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A245185 Triangle read by rows: T(n,k) = number of pseudo-square parallelogram (psp) polyominoes with semiperimeter n+1 and k columns.

%I A245185 #20 Mar 01 2020 20:44:25
%S A245185 1,1,1,1,1,1,1,2,2,1,1,2,5,2,1,1,3,7,7,3,1,1,3,11,15,11,3,1,1,4,15,25,
%T A245185 25,15,4,1,1,4,20,41,52,41,20,4,1,1,5,25,62,92,92,62,25,5,1,1,5,32,89,
%U A245185 159,179,159,89,32,5,1,1,6,38,122,249,342,342,249,122,38,6,1
%N A245185 Triangle read by rows: T(n,k) = number of pseudo-square parallelogram (psp) polyominoes with semiperimeter n+1 and k columns.
%H A245185 Andrew Howroyd, <a href="/A245185/b245185.txt">Table of n, a(n) for n = 1..300</a>
%H A245185 Srecko Brlek, Andrea Frosini, Simone Rinaldi, 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). See Table 2.
%e A245185 Triangle begins:
%e A245185   1;
%e A245185   1, 1;
%e A245185   1, 1,  1;
%e A245185   1, 2,  2,  1;
%e A245185   1, 2,  5,  2,  1;
%e A245185   1, 3,  7,  7,  3,  1;
%e A245185   1, 3, 11, 15, 11,  3,  1;
%e A245185   1, 4, 15, 25, 25, 15,  4, 1;
%e A245185   1, 4, 20, 41, 52, 41, 20, 4, 1;
%e A245185   ...
%o A245185 (PARI)
%o A245185 IsPos(v)={for(i=1, #v, if(v[i]<=0, return(0))); 1}
%o A245185 E(b)={my(v=vector(hammingweight(b)-1), h=0, k=0); if(bittest(b,0), b>>=1); while(k<#v, if(bittest(b,0), k++; v[k]=h, h++); b>>=1); v}
%o A245185 Row(n)={my(v=vector(n)); forstep(b=2^n, 2*2^n, 2, my(r=E(b), d=b); for(k=1, n, d=bitor(d>>1, bitand(d,1)<<n); if(bittest(d,0) && !bittest(d,n), v[1+#r]+=IsPos(r-E(d))))); v}
%o A245185 { for(n=1, 10, print(Row(n))) } \\ _Andrew Howroyd_, Mar 01 2020
%Y A245185 Row sums are A244521(n+1).
%K A245185 nonn,tabl
%O A245185 1,8
%A A245185 _N. J. A. Sloane_, Jul 20 2014
%E A245185 Name clarified and terms a(46) and beyond from _Andrew Howroyd_, Mar 01 2020