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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 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=c, if(r==c+1, -c, 1)))); (M^-1)[n+1, k+1] (PARI) T(n, k)=if(n

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