A102473 Triangle read by rows. Let S(k) be the sequence defined by F(0)=0, F(1)=1, F(n-1) + (n+k)*F(n) = F(n+1). E.g. S(0) = 0,1,1,3,10,43,225,1393,9976,81201, ... Then S(0), S(1), S(2), ... are written next to each other, vertically, with the initial term of each on the next row down. The order of the terms in the rows are then reversed.

1, 1, 1, 1, 2, 3, 1, 3, 7, 10, 1, 4, 13, 30, 43, 1, 5, 21, 68, 157, 225, 1, 6, 31, 130, 421, 972, 1393, 1, 7, 43, 222, 931, 3015, 6961, 9976, 1, 8, 57, 350, 1807, 7578, 24541, 56660, 81201, 1, 9, 73, 520, 3193, 16485, 69133, 223884, 516901, 740785, 1, 10, 91, 738

Offset: 1

Author

Russell Walsmith (russw(AT)lycos.com), Jan 09 2005

Keywords

• easy
• nonn
• tabl

Comments

For this triangle, the algorithm that generates the Bernoulli numbers gives 3/2, then 1/6, 1/24, ... 1/n!

T(n,n) = A001040(n); T(n,k) = A058294(n,k), k = 1..n. - Reinhard Zumkeller, Sep 14 2014

Examples

			Triangle begins:
0
0 1
0 1 1
0 1 2 3
0 1 3 7 10
0 1 4 13 30 43
...
(the zeros are omitted).
		

Links

• Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Crossrefs

Mirror image of triangle in A102472.

Cf. A001040, A058294, A247365 (central terms).

Programs

Haskell

a102473 n k = a102473_tabl !! (n-1) !! (k-1)
a102473_row n = a102473_tabl !! (n-1)
a102473_tabl = [1] : [1, 1] : f [1] [1, 1] 2 where
   f us vs x = ws : f vs ws (x + 1) where
               ws = 1 : zipWith (+) ([0] ++ us) (map (* x) vs)
-- Reinhard Zumkeller, Sep 14 2014

Extensions

Entry revised by N. J. A. Sloane, Jul 09 2005