A318805 - OEIS

%I #10 Sep 13 2018 04:23:39

%S 1,1,1,1,2,1,1,3,2,1,1,5,4,2,1,1,6,8,4,2,1,1,8,13,9,4,2,1,1,10,22,16,

%T 9,4,2,1,1,13,33,32,17,9,4,2,1,1,15,52,57,35,17,9,4,2,1,1,18,76,105,

%U 68,36,17,9,4,2,1,1,21,108,178,139,71,36,17,9,4,2,1

%N Array read by antidiagonals: T(n,k) is the number of inequivalent symmetric nonnegative integer n X n matrices with sum of elements equal to k, under row and column permutations.

%F T(n,k) = T(k,k) for n > k.

%e Array begins:

%e ===============================================

%e n\k| 1 2 3 4  5  6  7   8   9  10   11   12

%e ---+-------------------------------------------

%e 1  | 1 1 1 1  1  1  1   1   1   1    1    1 ...

%e 2  | 1 2 3 5  6  8 10  13  15  18   21   25 ...

%e 3  | 1 2 4 8 13 22 33  52  76 108  150  209 ...

%e 4  | 1 2 4 9 16 32 57 105 178 301  490  793 ...

%e 5  | 1 2 4 9 17 35 68 139 264 502  924 1695 ...

%e 6  | 1 2 4 9 17 36 71 151 303 619 1234 2473 ...

%e 7  | 1 2 4 9 17 36 72 154 315 661 1370 2885 ...

%e 8  | 1 2 4 9 17 36 72 155 318 673 1413 3034 ...

%e 9  | 1 2 4 9 17 36 72 155 319 676 1425 3078 ...

%e ...

%t permcount[v_List] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];

%t c[p_List, k_] := SeriesCoefficient[1/(Product[Product[(1 - x^(2*LCM[p[[i]], p[[j]] ]))^GCD[p[[i]], p[[j]]], {j, 1, i - 1}], {i, 2, Length[p]}]* Product[t = p[[i]]; (1 - x^t)^Mod[t, 2]*(1 - x^(2*t))^Quotient[t, 2], {i, 1, Length[p]}]), {x, 0, k}];

%t T[_, 1] = T[1, _] = 1; T[n_, k_] := (s = 0; Do[s += permcount[p]*c[p, k], {p, IntegerPartitions[n]}]; s/n!);

%t Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Sep 13 2018, after _Andrew Howroyd_ *)

%o (PARI)

%o permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}

%o c(p,k)={polcoef(1/(prod(i=2, #p, prod(j=1, i-1, (1 - x^(2*lcm(p[i],p[j])) + O(x*x^k))^gcd(p[i], p[j]))) * prod(i=1, #p, my(t=p[i]); (1 - x^t + O(x*x^k))^(t%2)*(1 - x^(2*t) + O(x*x^k))^(t\2) )), k)}

%o T(n,k)={if(n==0, k==0, my(s=0); forpart(p=n, s+=permcount(p)*c(p,k)); s/n!)}

%Y Cf. A318795.

%Y Main diagonal is A316983.

%K nonn,tabl

%O 1,5

%A _Andrew Howroyd_, Sep 03 2018