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A318795 Array read by antidiagonals: T(n,k) is the number of inequivalent nonnegative integer n X n matrices with sum of elements equal to k, under row and column permutations. 15
1, 1, 1, 1, 4, 1, 1, 5, 4, 1, 1, 11, 10, 4, 1, 1, 14, 24, 10, 4, 1, 1, 24, 51, 33, 10, 4, 1, 1, 30, 114, 78, 33, 10, 4, 1, 1, 45, 219, 224, 91, 33, 10, 4, 1, 1, 55, 424, 549, 277, 91, 33, 10, 4, 1, 1, 76, 768, 1403, 792, 298, 91, 33, 10, 4, 1, 1, 91, 1352, 3292, 2341, 881, 298, 91, 33, 10, 4, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Andrew Howroyd, Additional PARI Programs

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 4 5 11 14 24 30 45 55 76 91 119 ...

3 | 1 4 10 24 51 114 219 424 768 1352 2278 3759 ...

4 | 1 4 10 33 78 224 549 1403 3292 7677 16934 36581 ...

5 | 1 4 10 33 91 277 792 2341 6654 18802 51508 138147 ...

6 | 1 4 10 33 91 298 881 2825 8791 27947 87410 272991 ...

7 | 1 4 10 33 91 298 910 2974 9655 32287 108274 367489 ...

8 | 1 4 10 33 91 298 910 3017 9886 33767 116325 410298 ...

9 | 1 4 10 33 91 298 910 3017 9945 34124 118729 424498 ...

...

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, q_List, k_] := SeriesCoefficient[1/Product[(1 - x^LCM[p[[i]], q[[j]]])^GCD[p[[i]], q[[j]]], {j, 1, Length[q]}, {i, 1, Length[p]}], {x, 0, k}];

M[m_, n_, k_] := Module[{s=0}, Do[Do[s += permcount[p]*permcount[q]*c[p, q, k], {q, IntegerPartitions[n]}], {p, IntegerPartitions[m]}]; s/(m!*n!)];

Table[M[n-k+1, n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Sep 12 2018, after Andrew Howroyd *)

PROG

(PARI) \\ see also link.

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}

K(q, t, k)={1/prod(j=1, #q, (1-y^lcm(t, q[j]) + O(y*y^k))^gcd(t, q[j]))}

M(m, n, k)={my(s=0); forpart(q=m, s+=permcount(q)*polcoef(polcoef(exp(sum(t=1, n, K(q, t, k)/t*x^t) + O(x*x^n)), n), k)); s/m!}

for(n=1, 10, for(k=1, 12, print1(M(n, n, k), ", ")); print); \\ updated Andrew Howroyd, Mar 29 2020

CROSSREFS

Rows 2..7 are A053307, A052365, A052366, A052367, A052372, A052373.

Main diagonal is A007716.

Cf. A214398, A246106, A304942, A318805.

Sequence in context: A021247 A016522 A153843 * A099575 A173740 A028275

Adjacent sequences: A318792 A318793 A318794 * A318796 A318797 A318798

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Sep 03 2018

STATUS

approved

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Last modified February 6 04:21 EST 2023. Contains 360097 sequences. (Running on oeis4.)