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A318805 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. 5
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, 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, 68, 36, 17, 9, 4, 2, 1, 1, 21, 108, 178, 139, 71, 36, 17, 9, 4, 2, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

LINKS

Table of n, a(n) for n=1..78.

FORMULA

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

EXAMPLE

Array begins:

===============================================

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

---+-------------------------------------------

1  | 1 1 1 1  1  1  1   1   1   1    1    1 ...

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

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

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

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

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

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

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

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

...

MATHEMATICA

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];

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

Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-Fran├žois Alcover, Sep 13 2018, after Andrew Howroyd *)

PROG

(PARI)

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}

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)}

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

CROSSREFS

Cf. A318795.

Main diagonal is A316983.

Sequence in context: A104762 A152462 A180360 * A175331 A098805 A168396

Adjacent sequences:  A318802 A318803 A318804 * A318806 A318807 A318808

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Sep 03 2018

STATUS

approved

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Last modified November 20 10:09 EST 2019. Contains 329334 sequences. (Running on oeis4.)