|
Search: id:A093010
|
|
|
| A093010 |
|
Triangle, read by rows, such that the convolution of the n-th row with the natural numbers forms the n-th diagonal, for n>=0, where each row begins with 1. |
|
+0
3
|
|
| 1, 1, 2, 1, 4, 3, 1, 6, 7, 4, 1, 8, 14, 10, 5, 1, 10, 22, 22, 13, 6, 1, 12, 33, 40, 30, 16, 7, 1, 14, 45, 66, 58, 38, 19, 8, 1, 16, 60, 100, 104, 76, 46, 22, 9, 1, 18, 76, 146, 168, 142, 94, 54, 25, 10, 1, 20, 95, 202, 262, 242, 180, 112, 62, 28, 11, 1, 22, 115, 272, 386, 394, 316
(list; table; graph; listen)
|
|
|
OFFSET
|
0,3
|
|
|
COMMENT
|
Row sums form A000713, the number of partitions of n into parts of 3 kinds. Antidiagonal sums form A000990, the number of 2-line partitions of n.
|
|
FORMULA
|
T(n, k) = sum_{j=0..k} (k-j+1)*T(n-k, j), with T(0, n) = 1 for all n>=0. A000713(n) = sum_{k=0..n} T(n, k) (row sums). A000990(n) = sum_{k=0..floor(n/2)} T(n-k, k} (antidiagonal sums).
|
|
EXAMPLE
|
T(7,3) = 66 = 1*4+8*3+14*2+10*1 = T(4,0)*4+T(4,1)*3+T(4,2)*2+T(4,3)*1; this is also the third term of the 4-th diagonal.
The 6-th antidiagonal is {1,10,14,4}, which has a sum of 29 = A000990(6) = number of 2-line partitions of 6.
Rows begin:
{1},
{1,2},
{1,4,3},
{1,6,7,4},
{1,8,14,10,5},
{1,10,22,22,13,6},
{1,12,33,40,30,16,7},
{1,14,45,66,58,38,19,8},
{1,16,60,100,104,76,46,22,9},
{1,18,76,146,168,142,94,54,25,10},
{1,20,95,202,262,242,180,112,62,28,11},
{1,22,115,272,386,394,316,218,130,70,31,12},...
|
|
PROGRAM
|
(PARI) T(n, k)=if(n<k|k<0, 0, if(k==0, 1, sum(j=0, min(k, n-k), (k-j+1)*T(n-k, j))))
|
|
CROSSREFS
|
Cf. A000713, A000990, A092905.
Sequence in context: A112157 A093682 A134543 this_sequence A093966 A103406 A093190
Adjacent sequences: A093007 A093008 A093009 this_sequence A093011 A093012 A093013
|
|
KEYWORD
|
nonn,tabl
|
|
AUTHOR
|
Paul D. Hanna (pauldhanna(AT)juno.com), Mar 14 2004
|
|
|
Search completed in 0.002 seconds
|
