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.

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.

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.

A115085 Triangle, read by rows, where T(n,k) equals the dot product of the vector of terms in row n-1 from T(n-1,k) to T(n-1,n-1) with the vector of terms in column k+1 from T(k+1,k+1) to T(n,k+1): T(n,k) = Sum_{j=0..n-k-1} T(n-1,j+k)*T(j+k+1,k+1) 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, 12, 5, 3, 1, 58, 21, 7, 4, 1, 321, 102, 32, 9, 5, 1, 1963, 579, 158, 45, 11, 6, 1, 13053, 3601, 933, 226, 60, 13, 7, 1, 92946, 24426, 5939, 1395, 306, 77, 15, 8, 1, 702864, 176858, 41385, 9097, 1977, 398, 96, 17, 9, 1, 5599204, 1359906, 306070
Offset: 0

Views

Author

Paul D. Hanna, Jan 13 2006

Keywords

Comments

Triangle A115080 is the dual of this triangle.

Examples

			T(n,k)=[T(n-1,k),T(n-1,k+1),..,T(n-1,n-1)]*[T(k+1,k+1),T(k+2,k+1),..,T(n,k+1)]:
12 = [3,2,1]*[1,2,5] = 3*1 + 2*2 + 1*5;
21 = [5,3,1]*[1,3,7] = 5*1 + 3*3 + 1*7;
102 = [21,7,4,1]*[1,3,7,32] = 21*1 + 7*3 + 4*7 + 1*32;
158 = [32,9,5,1]*[1,4,9,45] = 32*1 + 9*4 + 5*9 + 1*45.
Triangle begins:
1;
1, 1;
3, 2, 1;
12, 5, 3, 1;
58, 21, 7, 4, 1;
321, 102, 32, 9, 5, 1;
1963, 579, 158, 45, 11, 6, 1;
13053, 3601, 933, 226, 60, 13, 7, 1;
92946, 24426, 5939, 1395, 306, 77, 15, 8, 1;
702864, 176858, 41385, 9097, 1977, 398, 96, 17, 9, 1;
5599204, 1359906, 306070, 65310, 13195, 2691, 502, 117, 19, 10, 1;
46746501, 10996740, 2403792, 494022, 97701, 18353, 3549, 618, 140, 21, 11, 1;
407019340, 93136545, 19799468, 3970878, 755834, 140178, 24691, 4563, 746, 165, 23, 12, 1; ...

Links

Crossrefs

Cf. A115086 (column 0), A115087 (column 1), A115088 (column 2), A115089 (row sums); A115080 (dual triangle).

Programs

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

A128320 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) for n>k+1>0, with the odd numbers in the secondary diagonal and all 1's in the main diagonal.

Original entry on oeis.org

1, 1, 1, 4, 3, 1, 17, 8, 5, 1, 98, 41, 12, 7, 1, 622, 234, 73, 16, 9, 1, 4512, 1602, 418, 113, 20, 11, 1, 35373, 11976, 3110, 650, 161, 24, 13, 1, 300974, 98541, 23920, 5242, 930, 217, 28, 15, 1, 2722070, 866942, 207549, 41304, 8094, 1258, 281, 32, 17, 1
Offset: 0

Views

Author

Paul D. Hanna, Feb 25 2007

Keywords

Examples

			Illustrate the recurrence by:
  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) = [8,5,1]*[1,1,4]~ = 8*1 + 5*1 + 1*4 = 17;
  T(4,1) = [12,7,1]*[1,3,8]~ = 12*1 + 7*3 + 1*8 = 41;
  T(5,1) = [73,16,9,1]*[1,3,8,41]~ = 73*1 + 16*3 + 9*8 + 1*41 = 234;
  T(6,2) = [113,20,11,1]*[1,5,12,73]~ = 113*1 + 20*5 + 11*12 + 1*73 = 418.
Triangle begins:
         1;
         1,       1;
         4,       3,       1;
        17,       8,       5,      1;
        98,      41,      12,      7,     1;
       622,     234,      73,     16,     9,     1;
      4512,    1602,     418,    113,    20,    11,    1;
     35373,   11976,    3110,    650,   161,    24,   13,   1;
    300974,   98541,   23920,   5242,   930,   217,   28,  15,  1;
   2722070,  866942,  207549,  41304,  8094,  1258,  281,  32, 17,  1;
  26118056, 8139602, 1885166, 377757, 65088, 11762, 1634, 353, 36, 19, 1;

Links

Crossrefs

Columns k: A128321 (k=0), A128322 (k=1), A128323 (k=2).
Sums: A128324 (row sums).
Variant of: A115080.

Programs

  • Magma
    function T(n,k) // T = A128320
   if k eq n then return 1;
   elif k eq n-1 then return 2*n-1;
   else return (&+[T(n, k+j+1)*T(k+j, k): j in [0..n-k-1]]);
   end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 25 2024
  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==n, 1, If[k==n-1, 2*n-1, Sum[T[n,k+j+1] *T[k+j,k], {j,0,n-k-1}]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 25 2024 *)
  • PARI
    {T(n,k)=if(n==k,1, if(n==k+1,2*n-1, sum(i=0,n-k-1, T(n,k+i+1)*T(k+i,k))))};
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))
  • SageMath
    @CachedFunction
def T(n,k): # T = A128320
    if k==n: return 1
    elif k==n-1: return 2*n-1
    else: return sum(T(n, k+j+1)*T(k+j, k) for j in range(n-k))
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jun 25 2024

Formula

T(n,k) = Sum_{j=0..n-1-k} T(n,k+j+1)*T(k+j,k) for n > k+1 > 0, with T(n,n) = 1 and T(n, n-1) = 2*n-1 for k >= 0.

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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