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A055801 Array T read by rows: T(i,0)=T(i,i)=1 for i >= 0; T(i,j)=Sum{T(i-2k,j-2k+1: k >= 1) for 1<=j<=i-1, where T(m,n) := 0 if m<0 or n<0. 7
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 3, 3, 1, 1, 1, 1, 2, 3, 4, 3, 1, 1, 1, 1, 2, 3, 5, 6, 4, 1, 1, 1, 1, 2, 3, 5, 7, 7, 4, 1, 1, 1, 1, 2, 3, 5, 8, 11, 10, 5, 1, 1, 1, 1, 2, 3, 5, 8, 12, 14, 11, 5, 1, 1, 1, 1, 2, 3, 5, 8, 13 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,14

COMMENTS

T(i+j,j)=number of strings (s(1),...,s(m)) of nonnegative integers s(k) such that m<=i+1, s(m)=j and s(k)-s(k-1) is an odd positive integer for k=2,3,...,m.

T(i+j,j)=number of compositions of numbers <=j using up to i parts, each an odd positive integer.

REFERENCES

C. Kimberling, Path-counting and Fibonacci numbers, Fib. Quart. 40 (4) (2002) 328-338, example 2B.

LINKS

Table of n, a(n) for n=0..85.

EXAMPLE

Rows:

  1

  1  1

  1  1  1

  1  1  1  1

  1  1  1  2  1

  1  1  1  2  2  1

  1  1  1  2  3  3  1

  1  1  1  2  3  4  3  1

  1  1  1  2  3  5  6  4  1

  1  1  1  2  3  5  7  7  4  1

  1  1  1  2  3  5  8 11 10  5  1

  1  1  1  2  3  5  8 12 14 11  5  1

  1  1  1  2  3  5  8 13 19 21 15  6  1

  1  1  1  2  3  5  8 13 20 26 25 16  6  1

  1  1  1  2  3  5  8 13 21 32 40 36 21  7  1

  1  1  1  2  3  5  8 13 21 33 46 51 41 22  7  1

T(9,6) counts the strings 3456, 1236, 1256, 1456, 036, 016, 056.

T(9,6) counts the compositions 111, 113, 131, 311, 33, 15, 51.

MAPLE

A055801 := proc(i, j)

    if j =0 or j = i then

        1;

    elif i < 0 or j < 0 then

        0;

    else

        add(procname(i-2*k, j-2*k+1), k=1..floor(i/2)) ;

    end if;

end proc: # R. J. Mathar, Feb 11 2018

CROSSREFS

Infinitely many of the columns are (1, 1, 1, 2, 3, 5, 8, ..., Fibonacci numbers)

Essentially a reflected version of A011794.

Sequence in context: A238384 A139040 A139147 * A155050 A140356 A119963

Adjacent sequences:  A055798 A055799 A055800 * A055802 A055803 A055804

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, May 28 2000

STATUS

approved

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Last modified January 18 20:57 EST 2019. Contains 319282 sequences. (Running on oeis4.)