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A068913 Square array read by antidiagonals of number of k step walks (each step +-1 starting from 0) which are never more than n or less than -n. 10
1, 0, 1, 0, 2, 1, 0, 2, 2, 1, 0, 4, 4, 2, 1, 0, 4, 6, 4, 2, 1, 0, 8, 12, 8, 4, 2, 1, 0, 8, 18, 14, 8, 4, 2, 1, 0, 16, 36, 28, 16, 8, 4, 2, 1, 0, 16, 54, 48, 30, 16, 8, 4, 2, 1, 0, 32, 108, 96, 60, 32, 16, 8, 4, 2, 1, 0, 32, 162, 164, 110, 62, 32, 16, 8, 4, 2, 1, 0, 64, 324, 328, 220, 124, 64, 32, 16, 8, 4, 2, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
FORMULA
Starting with T(n, 0) = 1, if (k-n) is negative or even then T(n, k) = 2*T(n, k-1), otherwise T(n, k) = 2*T(n, k-1) - A061897(n+1, (k-n-1)/2). So for n>=k, T(n, k) = 2^k. [Corrected by Sean A. Irvine, Mar 23 2024]
T(n,0) = 1, T(n,k) = (2^k/(n+1))*Sum_{r=1..n+1} (-1)^r*cos((Pi*(2*r-1))/(2*(n+1)))^k*cot((Pi*(1-2*r))/(4*(n+1))). - Herbert Kociemba, Sep 23 2020
EXAMPLE
Rows start:
1, 0, 0, 0, 0, ...
1, 2, 2, 4, 4, ...
1, 2, 4, 6, 12, ...
1, 2, 4, 8, 14, ...
...
MATHEMATICA
T[n_, 0]=1; T[n_, k_]:=2^k/(n+1) Sum[(-1)^r Cos[(Pi (2r-1))/(2 (n+1))]^k Cot[(Pi (1-2r))/(4 (n+1))], {r, 1, n+1}]; Table[T[r, n-r], {n, 0, 20}, {r, 0, n}]//Round//Flatten (* Herbert Kociemba, Sep 23 2020 *)
CROSSREFS
Cf. early rows: A000007, A016116 (without initial term), A068911, A068912, A216212, A216241, A235701.
Central and lower diagonals are A000079, higher diagonals include A000918, A028399.
Sequence in context: A289281 A212957 A035393 * A128306 A372626 A305152
KEYWORD
nonn,tabl
AUTHOR
Henry Bottomley, Mar 06 2002
STATUS
approved

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Last modified May 29 22:36 EDT 2024. Contains 372954 sequences. (Running on oeis4.)