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A175331 Array A092921(n,k) without the first two rows, read by antidiagonals.

%I A175331 #21 Jan 05 2025 04:45:54
%S A175331 1,1,1,1,2,1,1,3,2,1,1,5,4,2,1,1,8,7,4,2,1,1,13,13,8,4,2,1,1,21,24,15,
%T A175331 8,4,2,1,1,34,44,29,16,8,4,2,1,1,55,81,56,31,16,8,4,2,1,1,89,149,108,
%U A175331 61,32,16,8,4,2,1,1,144,274,208,120,63,32,16,8,4,2,1,1,233,504,401,236,125,64,32,16,8,4,2,1
%N A175331 Array A092921(n,k) without the first two rows, read by antidiagonals.
%C A175331 Antidiagonal sums are A048888. This is a transposed version of A048887, so the bivariate generating function is obtained by swapping the two arguments.
%C A175331 Brlek et al. (2006) call this table "number of psp-polyominoes with flat bottom". - _N. J. A. Sloane_, Oct 30 2018
%D A175331 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 125, 155.
%H A175331 Srecko Brlek, Andrea Frosini, Simone Rinaldi, and Laurent Vuillon, <a href="https://doi.org/10.37236/1041">Tilings by translation: enumeration by a rational language approach</a>, The Electronic Journal of Combinatorics, vol.13, (2006). Table 1 is essentially this array. - _N. J. A. Sloane_, Jul 20 2014
%F A175331 T(n,k) = A092921(n,k), n >= 2.
%F A175331 T(n,2) = A000045(n).
%F A175331 T(n,3) = A000073(n+2).
%F A175331 T(n,4) = A000078(n+2).
%e A175331 The array starts in row n=2 with columns k >= 1 as:
%e A175331   1   1   1   1   1   1   1   1   1   1
%e A175331   1   2   2   2   2   2   2   2   2   2
%e A175331   1   3   4   4   4   4   4   4   4   4
%e A175331   1   5   7   8   8   8   8   8   8   8
%e A175331   1   8  13  15  16  16  16  16  16  16
%e A175331   1  13  24  29  31  32  32  32  32  32
%e A175331   1  21  44  56  61  63  64  64  64  64
%e A175331   1  34  81 108 120 125 127 128 128 128
%e A175331   1  55 149 208 236 248 253 255 256 256
%p A175331 A092921 := proc(n,k) if k <= 0 or n <= 0 then 0; elif k = 1 or n = 1 then 1; else add( procname(n-i,k),i=1..k) ; end if; end proc:
%p A175331 A175331 := proc(n,k) A092921(n,k) ; end proc: # _R. J. Mathar_, Dec 17 2010
%t A175331 f[x_, n_] = (x - x^(m + 1))/(1 - 2*x + x^(m + 1))
%t A175331 a = Table[Table[SeriesCoefficient[
%t A175331       Series[f[x, m], {x, 0, 10}], n], {n, 0, 10}], {m, 1, 10}];
%t A175331 Table[Table[a[[m, n - m + 1]], {m, 1, n - 1}], {n, 1, 10}];
%t A175331 Flatten[%]
%Y A175331 Cf. A000045, A048887, A048004, A126198.
%K A175331 nonn,tabl,easy
%O A175331 2,5
%A A175331 _Roger L. Bagula_, Dec 03 2010