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A062135 Odd-numbered columns of Losanitsch triangle A034851 formatted as triangle with an additional first column. 1
1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 6, 3, 1, 0, 3, 10, 12, 4, 1, 0, 3, 19, 28, 20, 5, 1, 0, 4, 28, 66, 60, 30, 6, 1, 0, 4, 44, 126, 170, 110, 42, 7, 1, 0, 5, 60, 236, 396, 365, 182, 56, 8, 1, 0, 5, 85, 396, 868, 1001, 693, 280 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Because the sequence of column m=2*k, k >= 1, of A034851 is the partial sum sequence of the one of column m=2*k-1 the present triangle is essentially Losanitsch's triangle A034851.

Row sums give A051450 with A051450(0) := 1. Column sequences (without leading zeros) are for m=0..6: A000007, A008619, A005993, A005995, A018211, A018213, A061191.

LINKS

Table of n, a(n) for n=0..62.

FORMULA

a(n, m)= A034851(n-1+m, n-m), n >= m >= 0; A034851(n-1, n) := 0, n >= 1, A034851(-1, 0) := 1.

a(n, m)=0 if n<m; a(0, 0)=1, a(n, 0)=0 if n >= 1; a(n, m)= a(n-1, m)+sum(a(k, m-1), k=m-1..n-1) if n+m even and a(n, m)= a(n-1, m)+sum(a(k, m-1), k=m-1..n-1)-binomial((n+m-3)/2, m-1) if n+m odd, n >= m >= 1.

G.f. for column m: x^m*Pe(m, x^2)/(((1-x)^(2*m))*(1+x)^m), m >= 0, with Pe(m, x^2)= sum(A034839(m, k)*x^(2*k), k=0..floor(n/2)), the row polynomial of array A034839 (even-indexed entries of the rows of Pascal's triangle).

EXAMPLE

{1}; {0,1}; {0,1,1}; {0,2,2,1}; ...; Pe(4,x^2)=1+6*x^2+x^4.

CROSSREFS

Cf. A034851, A034839.

Sequence in context: A060086 A308680 A177975 * A190182 A068926 A276770

Adjacent sequences:  A062132 A062133 A062134 * A062136 A062137 A062138

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Jun 19 2001

STATUS

approved

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Last modified September 30 21:27 EDT 2020. Contains 337440 sequences. (Running on oeis4.)