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