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

Author

R. H. Hardin, Jan 26 2012

Keywords

• nonn
• tabl
• look

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

Examples

			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
		

Links

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

Crossrefs

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

Column 2 is A187430.

Row 3 is A002378(n-1).

Programs

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 *)

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 */

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