A247365 -id:A247365 - cp's OEIS frontend

A102472 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 vertically, next to each other, with the initial term of each on the next row down.

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

Offset: 1

Russell Walsmith, Jan 09 2005

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

This triangle results when the first column is removed from A062323. - Georg Fischer, Jul 26 2023

			Triangle begins:
[1] 1;
[2] 1, 1;
[3] 3, 2, 1;
[4] 10, 7, 3, 1;
[5] 43, 30, 13, 4, 1;
[6] 225, 157, 68, 21, 5, 1;
[7] 1393, 972, 421, 130, 31, 6, 1;
[8] 9976, 6961, 3015, 931, 222, 43, 7, 1;

Reinhard Zumkeller, Table of n, a(n) for n = 1..7875

Mirror image of triangle in A102473.

Cf. A001040, A058294, A062323, A247365 (central terms).

a102472 n k = a102472_tabl !! (n-1) !! (k-1)
a102472_row n = a102472_tabl !! (n-1)
a102472_tabl = map reverse a102473_tabl
-- Reinhard Zumkeller, Sep 14 2014

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

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.

Original entry on oeis.org

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

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

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

			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).

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

Mirror image of triangle in A102472.

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

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

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

cp's OEIS Frontend

A102472 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 vertically, next to each other, with the initial term of each on the next row down.

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