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.

A115080 Triangle, read by rows, where T(n,k) equals the dot product of the vector of terms in row n that are to the right of T(n,k) with the vector of terms in column k that are above T(n,k): T(n,k) = Sum_{j=0..n-k-1} T(n,j+k+1)*T(j+k,k) for n > k+1 > 0, with T(n,n) = 1 and T(n,n-1) = n (n>=1).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 11, 5, 3, 1, 50, 20, 7, 4, 1, 257, 94, 31, 9, 5, 1, 1467, 507, 150, 44, 11, 6, 1, 9081, 3009, 853, 218, 59, 13, 7, 1, 60272, 19350, 5251, 1307, 298, 76, 15, 8, 1, 424514, 132920, 35109, 8313, 1881, 390, 95, 17, 9, 1, 3151226, 966962, 249332
Offset: 0

Views

Author

Paul D. Hanna, Jan 13 2006

Keywords

Comments

Triangle A115085 is the dual of this triangle.

Examples

			T(n,k) = [T(n,k+1),T(n,k+2),...,T(n,n)]*[T(k,k),T(k+1,k),...,T(n-1,k)]:
  T(3,0) = [5,3,1]*[1,1,3] = 5*1 + 3*1 + 1*3 = 11;
  T(4,1) = [7,4,1]*[1,2,5] = 7*1 + 4*2 + 1*5 = 20;
  T(5,1) = [31,9,5,1]*[1,2,5,20] = 31*1 + 9*2 + 5*5 + 1*20 = 94;
  T(6,2) = [44,11,6,1]*[1,3,7,31] = 44*1 + 11*3 + 6*7 + 1*31 = 150.
Triangle begins:
         1;
         1,       1;
         3,       2,       1;
        11,       5,       3,      1;
        50,      20,       7,      4,     1;
       257,      94,      31,      9,     5,     1;
      1467,     507,     150,     44,    11,     6,    1;
      9081,    3009,     853,    218,    59,    13,    7,   1;
     60272,   19350,    5251,   1307,   298,    76,   15,   8,   1;
    424514,  132920,   35109,   8313,  1881,   390,   95,  17,   9,  1;
   3151226,  966962,  249332,  57738, 12315,  2587,  494, 116,  19, 10,  1;
  24510411, 7396366, 1873214, 422948, 88737, 17377, 3437, 610, 139, 21, 11, 1;
  ...

Links

Crossrefs

Cf. A115081 (column 0), A115082 (column 1), A115083 (column 2), A115084 (row sums); A115085 (dual triangle).

Programs

  • PARI
    {T(n,k)=if(n==k,1,if(n==k+1,n, sum(j=0,n-k-1,T(n,j+k+1)*T(j+k,k))))}
for(n=0,12,for(k=0,n, print1(T(n,k),", "));print(""))

A115082 Column 1 of triangle A115080.

Original entry on oeis.org

1, 2, 5, 20, 94, 507, 3009, 19350, 132920, 966962, 7396366, 59173897, 492995320, 4262193275, 38125138575, 351960913470, 3346157796060, 32700768584100, 327957494280000, 3370522049859990, 35451669429671520, 381183654441916290
Offset: 0

Views

Author

Paul D. Hanna, Jan 13 2006

Keywords

Crossrefs

Cf. A115080, A115081 (column 0), A115083 (column 2), A115084 (row sums); A115087.
Cf. A000108.

Programs

  • PARI
    {a(n)=sum(k=0,(n+1)\2,binomial(2*n-2*k,n-k)/(n-k+1)*binomial(2*k,k)/(k+1) *(k+1)!*binomial(n+1,2*k)/2^k)} \\ Paul D. Hanna, Feb 18 2007

Formula

From Paul D. Hanna, Feb 18 2007: (Start)
a(n) = A115081(n) + n*A115081(n-1).
a(n) = Sum_{k=0..[(n+1)/2]} A000108(n-k)*A000108(k)*(k+1)!*C(n+1,2*k)/2^k, where A000108(n) = C(2*n,n)/(n+1) is the n-th Catalan number. (End)

A115083 Column 2 of triangle A115080.

Original entry on oeis.org

1, 3, 7, 31, 150, 853, 5251, 35109, 249332, 1873214, 14754858, 121380083, 1037889452, 9197178411, 84207477581, 794810896751, 7717524260040, 76956293344620, 786822019659120, 8237319107295510, 88193381907972840
Offset: 0

Views

Author

Paul D. Hanna, Jan 13 2006

Keywords

Crossrefs

Cf. A115080, A115081 (column 0), A115082 (column 1), A115084 (row sums); A115088.

A115084 Row sums of triangle A115080.

Original entry on oeis.org

1, 2, 6, 20, 82, 397, 2186, 13241, 86578, 603249, 4440800, 34313272, 276904847, 2324369733, 20227700385, 181987602305, 1688734327673, 16129293322211, 158281104406067, 1593383284029479, 16431757177327175, 173372313294822983
Offset: 0

Views

Author

Paul D. Hanna, Jan 13 2006

Keywords

Crossrefs

Cf. A115080, A115081 (column 0), A115082 (column 1), A115083 (column 2); A115089.

