A072335 -id:A072335 - cp's OEIS frontend

A258730 T(n,k)=Number of length n+k 0..3 arrays with at most one downstep in every k consecutive neighbor pairs.

16, 60, 64, 190, 225, 256, 512, 608, 840, 1024, 1212, 1408, 2028, 3136, 4096, 2592, 2936, 4184, 6552, 11704, 16384, 5115, 5664, 7834, 12549, 20955, 43681, 65536, 9460, 10280, 13720, 21860, 35540, 68120, 163020, 262144, 16588, 17754, 22866, 35704

Offset: 1

R. H. Hardin, Jun 08 2015

Table starts
......16......60.....190.....512....1212....2592....5115....9460....16588
......64.....225.....608....1408....2936....5664...10280...17754....29416
.....256.....840....2028....4184....7834...13720...22866...36656....56925
....1024....3136....6552...12549...21860...35704...55660...83758...122584
....4096...11704...20955...35540...59188...92548..138196..199264...279560
...16384...43681...68120...98676..149960..228081..331584..465580...635992
...65536..163020..220854..281136..370510..526672..752180.1038256..1394568
..262144..608400..711432..819453..941024.1183616.1607656.2192682..2911776
.1048576.2270580.2300008.2358888.2487276.2727288.3343894.4392072..5783522
.4194304.8473921.7446144.6678576.6650600.6597449.7100132.8569478.10965340

			Some solutions for n=4 k=4
..1....1....0....0....3....0....1....3....2....0....2....3....0....3....2....0
..0....2....3....3....1....1....2....3....2....1....2....2....3....3....0....2
..2....0....3....1....1....1....3....3....0....1....3....3....0....0....2....2
..3....2....3....1....2....1....1....0....0....1....0....3....1....2....3....3
..3....3....3....1....2....1....1....0....1....1....0....3....3....3....3....0
..0....3....3....2....3....3....1....1....1....0....3....0....3....3....2....0
..0....3....1....3....0....0....2....3....2....0....3....2....1....0....2....2
..3....2....2....1....2....1....0....1....0....0....2....2....1....0....2....2

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A000302(n+1)

Column 2 is A072335(n+2)

Empirical for column k:
k=1: a(n) = 4*a(n-1)
k=2: a(n) = 4*a(n-1) -4*a(n-3) +a(n-4)
k=3: [order 8]
k=4: [order 12]
k=5: [order 16]
k=6: [order 19]
k=7: [order 22]
Empirical for row n:
n=1: [polynomial of degree 7]
n=2: [polynomial of degree 7]
n=3: [polynomial of degree 7] for n>1
n=4: [polynomial of degree 7] for n>2
n=5: [polynomial of degree 7] for n>3
n=6: [polynomial of degree 7] for n>4
n=7: [polynomial of degree 7] for n>5

A200785 T(n,k) is the number of arrays of n+2 elements from {0,1,...,k} with no two consecutive ascents.

Original entry on oeis.org

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

Offset: 1

R. H. Hardin Nov 22 2011

All the conjectured formulas are true, and follow from the Burstein-Mansour paper. - N. J. A. Sloane, May 21 2013

			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

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.

Column 1 is A000079
Column 2 is A076264
Column 3 is A072335
Row 1 is A002413
Cf. A200781.

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

A072279 Dimension of n-th graded section of a certain Lie algebra.

Original entry on oeis.org

1, 4, 6, 16, 45, 144, 440, 1440, 4680, 15600, 52344, 177840, 608160, 2095920, 7262640, 25300032, 88517520, 310927680, 1095923400, 3874804560, 13737892896, 48829153920, 173949483240, 620963048160, 2220904271040, 7956987570576, 28553731537320, 102617166646800

Offset: 0

N. J. A. Sloane, Jul 15 2002

Dimensions of Lie algebra associated to Yang-Lee algebra in the A. Connes and M. Dubois-Violette paper. - Roger L. Bagula, May 25 2007

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Latham Boyle, Paul J. Steinhardt, Self-Similar One-Dimensional Quasilattices, arXiv preprint arXiv:1608.08220 [math-ph], 2016.
A. Connes and M. Dubois-Violette, Yang-Mills Algebra, arXiv:math/0206205 [math.QA], 2002.
N. J. A. Sloane, Transforms

Inverse EULER transform of A072335 (with its initial 1 omitted).

Cf. A072337.

with(numtheory): f:= proc(n) option remember; `if`(n<1, `if`(n=0,1,0), 4*(f(n-1)-f(n-3)) +f(n-4)) end: c:= proc(n) option remember; local j; n*f(n) -add(c(j)*f(n-j), j=1..n-1) end: a:= proc(n) option remember; local d; `if`(n=0,1, add(mobius(n/d)*c(d), d=divisors(n))/n) end: seq(a(n), n=0..27); # Alois P. Heinz, Sep 09 2008

f[n_] := f[n] = If[n < 1, If[n == 0, 1, 0],  f[n-4] + 4*(f[n-1] - f[n-3])]; c[n_] := c[n] = n*f[n] - Sum[c[j]*f[n-j],  {j, 1, n-1}]; a[n_] := a[n] = If[n == 0, 1,  Sum[c[d]*MoebiusMu[n/d],  {d, Divisors[n]}]/n]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Mar 14 2014, after Alois P. Heinz *)

Product_{n=1..inf} 1/(1-x^n)^a(n) = 1/((1-x^2)*(1-4*x+x^2)).
a(n) = (1/n) * Sum_{k|n} moebius(n/k) (t1^k + t2^k), where t1, t2 are the roots of x^2-4x+1.
a(n) ~ (2+sqrt(3))^n / n. - Vaclav Kotesovec, Sep 11 2014

Edited by N. J. A. Sloane, May 16 2008 at the suggestion of R. J. Mathar

A225682 Triangle read by rows: T(n,k) (0 <= k <= n) = chi(k)*binomial(n,k), where chi(k) = 1,-1,0 according as k == 0,1,2 mod 3.

