A371967 - OEIS

%I #43 Jun 06 2024 03:50:32

%S 1,1,3,1,1,6,8,2,1,9,24,22,6,1,1,12,49,84,61,18,2,1,15,83,215,276,174,

%T 53,9,1,1,18,126,442,840,880,504,158,28,2,1,21,178,792,2023,3063,2763,

%U 1478,472,93,12,1,1,24,239,1292,4176,8406,10692,8604,4374,1416,297,38,2,1,27,309

%N Irregular triangle T(r,w) read by rows: number of ways of placing w non-attacking wazirs on a 3 X r board.

%H Alois P. Heinz, <a href="/A371967/b371967.txt">Rows r = 0..165</a> (first 17 rows from R. J. Mathar)

%H R. J. Mathar, <a href="http://viXra.org/abs/2404.0122">Bivariate Generating Functions for Non-attacking Wazirs on Rectangular Boards</a>

%H Jacob A. Siehler, <a href="https://arxiv.org/abs/1409.3869">Selections without adjacency on a rectangular grid</a>, arXiv:1409.3869 [math.CO] (2014) Table 2.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Wazir_(chess)">Wazir (chess)</a>

%F T(r,0) = 1.

%F T(r,1) = 3*r.

%F T(r,2) = A064225(r-1).

%F T(r,3) = A172229(r).

%F T(r,4) = 27*r^4/8 -117*r^3/4 +829*r^2/8 -715*r/4 +126. [Siehler Table 3]

%F T(3,w) = A232833(3,w).

%F G.f.: (1+x*y) *(1 +x*y +x*y^2 -x^2*y^3)/(1 -x -x*y -x^2*y^3 -2*x^2*y -3*x^2*y^2 -x^3*y^2 +x^3*y^4 +x^4*y^4). - _R. J. Mathar_, Apr 21 2024

%e The triangle starts with r>=0 rows and w>=0 wazirs as

%e   1 ;

%e   1 3 1 ;

%e   1 6 8 2  ;

%e   1 9 24 22 6 1 ;

%e   1 12 49 84 61 18 2  ;

%e   1 15 83 215 276 174 53 9 1 ;

%e   1 18 126 442 840 880 504 158 28 2  ;

%e   1 21 178 792 2023 3063 2763 1478 472 93 12 1 ;

%e   1 24 239 1292 4176 8406 10692 8604 4374 1416 297 38 2  ;

%e   1 27 309 1969 7731 19591 32716 36257 26674 13035 4264 945 142 15 1 ;

%e   ...

%p b:= proc(n, l) option remember; `if`(n=0, 1,

%p       add(`if`(Bits[And](j, l)>0, 0, expand(b(n-1, j)*

%p       x^add(i, i=Bits[Split](j)))), j=[0, 1, 2, 4, 5]))

%p     end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):

%p seq(T(n), n=0..10);  # _Alois P. Heinz_, Apr 14 2024

%t b[n_, l_] := b[n, l] = If[n == 0, 1, Sum[If[BitAnd[j, l] > 0, 0, Expand[b[n - 1, j]*x^DigitCount[j, 2, 1]]], {j, {0, 1, 2, 4, 5}}]];

%t T[n_] := CoefficientList[b[n, 0], x];

%t Table[T[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Jun 05 2024, after _Alois P. Heinz_ *)

%Y Cf. A051736 (row sums), A035607 (on 2Xr board), A011973 (on 1Xr board), A232833 (on rXr board).

%Y T(n,n) gives A371978.

%Y Row maxima give A371979.

%Y Cf. A007494.

%K nonn,tabf

%O 0,3

%A _R. J. Mathar_, Apr 14 2024