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

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

0,2

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

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Formula

T(n,k)=C(2n+1,n+1)-sum{j=1..k, product{i=0..j-2, (n-i)^2}/((j-1)!j!)}}*(n+1).

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(2n+1,n+1)-(n+1)*sum(j=1,k, C(n,j-1)^2/j). - M. F. Hasler, Jan 25 2012

Example

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

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 *)

Prog

(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

Crossrefs

Sequence in context: A337273 A376787 A116384 * A046658 A124574 A322383

Adjacent sequences: A117204 A117205 A117206 * A117208 A117209 A117210

Keyword

easy,nonn,tabl

Author

Paul Barry, Mar 02 2006

Status

approved