800315, 772718361911646, 8810943692302959, 100652401936330376, 1151695085791305180, 13197378357577994370
Offset: 0

Views

Author

Clark Kimberling

Keywords

A026076 a(n) = T(2n,n-1), where T is the array defined in A024996.

Original entry on oeis.org

1, 2, 12, 62, 339, 1882, 10594, 60216, 344846, 1986620, 11499940, 66835540, 389741279, 2279233530, 13362109050, 78505098150, 462109616730, 2724728263348, 16089931091944, 95141866318652
Offset: 1

Views

Author

Clark Kimberling

Keywords

A026077 a(n) = T(2n,n+1), where T is the array defined in A024996.

Original entry on oeis.org

1, 6, 40, 232, 1352, 7854, 45683, 266214, 1554684, 9098740, 53357985, 313495934, 1845063496, 10876094870, 64203285600, 379498497900, 2245861126596, 13305537121404, 78907795361228, 468393492593496
Offset: 1

Views

Author

Clark Kimberling

Keywords

A025177 Triangular array, read by rows: first differences in n,n direction of trinomial array A027907.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 2, 4, 4, 4, 2, 1, 1, 3, 7, 10, 12, 10, 7, 3, 1, 1, 4, 11, 20, 29, 32, 29, 20, 11, 4, 1, 1, 5, 16, 35, 60, 81, 90, 81, 60, 35, 16, 5, 1, 1, 6, 22, 56, 111, 176, 231, 252, 231, 176, 111, 56, 22, 6, 1, 1, 7, 29, 84, 189, 343, 518, 659, 714, 659, 518, 343
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

The Motzkin transforms of the rows starting (1, 2), (1, 3) and (1, 4), extended by zeros after their last element, are apparently in A026134, A026109 and A026110. - R. J. Mathar, Dec 11 2008

Examples

			               1
            1  0  1
         1  1  2  1  1
      1  2  4  4  4  2  1
   1  3  7 10 12 10  7  3  1
1  4 11 20 29 32 29 20 11  4  1

Links

Crossrefs

Columns include A025178, A025179, A025180, A025181, A025182.
Cf. A024996, A025192 (row sums).

Programs

  • Maple
    A025177 := proc(n,k)
    option remember;
    if k < 0 or k > 2*n then
        0;
    elif n = 0 then
        1 ;
    elif n = 1 then
        op(k+1,[1,0,1]) ;
    else
        procname(n-1,k-2)+procname(n-1,k-1)+procname(n-1,k) ;
    end if;
end proc:
seq(seq(A025177(n,k),k=0..2*n),n=0..20)  ; # R. J. Mathar, Feb 25 2015
  • Mathematica
    nmax = 10; CoefficientList[CoefficientList[Series[(1 - y*x)/(1 - x*(1 + y + y^2)), {x, 0, nmax}, {y, 0, 2*nmax}], x], y] // Flatten (* G. C. Greubel, May 22 2017; amended by Georg Fischer, Jun 24 2020 *)
  • PARI
    {T(n, k) = if( k<0 || k>2*n, 0, if( n==0, 1, if( n==1, [1,0,1][k+1], if( n==2, [1,1,2,1,1][k+1], T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)))))};
  • PARI
    T(n,k)=polcoeff(Ser(polcoeff(Ser((1-y*z)/(1-z*(1+y+y^2)),y),k,y),z),n,z)
  • PARI
    {T(n, k) = if( k<0 || k>2*n, 0, if( n==0, 1, polcoeff( (1 + x + x^2)^n, k) - polcoeff( (1 + x + x^2)^(n-1), k-1)))};
  • PARI
    g=matrix(33,65);
for(n=0,32,for(k=0,2*n,g[n+1,k+1]=0));
g[1,1]=1;
g[2,1]=1;g[2,2]=0;g[2,3]=1;
g[3,1]=1;g[3,2]=1;g[3,3]=2;g[3,4]=1;g[3,5]=1;
for(n=0,2,for(k=0,2*n,print(n," ",k," ",g[n+1,k+1])))
for(n=3,32,g[n+1,1]=1;print(n," 1 1");g[n+1,2]=n-1;print(n," 2 ",n-1);for(k=2,2*n,g[n+1,k+1]=g[n,k-1]+g[n,k]+g[n,k+1];print(n," ",k," ",g[n+1,k+1])))
\\ Michael B. Porter, Feb 02 2010

Formula

T(n, k) = T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), starting with [1], [1, 0, 1].
G.f.: (1-y*z)/[1-z*(1+y+y^2)].

