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A117396 Triangle, read by rows, defined by: T(n,k) = (k+1)*T(n,k+1) - Sum_{j=1..n-k-1} T(j,0)*T(n,j+k+1) for n>k with T(n,n)=1 for n>=0. 3
1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 17, 11, 4, 1, 1, 77, 51, 19, 5, 1, 1, 437, 291, 109, 29, 6, 1, 1, 2957, 1971, 739, 197, 41, 7, 1, 1, 23117, 15411, 5779, 1541, 321, 55, 8, 1, 1, 204557, 136371, 51139, 13637, 2841, 487, 71, 9, 1, 1, 2018957, 1345971, 504739, 134597 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Columns equal the partial sums of columns of triangle A092582 for k>0: T(n,k) - T(n-1,k) = A092582(n,k) = number of permutations p of [n] having length of first run equal to k.

LINKS

Table of n, a(n) for n=0..59.

FORMULA

T(n,k) = k*Sum_{j=k-1..n} j!/(k+1)! for n>=k>0, with T(n,0) = 1 for n>=0. - Paul D. Hanna, Jun 20 2006

EXAMPLE

Triangle begins:

1;

1,1;

1,2,1;

1,5,3,1;

1,17,11,4,1;

1,77,51,19,5,1;

1,437,291,109,29,6,1;

1,2957,1971,739,197,41,7,1;

1,23117,15411,5779,1541,321,55,8,1;

1,204557,136371,51139,13637,2841,487,71,9,1; ...

Matrix inverse is:

1;

-1,1;

1,-2,1;

1,1,-3,1;

1,1,1,-4,1;

1,1,1,1,-5,1; ...

Matrix log is the integer triangle A117398:

0;

1,0;

0,2,0;

-1,2,3,0;

-3,4,5,4,0;

-9,14,15,9,5,0;

-33,68,65,34,14,6,0; ...

PROG

(PARI) T(n, k)=if(n<k || k<0, 0, if(n==k, 1, (k+1)*T(n, k+1)-sum(j=1, n-k-1, T(j, 0)*T(n, j+k+1))))

(PARI) /* Definition by Matrix Inverse: * / T(n, k)=local(M=matrix(n+1, n+1, r, c, if(r>=c, if(r==c+1, -c, 1)))); (M^-1)[n+1, k+1]

(PARI) T(n, k)=if(n<k || k<0, 0, if(k==0, 1, k*sum(j=k-1, n, j!)/(k+1)!)) \\ Paul D. Hanna, Jun 20 2006

CROSSREFS

Cf. A014288 (column 1), A056199 (column 2), A117397 (column 3), A003422 (row sums), A117398 (matrix log); A092582.

Sequence in context: A186020 A241579 A308292 * A125860 A294585 A283674

Adjacent sequences:  A117393 A117394 A117395 * A117397 A117398 A117399

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Mar 11 2006

STATUS

approved

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Last modified June 24 18:43 EDT 2021. Contains 345419 sequences. (Running on oeis4.)