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

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.

A115089 Row sums of triangle A115085.

Original entry on oeis.org

1, 2, 6, 21, 91, 470, 2763, 17894, 125113, 932702, 7347025, 60761449, 524852444, 4716259252, 43936770258, 423178496553, 4203717419747, 42980494150963, 451494062943969, 4865135046178557, 53701937703205383, 606454041389250791
Offset: 0

Views

Author

Paul D. Hanna, Jan 13 2006

Keywords

Crossrefs

Cf. A115085, A115086 (column 0), A115087 (column 1), A115088 (column 2); A115084.

A128324 Row sums of triangle A128320.

Original entry on oeis.org

1, 2, 8, 31, 159, 955, 6677, 51308, 429868, 3847548, 36599474, 366515450, 3848812068, 42154442638, 480103999452, 5666303543327, 69139729751267, 870092119451903, 11272494698299169, 150074841511853609
Offset: 0

Views

Author

Paul D. Hanna, Feb 25 2007

Keywords

Crossrefs

Cf. A128320 (triangle), A128321 (column 0), A128322 (column 1), A128323 (column 2); variant: A115084.

Programs

  • PARI
    
				
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