A059431 -id:A059431

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A059429

Cumulative boustrophedon transform of 1, 0, 0, 0, ...

+10
3

1, 1, 2, 8, 51, 478, 6178, 105330, 2290069, 61839897, 2030449500, 79661186168, 3680458880352, 197781841355220, 12231649482909444, 862560715175755168, 68799732139319891208, 6162698115430291654438, 615995773861169229993018

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

For n>0, a(n) equals the element in the upper left corner of the matrix equal to the product of n X n matrices given by: Product_{k=1..n} M_k where M_k(r,c)=k+2-r-c when r+c<=k+1 and zeros elsewhere (see example). - Paul D. Hanna, Feb 08 2007

LINKS

Table of n, a(n) for n=0..18.

FORMULA

See Maple code for precise description.

EXAMPLE

For n=5, the matrix product:

[1 0 0 0 0] [2 1 0 0 0] [3 2 1 0 0] [4 3 2 1 0] [5 4 3 2 1]

[0 0 0 0 0] [1 0 0 0 0] [2 1 0 0 0] [3 2 1 0 0] [4 3 2 1 0]

[0 0 0 0 0] [0 0 0 0 0] [1 0 0 0 0] [2 1 0 0 0] [3 2 1 0 0]

[0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [1 0 0 0 0] [2 1 0 0 0]

[0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0] [1 0 0 0 0]

equals the matrix below with a(5)=478 in the upper left corner:

[478 362 246 138 51]

[__0 __0 __0 __0 _0]

also a(4)=51 will be in the upper right corner.

MAPLE

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

MATHEMATICA

m[n_, k_] := Table[If[r+c <= k+1, k+2-r-c, 0], {r, 1, n}, {c, 1, n}]; a[0] = 1; a[n_] := (Dot @@ Table[m[n, k], {k, 1, n}])[[1, 1]]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Jul 18 2012, after Paul D. Hanna *)

PROG

(PARI) {a(n)=if(n==0, 1, prod(k=1, n, matrix(n, n, r, c, if(r+c<=k+1, k+2-r-c)))[1, 1])} \\ Paul D. Hanna, Feb 08 2007

CROSSREFS

See the triangles in A059431 and A059432.

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Jan 31 2001

STATUS

approved

A059432

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

+10
3

1, 0, 1, 0, 1, 2, 0, 2, 5, 8, 0, 8, 21, 36, 51, 0, 51, 138, 246, 362, 478, 0, 478, 1318, 2404, 3628, 4903, 6178, 0, 6178, 17259, 31968, 49081, 67512, 86421, 105330, 0, 105330, 297081, 556344, 864688, 1205000, 1562571, 1926320, 2290069

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,6

LINKS

Table of n, a(n) for n=0..44.

FORMULA

From Petros Hadjicostas, Feb 16 2021: (Start)

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.)

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

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

EXAMPLE

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

0, 1;

0, 1, 2;

0, 2, 5, 8;

0, 8, 21, 36, 51;

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

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

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

... - Petros Hadjicostas, Feb 16 2021

MAPLE

# This is a modification of N. J. A. Sloane's program from A059429:

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:

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

CBOUS2([1, 0, 0, 0, 0, 0, 0, 0, 0, 0]); # Petros Hadjicostas, Feb 16 2021

CROSSREFS

Cf. A059429, A059431.

KEYWORD

nonn,tabl,easy

AUTHOR

N. J. A. Sloane, Jan 31 2001

STATUS

approved

A341647

a(n) = Sum_{k=0..n} A059432(n,k).

+10
0

1, 1, 3, 15, 116, 1275, 18909, 363749, 8807403, 262089849, 9401686696, 400089347172, 19927273211442, 1148368173455106, 75812694541312803, 5684165502304973376, 480334585284310506044, 45439833370389830274384, 4783252893141511378449783

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Row sums of both triangular arrays A059431 and A059432.

LINKS

Table of n, a(n) for n=0..18.

FORMULA

a(n) = Sum_{k=0..n} A059431(n,k).

EXAMPLE

a(6) = 0 + 478 + 1318 + 2404 + 3628 + 4903 + 6178 = 18909.

CROSSREFS

Cf. A059431, A059432.

KEYWORD

nonn

AUTHOR

Petros Hadjicostas, Feb 16 2021

STATUS

approved

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