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A371967 Irregular triangle T(r,w) read by rows: number of ways of placing w non-attacking wazirs on a 3 X r board.

%I A371967 #43 Jun 06 2024 03:50:32
%S A371967 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 A371967 53,9,1,1,18,126,442,840,880,504,158,28,2,1,21,178,792,2023,3063,2763,
%U A371967 1478,472,93,12,1,1,24,239,1292,4176,8406,10692,8604,4374,1416,297,38,2,1,27,309
%N A371967 Irregular triangle T(r,w) read by rows: number of ways of placing w non-attacking wazirs on a 3 X r board.
%H A371967 Alois P. Heinz, <a href="/A371967/b371967.txt">Rows r = 0..165</a> (first 17 rows from R. J. Mathar)
%H A371967 R. J. Mathar, <a href="http://viXra.org/abs/2404.0122">Bivariate Generating Functions for Non-attacking Wazirs on Rectangular Boards</a>
%H A371967 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 A371967 Wikipedia, <a href="https://en.wikipedia.org/wiki/Wazir_(chess)">Wazir (chess)</a>
%F A371967 T(r,0) = 1.
%F A371967 T(r,1) = 3*r.
%F A371967 T(r,2) = A064225(r-1).
%F A371967 T(r,3) = A172229(r).
%F A371967 T(r,4) = 27*r^4/8 -117*r^3/4 +829*r^2/8 -715*r/4 +126. [Siehler Table 3]
%F A371967 T(3,w) = A232833(3,w).
%F A371967 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 A371967 The triangle starts with r>=0 rows and w>=0 wazirs as
%e A371967   1 ;
%e A371967   1 3 1 ;
%e A371967   1 6 8 2  ;
%e A371967   1 9 24 22 6 1 ;
%e A371967   1 12 49 84 61 18 2  ;
%e A371967   1 15 83 215 276 174 53 9 1 ;
%e A371967   1 18 126 442 840 880 504 158 28 2  ;
%e A371967   1 21 178 792 2023 3063 2763 1478 472 93 12 1 ;
%e A371967   1 24 239 1292 4176 8406 10692 8604 4374 1416 297 38 2  ;
%e A371967   1 27 309 1969 7731 19591 32716 36257 26674 13035 4264 945 142 15 1 ;
%e A371967   ...
%p A371967 b:= proc(n, l) option remember; `if`(n=0, 1,
%p A371967       add(`if`(Bits[And](j, l)>0, 0, expand(b(n-1, j)*
%p A371967       x^add(i, i=Bits[Split](j)))), j=[0, 1, 2, 4, 5]))
%p A371967     end:
%p A371967 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
%p A371967 seq(T(n), n=0..10);  # _Alois P. Heinz_, Apr 14 2024
%t A371967 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 A371967 T[n_] := CoefficientList[b[n, 0], x];
%t A371967 Table[T[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Jun 05 2024, after _Alois P. Heinz_ *)
%Y A371967 Cf. A051736 (row sums), A035607 (on 2Xr board), A011973 (on 1Xr board), A232833 (on rXr board).
%Y A371967 T(n,n) gives A371978.
%Y A371967 Row maxima give A371979.
%Y A371967 Cf. A007494.
%K A371967 nonn,tabf
%O A371967 0,3
%A A371967 _R. J. Mathar_, Apr 14 2024