The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 60th year, we have over 367,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A086856 Triangle read by rows: T(n,k) = one-half number of permutations of length n with exactly k rising or falling successions, for n >= 1, 0 <= k <= n-1. T(1,0) = 1 by convention. 3
 1, 0, 1, 0, 2, 1, 1, 5, 5, 1, 7, 20, 24, 8, 1, 45, 115, 128, 60, 11, 1, 323, 790, 835, 444, 113, 14, 1, 2621, 6217, 6423, 3599, 1099, 183, 17, 1, 23811, 55160, 56410, 32484, 11060, 2224, 270, 20, 1, 239653, 545135, 554306, 325322, 118484, 27152, 3950, 374, 23, 1, 2648395 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS (1/2) times number of permutations of 12...n such that exactly k of the following occur: 12, 23, ..., (n-1)n, 21, 32, ..., n(n-1). REFERENCES F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263. LINKS Alois P. Heinz, Rows n = 1..141, flattened J. Riordan, A recurrence for permutations without rising or falling successions, Ann. Math. Statist. 36 (1965), 708-710. EXAMPLE Triangle T(n,k) begins: 1; 0, 1; 0, 2, 1; 1, 5, 5, 1; 7, 20, 24, 8, 1; 45, 115, 128, 60, 11, 1; 323, 790, 835, 444, 113, 14, 1; ... MAPLE S:= proc(n) option remember; `if`(n<4, [1, 1, 2*t, 4*t+2*t^2] [n+1], expand((n+1-t)*S(n-1) -(1-t)*(n-2+3*t)*S(n-2) -(1-t)^2*(n-5+t)*S(n-3) +(1-t)^3*(n-3)*S(n-4))) end: T:= (n, k)-> ceil(coeff(S(n), t, k)/2): seq(seq(T(n, k), k=0..n-1), n=1..10); # Alois P. Heinz, Jan 11 2013 MATHEMATICA S[n_] := S[n] = If[n < 4, {1, 1, 2*t, 4*t+2*t^2}[[n+1]], Expand[(n+1-t)*S[n-1] - (1-t)*(n-2+3*t)*S[n-2] - (1-t)^2*(n-5+t)*S[n-3] + (1-t)^3*(n-3)*S[n-4]]]; T[n_, k_] := Ceiling[Coefficient[S[n], t, k]/2]; Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 11}] // Flatten (* Jean-François Alcover, Jan 14 2014, translated from Alois P. Heinz's Maple code *) CROSSREFS Diagonals give A001266 (and A002464), A000130, A000349, A001267, A001268. Triangle in A001100 divided by 2. A010028 transposed. Row sums give A001710. Sequence in context: A091378 A156045 A119687 * A052916 A326048 A156576 Adjacent sequences: A086853 A086854 A086855 * A086857 A086858 A086859 KEYWORD tabl,nonn AUTHOR N. J. A. Sloane, Aug 19 2003 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 7 11:41 EST 2023. Contains 367656 sequences. (Running on oeis4.)