A060556 Bisection of triangle A060098: odd-indexed members of column sequences of A060098 (not counting leading zeros).

1, 1, 2, 1, 6, 3, 1, 12, 16, 4, 1, 20, 50, 32, 5, 1, 30, 120, 140, 55, 6, 1, 42, 245, 448, 316, 86, 7, 1, 56, 448, 1176, 1284, 622, 126, 8, 1, 72, 756, 2688, 4170, 3102, 1113, 176, 9, 1, 90, 1200, 5544, 11550, 12122

Offset

0,3

Comments

Row sums give A060557. Column sequences without leading zeros give for m=0..5: A000012 (powers of 1), A002378 = 2*A000217, A004320, 4*A040977, A060558, 2*A060559.

Companion triangle (even-indexed members) A060102.

With offset 1 for n and k, T(n,k) is the number of (1-2-3)-avoiding trapezoidal words of length n that contain n+1-k 1s. A trapezoidal word (following Riordan) is a sequence (a_1,a_2,...,a_n) of integers with 1 <= a_i <= 2i-1. For example, T(3,3)=3 counts 122, 132, 133 and T(4,2)=12 counts 1112, 1113, 1114, 1115, 1116, 1117, 1121, 1131, 1141, 1151, 1211, 1311. - David Callan, Aug 25 2009

Links

Table of n, a(n) for n=0..50.

Formula

a(n, m)= A060098(2*n+1-m, m).

G.f. for column m: (x^m)*Po(m+1, x)/(1-x)^(2*m+1), with Po(n, x) = Sum_{j=0..floor(n/2)} binomial(n, 2*j+1)*x^j (odd members of row n of Pascal triangle A007318).

a(n, m) = Sum_{j=0..floor((m+1)/2)} binomial(n-j+m, 2*m)*binomial(m+1, 2*j+1), n >= m >= 0, otherwise zero.

Example

{1}; {1,2}; {1,6,3}; {1,12,16,4}; ...; Po(3,x) = 3 + x.

Crossrefs

Sequence in context: A362997 A120108 A350292 * A222969 A132813 A034898

Adjacent sequences: A060553 A060554 A060555 * A060557 A060558 A060559

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Apr 06 2001

Status

approved