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 A200785 T(n,k) is the number of arrays of n+2 elements from {0,1,...,k} with no two consecutive ascents. 11
 8, 26, 16, 60, 75, 32, 115, 225, 216, 64, 196, 530, 840, 622, 128, 308, 1071, 2425, 3136, 1791, 256, 456, 1946, 5796, 11100, 11704, 5157, 512, 645, 3270, 12152, 31395, 50775, 43681, 14849, 1024, 880, 5175, 23136, 75992, 169884, 232275, 163020, 42756, 2048 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS All the conjectured formulas are true, and follow from the Burstein-Mansour paper. - N. J. A. Sloane, May 21 2013 LINKS R. H. Hardin, Table of n, a(n) for n = 1..10010 A. Burstein and T. Mansour, Words restricted by 3-letter generalized multipermutation patterns, Annals. Combin., 7 (2003), 1-14. See Th. 3.13. FORMULA T(n-2,k) = \sum_{L=0}^n (-1)^L / L! * \sum_{M=0}^{min(L,[(n-L)/2])} binomial(n-L-M,M) * M! * (k+1)^(n-L-2*M) B_{L,M}(x_1,x_2,...), where B_{L,M}() are Bell polynomials, x_i = binomial(k+1,i+2) * i! * f(i), i=1,2,..., and f(i) has period of length 6: [0,1,1,0,-1,-1] (i.e., f(0)=0, f(1)=1, etc.). This formula implies that for a fixed n, T(n,k) is a polynomial in k, which is easy to compute. - Max Alekseyev, Dec 12 2011 Empirical formulas for columns: k=1: a(n) = 2*a(n-1) k=2: a(n) = 3*a(n-1) -a(n-3) k=3: a(n) = 4*a(n-1) -4*a(n-3) +a(n-4) k=4: a(n) = 5*a(n-1) -10*a(n-3) +5*a(n-4) k=5: a(n) = 6*a(n-1) -20*a(n-3) +15*a(n-4) -a(n-6) k=6: a(n) = 7*a(n-1) -35*a(n-3) +35*a(n-4) -7*a(n-6) +a(n-7) k=7: a(n) = 8*a(n-1) -56*a(n-3) +70*a(n-4) -28*a(n-6) +8*a(n-7) Empirical recurrence for general column k: 0 = sum{i=0..floor(k/3) (binomial(k+1,3*i+1)*T(n-(3*i+1),k))} - sum{i=0..floor((k+1)/3) (binomial(k+1,3*i)*T(n-3*i,k))} Formulae for rows: T(1,k) = (5/6)*k^3 + 3*k^2 + (19/6)*k + 1 T(2,k) = (17/24)*k^4 + (43/12)*k^3 + (151/24)*k^2 + (53/12)*k + 1 T(3,k) = (7/12)*k^5 + (47/12)*k^4 + (39/4)*k^3 + (133/12)*k^2 + (17/3)*k + 1 T(4,k) = (349/720)*k^6 + (321/80)*k^5 + (1883/144)*k^4 + (1013/48)*k^3 + (3139/180)*k^2 + (413/60)*k + 1 T(5,k) = (2017/5040)*k^7 + (1427/360)*k^6 + (5759/360)*k^5 + (607/18)*k^4 + (28459/720)*k^3 + (9113/360)*k^2 + (848/105)*k + 1 T(6,k) = (6679/20160)*k^8 + (4799/1260)*k^7 + (26449/1440)*k^6 + (2162/45)*k^5 + (212153/2880)*k^4 + (6019/90)*k^3 + (174571/5040)*k^2 + (3893/420)*k + 1 T(7,k) = (99377/362880)*k^9 + (48247/13440)*k^8 + (243673/12096)*k^7 + (60529/960)*k^6 + (2076437/17280)*k^5 + (274529/1920)*k^4 + (952027/9072)*k^3 + (152461/3360)*k^2 + (26399/2520)*k + 1 EXAMPLE Table starts ....8.....26......60.......115.......196........308.........456.........645 ...16.....75.....225.......530......1071.......1946........3270........5175 ...32....216.....840......2425......5796......12152.......23136.......40905 ...64....622....3136.....11100.....31395......75992......164004......324087 ..128...1791...11704.....50775....169884.....474566.....1160616.....2562633 ..256...5157...43681....232275....919413....2964416.....8216484....20273247 ..512..14849..163020...1062500...4975322...18514405....58154912...160338680 .1024..42756..608400...4860250..26924106..115637431...411637168..1268210421 .2048.123111.2270580..22232375.145698840..722234149..2913595712.10030582998 .4096.354484.8473921.101698250.788446400.4510869636.20622837480.79335475611 Some arrays for n=4, k=3: ..0....1....0....0....1....0....3....3....0....1....3....0....2....2....2....2 ..3....0....2....2....0....2....0....0....3....1....0....0....0....3....3....3 ..2....3....2....2....2....2....3....3....1....0....1....0....2....1....3....3 ..1....0....2....1....0....0....2....2....2....2....1....2....2....0....0....2 ..0....3....0....0....1....2....1....2....0....0....3....2....0....3....1....3 ..3....3....0....3....0....2....3....2....0....3....0....0....2....2....1....3 CROSSREFS Column 1 is A000079 Column 2 is A076264 Column 3 is A072335 Row 1 is A002413 Cf. A200781. Sequence in context: A060718 A060743 A029617 * A120743 A063560 A265104 Adjacent sequences:  A200782 A200783 A200784 * A200786 A200787 A200788 KEYWORD nonn,tabl AUTHOR R. H. Hardin Nov 22 2011 STATUS approved

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