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.

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A117207 Number triangle read by rows: T(n,k) = Sum_{j=0..n-k} C(n+j,j+k)*C(n-j,k).

Original entry on oeis.org

1, 3, 1, 10, 7, 1, 35, 31, 13, 1, 126, 121, 81, 21, 1, 462, 456, 381, 181, 31, 1, 1716, 1709, 1583, 1058, 358, 43, 1, 6435, 6427, 6231, 5055, 2605, 645, 57, 1, 24310, 24301, 24013, 21661, 14605, 5785, 1081, 73, 1, 92378, 92368, 91963, 87643, 70003, 38251, 11791
Offset: 0

Views

Author

Paul Barry, Mar 02 2006

Keywords

Comments

Row sums are A037965(n+1).
Second column is A048775. - Paul Barry, Oct 01 2010
First column is A001700. - Dan Uznanski, Jan 23 2012
The number of different ordered partitions of n+1 into n+1 bins (as with A001700), such that more than k bins are nonempty. - Dan Uznanski, Jan 23 2012
Second diagonal is A002061. - Franklin T. Adams-Watters, Jan 24 2012

Examples

			Triangle begins:
     1,
     3,    1,
    10,    7,    1,
    35,   31,   13,    1,
   126,  121,   81,   21,   1,
   462,  456,  381,  181,  31,  1,
  1716, 1709, 1583, 1058, 358, 43, 1

Links

Programs

  • Mathematica
    Table[Sum[Binomial[n+j,j+k]Binomial[n-j,k],{j,0,n-k}],{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Apr 23 2016 *)
  • PARI
    T(n,k)=sum(j=0,n-k, binomial(n+j,j+k)*binomial(n-j,k))
T(n,k)=binomial(2*n+1,n+1)-(n+1)*sum(j=1,k, binomial(n,j-1)^2/j)
A117207(k)=my(n=sqrtint(2*k-sqrtint(2*k))); T(n,k-n*(n+1)/2) \\ M. F. Hasler, Jan 25 2012

Formula

T(n,k) = C(2*n+1,n+1) - (n+1)*Sum_{j=1..k} (Product_{i=0..j-2} (n-i)^2)/((j-1)!*j!).
T(n,k) = [x^(n-k)](1+x)^(n-k)*F(-n-1,-n,1,x/(1+x)). - Paul Barry, Oct 01 2010
T(n,k) = C(2*n+1,n+1) - (n+1)*Sum_{j=1..k} C(n,j-1)^2/j. - M. F. Hasler, Jan 25 2012

A119574 a(n) = binomial(2*n,n)*(n+2)^2/(n+1).

Original entry on oeis.org

4, 9, 32, 125, 504, 2058, 8448, 34749, 143000, 588302, 2418624, 9934834, 40770352, 167152500, 684656640, 2801810205, 11455885080, 46801769190, 191055480000, 779363066790, 3177034283280, 12942655253580, 52693956656640, 214412258531250, 871975203591024
Offset: 0

Views

Author

Zerinvary Lajos, May 31 2006

Keywords

Links

Crossrefs

Cf. A000108, A000290, A000984, A037965, A110609.

Programs

  • Maple
    [seq (binomial(2*n,n)*(n+2)^2/(n+1),n=0..25)];
  • Mathematica
    Table[Binomial[2n,n] (n+2)^2/(n+1),{n,0,30}] (* Harvey P. Dale, Jun 02 2024 *)

Formula

Conjectured g.f.: (-1 + 14*x - 36*x^2 + (1 - 4*x)^(3/2))/(2*x*(1 - 4*x)^(3/2)). - Harvey P. Dale, Jun 02 2024.
The conjecture is true (see links). - Sela Fried, Oct 02 2024.
a(n) = A000108(n)*A000290(n+2). - Alois P. Heinz, Oct 02 2024

A176564 Triangle T(n,m)= binomial(2*n,m) + binomial(2*n,n-m) -binomial(2*n,n) read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, -6, -14, -6, 1, 1, -32, -87, -87, -32, 1, 1, -120, -363, -484, -363, -120, 1, 1, -415, -1339, -2067, -2067, -1339, -415, 1, 1, -1414, -4742, -7942, -9230, -7942, -4742, -1414, 1, 1, -4844, -16643, -29240, -36992, -36992, -29240
Offset: 0

Views

Author

Roger L. Bagula, Apr 20 2010

Keywords

Comments

Row sums are 1, 2, 4, 4, -24, -236, -1448, -7640, -37424, -175436,... = 2*A032443(n) -A037965(n+1).

