A205341 T(n,k)=Number of length n+1 nonnegative integer arrays starting and ending with 0 with adjacent elements unequal but differing by no more than k.

0, 0, 1, 0, 2, 0, 0, 3, 2, 2, 0, 4, 6, 11, 0, 0, 5, 12, 35, 24, 5, 0, 6, 20, 82, 138, 93, 0, 0, 7, 30, 160, 454, 689, 272, 14, 0, 8, 42, 277, 1130, 2912, 3272, 971, 0, 0, 9, 56, 441, 2370, 8927, 18652, 16522, 3194, 42, 0, 10, 72, 660, 4424, 22297, 71630, 124299, 83792, 11293, 0, 0

Offset

1,5

Comments

Table starts

..0...0.....0......0......0.......0.......0........0........0........0

..1...2.....3......4......5.......6.......7........8........9.......10

..0...2.....6.....12.....20......30......42.......56.......72.......90

..2..11....35.....82....160.....277.....441......660......942.....1295

..0..24...138....454...1130....2370....4424.....7588....12204....18660

..5..93...689...2912...8927...22297...48335....94456...170529...289229

..0.272..3272..18652..71630..214724..542850..1211784..2459988..4633800

.14.971.16522.124299.594405.2133784.6285127.16018970.36557640.76469705

Links

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

Formula

Empirical for row n:

n=2: T(2,k) = k

n=3: T(3,k) = k^2 - k

n=4: T(4,k) = (4/3)*k^3 - (1/2)*k^2 + (7/6)*k

n=5: T(5,k) = (23/12)*k^4 - (1/2)*k^3 + (1/12)*k^2 - (3/2)*k

n=6: T(6,k) = (44/15)*k^5 - (5/12)*k^4 + (5/12)*k^2 + (31/15)*k

n=7: T(7,k) = (841/180)*k^6 - (1/3)*k^5 - (19/36)*k^4 + (1/3)*k^3 - (103/90)*k^2 - 3*k

T(n,m) = 1/n*Sum_{i=1..n} (Sum_{,l,0,i} (binomial(i,l)*(-1)^l *Sum_{j=0..(i-l)* m/(2*m+1)}((-1)^j*binomial(i-l,j)*binomial((-l-2*j+i)*m-l-j+i-1,(-l-2*j+i)*m-j)))*T(n-i,m)), T(0,m)=1. - Vladimir Kruchinin, Apr 07 2017

Example

Some solutions for n=5, k=3:
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..2....2....2....2....3....2....1....2....2....2....2....2....1....3....2....3
..4....5....4....0....2....4....4....4....1....4....3....1....2....5....5....5
..6....4....3....1....4....1....2....2....0....1....0....2....4....4....4....4
..3....3....2....3....1....2....1....3....3....3....2....3....2....2....2....1
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0

Mathematica

T[n_, m_] := T[n, m] = If[n == 0, 1, 1/(n)*Sum[Sum[Binomial[i, l]*(-1)^l* Sum[(-1)^j*Binomial[i-l, j]*Binomial[(-l - 2*j + i)*m - l - j + i - 1, (-l - 2*j + i)*m-j], {j, 0, (i-l)*m/(2*m+1)}], {l, 0, i}]*T[n-i, m], {i, 1, n}]];
Table[T[n-m+1, m], {n, 1, 11}, {m, n, 1, -1}] // Flatten (* Jean-François Alcover, Sep 24 2019, after Vladimir Kruchinin *)

Prog

(Maxima)
T(n, m):=if n=0 then 1 else 1/(n)*sum(sum(binomial(i, l)*(-1)^l*sum((-1)^j*binomial(i-l, j)*binomial((-l-2*j+i)*m-l-j+i-1, (-l-2*j+i)*m-j), j, 0, (i-l)*m/(2*m+1)), l, 0, i)*T(n-i, m), i, 1, n); /* Vladimir Kruchinin, Apr 07 2017 */

Crossrefs

Column 1 odd n is A000108((n+5)/2).

Column 2 is A187430.

Row 3 is A002378(n-1).

Sequence in context: A374019 A099026 A341410 * A195664 A053202 A188122

Adjacent sequences: A205338 A205339 A205340 * A205342 A205343 A205344

Keyword

nonn,tabl,look

Author

R. H. Hardin, Jan 26 2012

Status

approved