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A026105 Triangle T read by rows: differences of Motzkin triangle (A026300). 15
1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 3, 6, 7, 5, 1, 4, 10, 16, 18, 12, 1, 5, 15, 30, 44, 46, 30, 1, 6, 21, 50, 89, 120, 120, 76, 1, 7, 28, 77, 160, 259, 329, 316, 196, 1, 8, 36, 112, 265, 496, 748, 904, 841, 512, 1, 9, 45, 156, 413, 873, 1509, 2148, 2493, 2257, 1535, 1, 10, 55, 210 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

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

LINKS

Table of n, a(n) for n=1..69.

FORMULA

T(n, k) = A026300(n, k) - A026300(n-1, k-1), T(1, 1) = 1.

T(i, 0)=1 for i >= 0, T(2, 1)=1, T(2, 2)=1, T(3, 1)=2, T(3, 2)=3, T(3, 3)=2; and for i >= 4, T(i, 1)=i-1, T(i, i)=T(i-1, i-2)+T(i-1, i-1) and T(i, j)=T(i-1, j-2)+T(i-1, j-1)+T(i-1, j) for j=2, 3, ...., i-1.

Right-hand columns have g.f. (1-z)*M^k, where M is g.f. of Motzkin numbers (A001006).

EXAMPLE

1

1,1

1,1,1

1,2,3,2

1,3,6,7,5

1,4,10,16,18,12

1,5,15,30,44,46,30

CROSSREFS

Cf. Right-hand columns include A002026, A026107, A026134, A026109, A026110. Row sums are in A025566. Central column is in A026112.

Sequence in context: A107357 A251718 A182457 * A060475 A168069 A231725

Adjacent sequences:  A026102 A026103 A026104 * A026106 A026107 A026108

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling

EXTENSIONS

Edited by Ralf Stephan, Dec 18 2004

STATUS

approved

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Last modified December 7 03:45 EST 2016. Contains 278841 sequences.