Examples

			The triangle starts in row n=0 with columns 0<=m<=n as:
1;
1, 1;
1, 2, 1;
1, 1, 1, 1;
1, -6, -14, -6, 1;
1, -32, -87, -87, -32, 1;
1, -120, -363, -484, -363, -120, 1;
1, -415, -1339, -2067, -2067, -1339, -415, 1;
1, -1414, -4742, -7942, -9230, -7942, -4742, -1414, 1;
1, -4844, -16643, -29240, -36992, -36992, -29240, -16643, -4844, 1;
1, -16776, -58596, -106096, -141151, -153748, -141151, -106096, -58596, -16776, 1;

Programs

  • Maple
    A176564 := proc(n,m) binomial(2*n,m)+binomial(2*n,n-m) -binomial(2*n,n) ; end proc:
  • Mathematica
    t[n_, m_] = Binomial[2*n, m] + Binomial[2*n, n - m] - (Binomial[2*n, 0] + Binomial[2*n, n]) + 1;
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%]

Formula

T(n,m) = T(n,n-m).

A349147 Triangle T(n,m) read by rows: the sum of runs of all sequences arranging n objects of one type and m objects of another type.

Original entry on oeis.org

1, 1, 4, 1, 7, 18, 1, 10, 34, 80, 1, 13, 55, 155, 350, 1, 16, 81, 266, 686, 1512, 1, 19, 112, 420, 1218, 2982, 6468, 1, 22, 148, 624, 2010, 5412, 12804, 27456, 1, 25, 189, 885, 3135, 9207, 23595, 54483, 115830, 1, 28, 235, 1210, 4675, 14872, 41041, 101530, 230230, 486200, 1, 31
Offset: 0

Views

Author

R. J. Mathar, Nov 08 2021

Keywords

Examples

			The triangle starts
  1,
  1,  4,
  1,  7,  18,
  1, 10,  34,   80,
  1, 13,  55,  155,  350,
  1, 16,  81,  266,  686,  1512,
  1, 19, 112,  420, 1218,  2982,  6468,
  1, 22, 148,  624, 2010,  5412, 12804,  27456,
  1, 25, 189,  885, 3135,  9207, 23595,  54483, 115830,
  1, 28, 235, 1210, 4675, 14872, 41041, 101530, 230230, 486200,
  1, 31, 286, 1606, 6721, 23023, 68068, 179608, 432718, 967538, 2032316
For n=m=1 the sequences are ab (2 runs) and ba (2 runs), so T(1,1)=2+2=4.
For n=1, m=2 the sequences are aab (2 runs), aba (3 runs), baa (2 runs), so T(1,2)=2+3+2=7.
For n=m=2 the sequences are aabb (2 runs), abab (4 runs), abba (3 runs), baab (3 runs), baba (4 runs), bbaa (2 runs), so T(2,2) = 2+4+3+3+4+2=18.

Links

Crossrefs

Cf. A016777 (row/col 1), A000566 (row/col 2), A007584 (row/col 3), A051798 (row/col 4).
Diagonal gives A037965(n+1).

Formula

T(n,m) = T(m,n).
Sum_{m=0..n} T(n,m) = A000917(n-1) + A000984(n) = 1, 5, 26, 125, 574, ... - R. J. Mathar, Nov 09 2021
T(n,m) = binomial(n+m,n)*(2*n*m+n+m)/(n+m) for n+m >= 1.

A178343 Triangle T(n,m)= binomial(n, m)/Beta(m + 1, n - m + 1) read by rows.

Original entry on oeis.org

1, 2, 2, 3, 12, 3, 4, 36, 36, 4, 5, 80, 180, 80, 5, 6, 150, 600, 600, 150, 6, 7, 252, 1575, 2800, 1575, 252, 7, 8, 392, 3528, 9800, 9800, 3528, 392, 8, 9, 576, 7056, 28224, 44100, 28224, 7056, 576, 9, 10, 810, 12960, 70560, 158760, 158760, 70560, 12960, 810, 10
Offset: 0

Views

Author

Roger L. Bagula, May 25 2010

Keywords

Comments

Beta(x,y) = Gamma(x)*Gamma(y)/Gamma(x+y) is the Beta-function.
Row sums are A037965(n+1). The second column is A011379.

Examples

			1;
2, 2;
3, 12, 3;
4, 36, 36, 4;
5, 80, 180, 80, 5;
6, 150, 600, 600, 150, 6;
7, 252, 1575, 2800, 1575, 252, 7;
8, 392, 3528, 9800, 9800, 3528, 392, 8;
9, 576, 7056, 28224, 44100, 28224, 7056, 576, 9;
10, 810, 12960, 70560, 158760, 158760, 70560, 12960, 810, 10;
11, 1100, 22275, 158400, 485100, 698544, 485100, 158400, 22275, 1100, 11;

Crossrefs

Cf. A037965

Programs

  • Mathematica
    Flatten[Table[Table[Binomial[n, m]/Beta[m + 1, n - m + 1], {m, 0, n}], {n, 0, 10}]]

Formula

T(n,m)=T(n,n-m) = (n+1)*( binomial(n,m))^2 = (n+1)*A008459(n).

Extensions

Edited by the Assoc. Eds. of the OEIS - Jun 27 2010
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