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 1-6 of 6 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(""))

A115081 Column 0 of triangle A115080.

Original entry on oeis.org

1, 1, 3, 11, 50, 257, 1467, 9081, 60272, 424514, 3151226, 24510411, 198870388, 1676878231, 14648843341, 132228263355, 1230505582380, 11782173683640, 115878367974480, 1168833058344870, 12075008262774120, 127608480923659770
Offset: 0

Views

Author

Paul D. Hanna, Jan 13 2006, Nov 19 2006

Keywords

Comments

Also equals row sums of triangle A125080.

Examples

			At n=5, a(5) = Sum_{k=0..2} A000108(5-k)*A001147(k)*C(5,2*k) so that a(5) = 42*1*C(5,0) + 14*1*C(5,2) + 5*3*C(5,4) = 42*1*1 + 14*1*10 + 5*3*5 = 42 + 140 + 75 = 257.

Crossrefs

Cf. A115080, A115082 (column 1), A115083 (column 2), A115084 (row sums); A115086.
Cf. A125080 (related triangle); A000108, A001147.

Programs

  • PARI
    {a(n)=sum(k=0,n\2,binomial(2*n-2*k,n-k)/(n-k+1)*binomial(2*k,k)*k!/2^k*binomial(n,2*k))}
  • PARI
    {a(n)=sum(k=0,n\2,(2*n-2*k)!*n!/k!/(n-k)!/(n-k+1)!/(n-2*k)!/2^k )}

Formula

a(n) = Sum_{k=0..[n/2]} A000108(n-k)*A001147(k)*C(n,2*k), where A000108 is the Catalan numbers and A001147 is the double factorials.
a(n) = Sum_{k=0..[n/2]} A000108(n-k)*A000108(k)*(k+1)!*C(n,2k)/2^k where A000108(n) = C(2n,n)/(n+1) are the Catalan numbers. a(n) = Sum_{k=0..n} (-1)^(n-k)*n!/k!*A115082(k) . - Paul D. Hanna, Feb 19 2007

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.

A115087 Column 1 of triangle A115085.

Original entry on oeis.org

1, 2, 5, 21, 102, 579, 3601, 24426, 176858, 1359906, 10996740, 93136545, 821734072, 7529402477, 71404315785, 699210393923, 7053151126260, 73158494370744, 778922147312076, 8500606925203992, 94961268583354364, 1084632255967292304
Offset: 0

Views

Author

Paul D. Hanna, Jan 13 2006

Keywords

Crossrefs

Cf. A115085, A115086 (column 0), A115088 (column 2), A115089 (row sums); A115082.

A128322 Column 1 of triangle A128320; a(n) = A128321(n) + 2n*A128321(n-1), where A128321 is column 0 of triangle A128320.

Original entry on oeis.org

1, 3, 8, 41, 234, 1602, 11976, 98541, 866942, 8139602, 80559456, 837863578, 9098447188, 102867879636, 1206145137840, 14632952210685, 183197674060470, 2362463132266770, 31320354882679440, 426245968192108590
Offset: 0

Views

Author

Paul D. Hanna, Feb 25 2007

Keywords

Crossrefs

Cf. A128320 (triangle), A128321 (column 0), A128323 (column 2), A128324 (row sums); variant: A115082.

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))}
for(n=0,25, print1(a(n),", "))

Formula

a(n) = Sum_{k=0..[(n+1)/2]} A000108(n-k)*A000108(k)*(k+1)!*C(n+1,2k) where A000108 is the Catalan numbers.
a(n) = Sum_{k=0..[(n+1)/2]} C(2(n-k),n-k)/(n-k+1)*C(2k,k)/(k+1)*(k+1)!*C(n+1,2k).
Showing 1-6 of 6 results.

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