Extensions

Edited by Ralf Stephan, Jan 09 2005
Offset corrected by R. J. Mathar, Feb 25 2015

A026082 Irregular triangular array T read by rows: T(n,k) = C(n,k) for k=0..n for n = 0,1,2,3. For n >= 4, T(n,0) = T(n,2n)=1, T(n,1) = T(n,2n-1) = n - 3, T(4,2) = 4, T(4,3) = 3, T(4,4) = 6; T(4,5) = 3, T(4,6)=4; for n >= 5, T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k) for k=2..2n-2.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 1, 4, 3, 6, 3, 4, 1, 1, 1, 2, 6, 8, 13, 12, 13, 8, 6, 2, 1, 1, 3, 9, 16, 27, 33, 38, 33, 27, 16, 9, 3, 1, 1, 4, 13, 28, 52, 76, 98, 104, 98, 76, 52, 28, 13, 4, 1, 1, 5, 18, 45, 93, 156, 226, 278, 300, 278, 226, 156, 93, 45, 18, 5, 1, 1, 6, 24, 68, 156, 294, 475, 660, 804
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

For n >= 4, T(n,k) = number of strings s(0)..s(n) such that s(n) = n - k, s(0) = 0, |s(i)-s(i-1)| = 1 for i=1,2,3 and |s(i)-s(i-1)| <= 1 for i >= 4.

Examples

			First 6 rows:
  1
  1  1
  1  2  1
  1  3  3  1
  1  1  4  3  6  3  4  1  1
  1  2  6  8 12 12 13  8  6  2  1

Links

Crossrefs

First differences of A024996.

Programs

  • Maple
    A026082 := proc(n,k)
    option remember;
    if n < 0 or k < 0 or k > 2*n then
        0 ;
    elif n <= 3 then
        binomial(n,k) ;
    elif n = 4 then
        op(k+1,[1,1,4,3,6,3,4,1,1]) ;
    elif k =0 or k=2*n then
        1 ;
    else
        procname(n-1,k-2)+procname(n-1,k-1)+procname(n-1,k) ;
    end if;
end proc: # R. J. Mathar, Jun 23 2013
  • Mathematica
    z = 15; t[n_, 0] := 1 /; n >= 4; t[n_, 1] := n - 3 /; n >= 4;
t[4, 2] = 4; t[4, 3] = 3; t[4, 4] = 6; t[4, 5] = 3; t[4, 6] = 4;
t[n_, k_] := t[n, k] = Which[0 <= k <= n && 0 <= n <= 3, Binomial[n, k], n
>= 4 && k == 2 n, 1, k == 2 n - 1, n - 3, 2 <= k <= 2 n - 2, t[n - 1, k -
2] + t[n - 1, k - 1] + t[n - 1, k]]; s = Table[Binomial[n, k], {n, 0, 3},
{k, 0, n}]; u = Join[s, Table[t[n, k], {n, 4, z}, {k, 0, 2 n}]];
TableForm[u] (* A026082 array *)
Flatten[u]   (* A026082 sequence *)

Formula

G.f.: (1-y*z)^3 / (1-z*(1+y+y^2)).

Extensions

Updated by Clark Kimberling, Aug 28 2014

A025579 a(1)=1, a(2)=2, a(n) = 4*3^(n-3) for n >= 3.

Original entry on oeis.org

1, 2, 4, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78732, 236196, 708588, 2125764, 6377292, 19131876, 57395628, 172186884, 516560652, 1549681956, 4649045868, 13947137604, 41841412812, 125524238436, 376572715308, 1129718145924
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

a(n) is the sum of the numbers in row n+1 of the array defined in A025564 (and of the array in A024996).
a(n) is the number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, |s(i) - s(i-1)| = 1 for i = 1,2; |s(i) - s(i-1)| <= 1 for i >= 3.
Equals binomial transform of A095342: (1, 1, 5, 5, 17, 25, 61, ...). - Gary W. Adamson, Mar 04 2010

Links

Crossrefs

Cf. A003946, A024996, A025564, A095342.

Programs

  • GAP
    Concatenation([1,2], List([3..30], n-> 4*3^(n-3) )); # G. C. Greubel, Dec 26 2019
  • Magma
    [1,2] cat [4*3^(n-3): n in [3..30]]; // G. C. Greubel, Dec 26 2019
  • Maple
    seq( `if`(n<3, n, 4*3^(n-3)), n=1..30); # G. C. Greubel, Dec 26 2019
  • Mathematica
    Join[{1,2},4*3^Range[0,30]] (* or *) Join[{1,2},NestList[3#&,4,30]] (* Harvey P. Dale, Jun 27 2011 *)
  • PARI
    a(n)=max(n,4*3^(n-3)) \\ Charles R Greathouse IV, Jun 28 2011
  • PARI
    Vec(x*(1+x)*(1-2*x)/(1-3*x) + O(x^30)) \\ Colin Barker, Oct 29 2019
  • Sage
    [1,2]+[4*3^(n-3) for n in (3..30)] # G. C. Greubel, Dec 26 2019

Formula

a(n) = A003946(n-2), n>2. - R. J. Mathar, May 28 2008
From Colin Barker, Oct 29 2019: (Start)
G.f.: x*(1 + x)*(1 - 2*x) / (1 - 3*x).
a(n) = 3*a(n-1) for n>3. (End)

Extensions

Definition corrected by R. J. Mathar, May 28 2008
Previous Showing 11-17 of 17 results.

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