Original entry on oeis.org

1, 1, -1, 1, -2, 0, 1, -3, 0, 1, 1, -4, 0, 4, -1, 1, -5, 0, 10, -5, 0, 1, -6, 0, 20, -15, 0, 1, 1, -7, 0, 35, -35, 0, 7, -1, 1, -8, 0, 56, -70, 0, 28, -8, 0, 1, -9, 0, 84, -126, 0, 84, -36, 0, 1, 1, -10, 0, 120, -210, 0, 210, -120, 0, 10, -1, 1, -11, 0, 165, -330, 0, 462, -330, 0, 55, -11, 0, 1, -12, 0, 220, -495, 0, 924, -792, 0, 220, -66, 0, 1

Offset: 0

Murray R. Bremner and N. J. A. Sloane, May 17 2013

Corresponding to row n of this triangle, define a generating function G_n(x) = 1/(Sum_{k=0..n} T(n,k)*x^k).
Then G_n(x) is the g.f. for the number of words of length n over an alphabet of size n which do not contain any strictly decreasing factor (consecutive subword) of length 3.
For example, G_2, G_3, G_4, G_5, G_6 are g.f.'s for A000079, A076264, A072335, A200781, A200782.

			Triangle begins:
[1],
[1, -1],
[1, -2, 0],
[1, -3, 0, 1],
[1, -4, 0, 4, -1],
[1, -5, 0, 10, -5, 0],
[1, -6, 0, 20, -15, 0, 1],
[1, -7, 0, 35, -35, 0, 7, -1],
[1, -8, 0, 56, -70, 0, 28, -8, 0],
...

Cf. A000079, A076264, A072335, A200781, A200782.

f:=proc(n) local k,s;
s:=k->if k mod 3 = 0 then 1 elif k mod 3 = 1 then -1 else 0; fi;
[seq(s(k)*binomial(n,k),k=0..n)];
end;
[seq(f(n),n=0..12)];

chi[k_] := Switch[Mod[k, 3], 0, 1, 1, -1, 2, 0]; t[n_, k_] := chi[k]*Binomial[n, k]; Table[t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 14 2014 *)

A200781 G.f.: 1/(1-5x+10x^3-5*x^4).

Original entry on oeis.org

1, 5, 25, 115, 530, 2425, 11100, 50775, 232275, 1062500, 4860250, 22232375, 101698250, 465201250, 2127983750, 9734098125, 44526969375, 203681015625, 931704015625, 4261920875000, 19495429065625, 89178510250000, 407931862578125, 1866014626609375, 8535765175875000, 39045399804843750, 178606512071015625, 817004981729375000

Offset: 0

R. H. Hardin, Nov 22 2011

nonn

Number of words of length n over an alphabet of size 5 which do not contain any strictly decreasing factor (consecutive subword) of length 3. For alphabets of size 2, 3, 4, 6 see A000079, A076264, A072335, A200782.

Equivalently, number of 0..4 arrays x(0..n-1) of n elements without any two consecutive increases.

			Some solutions for n=5:
..1....3....4....0....1....0....4....0....2....1....4....1....2....2....4....4
..3....4....4....2....1....0....3....3....1....4....1....1....4....4....3....3
..3....1....0....2....0....2....0....3....3....0....4....3....0....1....4....4
..2....0....2....4....4....0....3....2....0....0....3....2....0....2....1....3
..4....4....2....2....0....3....3....2....1....0....4....1....3....1....0....2

R. H. Hardin and N. J. A. Sloane, Table of n, a(n) for n = 0..249 [The first 210 terms were computed by R. H. Hardin]
A. Burstein and T. Mansour, Words restricted by 3-letter generalized multipermutation patterns, Annals. Combin., 7 (2003), 1-14. See Th. 3.13.
Index entries for linear recurrences with constant coefficients, signature (5,0,-10,5).

The g.f. corresponds to row 5 of triangle A225682.
Column 4 of A200785.
Cf. A076264, A072335, A200782.

Vec(1/(1-5*x+10*x^3-5*x^4) + O(x^30)) \\ Jinyuan Wang, Mar 10 2020

a(n) = 5*a(n-1) - 10*a(n-3) + 5*a(n-4).

Edited by N. J. A. Sloane, May 21 2013

cp's OEIS Frontend

A258730 T(n,k)=Number of length n+k 0..3 arrays with at most one downstep in every k consecutive neighbor pairs.

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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.

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A072279 Dimension of n-th graded section of a certain Lie algebra.

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A225682 Triangle read by rows: T(n,k) (0 <= k <= n) = chi(k)*binomial(n,k), where chi(k) = 1,-1,0 according as k == 0,1,2 mod 3.

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A200781 G.f.: 1/(1-5*x+10*x^3-5*x^4).

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A200781 G.f.: 1/(1-5x+10x^3-5*x^4).