A026120 Triangle T by rows: second differences of Motzkin triangle (A026300), (i >= -1, -1<=j<=i).

1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 1, 2, 4, 4, 3, 1, 3, 7, 10, 11, 7, 1, 4, 11, 20, 28, 28, 18, 1, 5, 16, 35, 59, 76, 74, 46, 1, 6, 22, 56, 110, 170, 209, 196, 120, 1, 7, 29, 84, 188, 336, 489, 575, 525, 316, 1, 8, 37, 120, 301, 608, 1013, 1400, 1589, 1416, 841, 1, 9, 46, 165, 458, 1029, 1922, 3021, 4002, 4405, 3846, 2257

Offset

-1,9

Comments

For n >= 2, T(n,k) = number of nonnegative integer strings s(0),...,s(n) such that s(0)=1, s(n)=n-k, |s(1)-1|=1, |s(i)-s(i-1)|<=1 for i >= 2.

Links

Table of n, a(n) for n=-1..76.

Formula

T(n, k) = A026105(n+1, k+1) - A026105(n, k), T(0, 0) = 0.

T(i, 0)=1 for i >= -1; T(0, 1)=0; T(1, 1)=1, T(1, 2)=0; for i >= 2, T(i, 1)=i-1, T(i, i+1)=T(i-1, i-1)+T(i-1, i), T(i, j)=T(i-1, j-2)+T(i-1, j-1)+T(i-1, j) for j=2, 3, ...., i.

G.f. of right-hand columns is (1-z)^2*M^k, with M the g.f. of the Motzkin numbers (A001006).

Example

1;
1,0;
1,1,0;
1,1,2,1;
1,2,4,4,3;
1,3,7,10,11,7;
...

Prog

(PARI) T(i, j)=if(j<0||j>i+1, 0, if(j==0, 1, if(j==1, if(i>1, i-1, i>0), if(i+1==j, if(i==1, 0, T(i-1, i-1)+T(i-1, i)), T(i-1, j-2)+T(i-1, j-1)+T(i-1, j))))) \\ Ralf Stephan

Crossrefs

Cf. Right-hand columns include A026107, A026122, A026123, A026124, A026125, A026126. Row sums are A026135. Central column is A026127.

Sequence in context: A085143 A321029 A253473 * A108746 A333270 A333272

Adjacent sequences: A026117 A026118 A026119 * A026121 A026122 A026123

Keyword

nonn,tabl

Author

Clark Kimberling

Extensions

Edited by Ralf Stephan, Dec 18 2004

a(26) corrected and more terms from Sean A. Irvine, Sep 17 2019

Status

approved