

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
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OFFSET

0,5


LINKS



FORMULA

Starting with T(n, 0) = 1, if (kn) is negative or even then T(n, k) = 2*T(n, k1), otherwise T(n, k) = 2*T(n, k1)  A061897(n+1, (kn1)/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*r1))/(2*(n+1)))^k*cot((Pi*(12*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 (2r1))/(2 (n+1))]^k Cot[(Pi (12r))/(4 (n+1))], {r, 1, n+1}]; Table[T[r, nr], {n, 0, 20}, {r, 0, n}]//Round//Flatten (* Herbert Kociemba, Sep 23 2020 *)


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