A115086 Column 0 of triangle A115085.

Original entry on oeis.org

1, 1, 3, 12, 58, 321, 1963, 13053, 92946, 702864, 5599204, 46746501, 407019340, 3682922657, 34519445611, 334288656446, 3336972646500, 34270606413428, 361466138304372, 3909661505975148, 43305438098350844, 490632571238216052
Offset: 0

Views

Author

Paul D. Hanna, Jan 13 2006

Keywords

Crossrefs

Cf. A115085, A115087 (column 1), A115088 (column 2), A115089 (row sums); A115081.

A128321 Column 0 of triangle A128320.

Original entry on oeis.org

1, 1, 4, 17, 98, 622, 4512, 35373, 300974, 2722070, 26118056, 263266346, 2780054884, 30586452652, 349724463584, 4141218303165, 50678688359190, 639387728054310, 8302396672724280, 110754894628585950
Offset: 0

Views

Author

Paul D. Hanna, Feb 25 2007

Keywords

Links

Crossrefs

Cf. A128320 (triangle), A128322 (column 1), A128323 (column 2), A128324 (row sums); variant: A115081.
Cf. A000108 (Catalan numbers).

Programs

  • Magma
    I:=[1,1,4]; [n le 3 select I[n] else (-(n-2)*(n-3)*Self(n-1) + 4*(3*(n-2)^2+n-3)*Self(n-2) + 8*(n-3)^2*(n-1)*Self(n-3))/n: n in [1..30]]; // G. C. Greubel, Jun 25 2024
  • Mathematica
    a[n_]:= a[n]= If[n<3, (n!)^2, (-(n-1)*(n-2)*a[n-1] +4*(3*n^2-5*n +1)*a[n-2] + 8*(n-2)^2*n*a[n-3])/(n+1)];
Table[a[n], {n,0,40}] (* G. C. Greubel, Jun 25 2024 *)
  • PARI
    {a(n)=sum(k=0,n\2,binomial(2*n-2*k,n-k)/(n-k+1)*binomial(2*k,k)/(k+1) *(k+1)!*binomial(n,2*k))}
  • SageMath
    @CachedFunction
def a(n): # a = A128321
    if n<3: return (1,1,4)[n]
    else: return (-(n-1)*(n-2)*a(n-1) + 4*(3*n^2-5*n+1)*a(n-2) + 8*(n-2)^2*n*a(n-3))/(n+1)
[a(n) for n in range(31)] # G. C. Greubel, Jun 25 2024

Formula

a(n) = Sum_{k=0..floor(n/2)} A000108(n-k)*A000108(k)*(k+1)!*C(n,2*k).
a(n) = Sum_{k=0..floor((n+1)/2)} ((k+1)!*C(2*(n-k), n-k)*C(2*k, k)*C(n, 2*k))/((k+1)*(n-k+1)).
a(n) = ( -(n-1)*(n-2)*a(n-1) + 4*(3*n^2 -5*n +1)*a(n-2) + 8*n*(n-2)^2* a(n-3) )/(n+1), with a(0) = 1, a(1) = 1, a(2) = 4. - G. C. Greubel, Jun 25 2024
a(n) ~ 2^(3*n/2 + 1) * exp(sqrt(2*n) - n/2 - 1/2) * n^((n-3)/2) / sqrt(Pi) * (1 - 7/(3*sqrt(2*n))). - Vaclav Kotesovec, Jun 25 2024

A125080 Triangle, read by rows, defined by T(n,k) = A000108(n-k)*A001147(k)*C(n,2*k), for k=0..[n/2], n>=0, where A000108 is the Catalan numbers and A001147 is the double factorials.

Original entry on oeis.org

1, 1, 2, 1, 5, 6, 14, 30, 6, 42, 140, 75, 132, 630, 630, 75, 429, 2772, 4410, 1470, 1430, 12012, 27720, 17640, 1470, 4862, 51480, 162162, 166320, 39690, 16796, 218790, 900900, 1351350, 623700, 39690, 58786, 923780, 4813380, 9909900, 7432425
Offset: 0

Views

Author

Paul D. Hanna, Nov 19 2006

Keywords

Examples

			Table begins:
1;
1;
2, 1;
5, 6;
14, 30, 6;
42, 140, 75;
132, 630, 630, 75;
429, 2772, 4410, 1470;
1430, 12012, 27720, 17640, 1470;
4862, 51480, 162162, 166320, 39690;
16796, 218790, 900900, 1351350, 623700, 39690; ...

Crossrefs

Cf. A115081 (row sums), A115080; A000108, A001147.

Programs

  • PARI
    T(n,k)=binomial(2*n-2*k,n-k)/(n-k+1)*binomial(2*k,k)*k!/2^k*binomial(n,2*k)
  • PARI
    T(n,k)=(2*n-2*k)!*n!/k!/(n-k)!/(n-k+1)!/(n-2*k)!/2^k

Formula

Row sums equals A115081, which is column 0 of triangle A115080.
Showing 1-7 of 7 results.

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

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