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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. 2
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; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..72.

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 4th-diagonal.

The 6th 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},...

PROG

(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: A134543 A305540 A197871 * A179000 A210559 A180803

Adjacent sequences:  A093007 A093008 A093009 * A093011 A093012 A093013

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Mar 14 2004

STATUS

approved

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Last modified April 23 03:26 EDT 2019. Contains 322380 sequences. (Running on oeis4.)