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A059432 Triangle formed when cumulative boustrophedon transform is applied to 1, 0, 0, 0, ..., read by rows in natural order. 3

%I

%S 1,0,1,0,1,2,0,2,5,8,0,8,21,36,51,0,51,138,246,362,478,0,478,1318,

%T 2404,3628,4903,6178,0,6178,17259,31968,49081,67512,86421,105330,0,

%U 105330,297081,556344,864688,1205000,1562571,1926320,2290069

%N Triangle formed when cumulative boustrophedon transform is applied to 1, 0, 0, 0, ..., read by rows in natural order.

%F From _Petros Hadjicostas_, Feb 16 2021: (Start)

%F T(i,j) = T(i,j-1) + Sum_{r=1..j} T(i-1,i-r) for i >= 1 and 1 <= j <= i with T(i,0) = b(i+1) for i >= 0, where b(1) = 1 and b(i) = 0 for i >= 2. (The sequence b = (b(i): i >= 1) is the input sequence.)

%F T(i,j) = 2*T(i,j-1) - T(i,j-2) + T(i-1,i-j) for i >= 2 and 2 <= j <= i.

%F T(i,i) = A059429(i) = T(i+1,1) for i >= 0. (End)

%e Triangle T(i,j) (with rows i >= 0 and columns j = 0..i) begins:

%e 1;

%e 0, 1;

%e 0, 1, 2;

%e 0, 2, 5, 8;

%e 0, 8, 21, 36, 51;

%e 0, 51, 138, 246, 362, 478;

%e 0, 478, 1318, 2404, 3628, 4903, 6178;

%e 0, 6178, 17259, 31968, 49081, 67512, 86421, 105330;

%e ... - _Petros Hadjicostas_, Feb 16 2021

%p # This is a modification of _N. J. A. Sloane_'s program from A059429:

%p CBOUS2 := proc(a) local c, i, j, n, r: option remember: if whattype(a) <> list then RETURN([]): end if: n := min(nops(a), 60): for i from 0 to n - 1 do c[i, 0] := a[i + 1]: end do: for i to n - 1 do for j to i do c[i, j] := c[i, j - 1] + add(c[i - 1, i - r], r = 1 .. j): end do: end do: RETURN([seq(seq(c[i, j], j = 0 .. i), i = 0 .. n - 1)]): end proc:

%p # To get the flattened triangle up to the 9th row, we type

%p CBOUS2([1, 0, 0, 0, 0, 0, 0, 0, 0, 0]); # _Petros Hadjicostas_, Feb 16 2021

%Y Cf. A059429, A059431.

%K nonn,tabl,easy

%O 0,6

%A _N. J. A. Sloane_, Jan 31 2001

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Last modified August 3 15:37 EDT 2021. Contains 346439 sequences. (Running on oeis4.)