A062323 -id:A062323 - 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

A070895 Triangle read by rows where T(n+1,k)=T(n,k)+n*T(n-1,k) starting with T(n,n)=1 and T(n,k)=0 if n

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 10, 6, 4, 1, 1, 26, 18, 8, 5, 1, 1, 76, 48, 28, 10, 6, 1, 1, 232, 156, 76, 40, 12, 7, 1, 1, 764, 492, 272, 110, 54, 14, 8, 1, 1, 2620, 1740, 880, 430, 150, 70, 16, 9, 1, 1, 9496, 6168, 3328, 1420, 636, 196, 88, 18, 10, 1, 1, 35696, 23568

Offset: 0

Henry Bottomley, May 23 2002

For n>k+1, T(n,k) is a multiple of k+2.

Eigentriangle of inverse of (-1)^(n-k)*A094587. Row sums are A187044. - Paul Barry, Mar 02 2011

			Rows start: 1; 1,1; 2,1,1; 4,3,1,1; 10,6,4,1,1; etc.
Triangle begins
1,
1, 1,
2, 1, 1,
4, 3, 1, 1,
10, 6, 4, 1, 1,
26, 18, 8, 5, 1, 1,
76, 48, 28, 10, 6, 1, 1,
232, 156, 76, 40, 12, 7, 1, 1
Production matrix begins
1, 1,
1, 0, 1,
1, 1, 0, 1,
2, 1, 1, 0, 1,
4, 3, 1, 1, 0, 1,
10, 6, 4, 1, 1, 0, 1,
26, 18, 8, 5, 1, 1, 0, 1,
76, 48, 28, 10, 6, 1, 1, 0, 1,
232, 156, 76, 40, 12, 7, 1, 1, 0, 1
Inverse begins
1,
-1, 1,
-1, -1, 1,
0, -2, -1, 1,
0, 0, -3, -1, 1,
0, 0, 0, -4, -1, 1,
0, 0, 0, 0, -5, -1, 1,
0, 0, 0, 0, 0, -6, -1, 1
- _Paul Barry_, Mar 02 2011

Columns include A000085, A000932, A059480. Right hand columns effectively include A000012 (twice), A000027, A005843, A028552. Cf. A062323 for a triangle with similar formulas.

T(n, k+1)=(T(n, k-1)-T(n-1, k))/k for 0

Showing 1-2 of 2 results.

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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