A048775 -id:A048775 - cp's OEIS frontend

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

Paul Barry, Mar 02 2006

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

			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

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

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 *)

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

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

A147984 Column 5 of A144512.

Original entry on oeis.org

1, 2431, 4061871, 7528988476, 15467641899285, 34155922905682979, 79397199549271412737, 191739533381111401455478, 476872353039366288373555323

Offset: 0

N. J. A. Sloane, May 13 2009

nonn

Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394, 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem (arXiv:0907.0513, 2009)

Cf. A144512, A048775, A144511, A144662.

A371036 a(n) is the number of ways to place n indistinguishable balls into n distinguishable boxes with at least one box remaining empty and not all balls placed in a single box.

Original entry on oeis.org

0, 0, 6, 30, 120, 455, 1708, 6426, 24300, 92367, 352704, 1352065, 5200286, 20058285, 77558744, 300540178, 1166803092, 4537567631, 17672631880, 68923264389, 269128937198, 1052049481837, 4116715363776, 16123801841525, 63205303218850, 247959266474025, 973469712824028

Offset: 1

Enrique Navarrete, Mar 08 2024

nonn

a(n) is also the number of weak compositions of n into n parts in which at least one part is zero and the composition does not contain a single nonzero part.

			a(4)=30 since 4 can be written as 3+1+0+0, 0+3+1+0, etc. (12 such compositions); 2+2+0+0 (6 such compositions); 2+1+1+0 (12 such compositions).

Cf. A001700, A010763, A048775, A352027, A359435.

a(n) = binomial(2n-1,n)-n-1, n > 1; a(1)=0.
a(n) = A048775(n-1)-1, n > 1.
a(n) = A001700(n-1)-(n+1), n > 1.

cp's OEIS Frontend

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

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A147984 Column 5 of A144512.

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A371036 a(n) is the number of ways to place n indistinguishable balls into n distinguishable boxes with at least one box remaining empty and not all balls placed in a single box.

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