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A318951 Array read by rows: T(n,k) is the number of nonisomorphic n X n matrices with nonnegative integer entries and row sums k under row and column permutations, (n >= 1, k >= 0). 9
1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 14, 5, 1, 1, 9, 44, 53, 7, 1, 1, 12, 129, 458, 198, 11, 1, 1, 16, 316, 3411, 5929, 782, 15, 1, 1, 20, 714, 19865, 145168, 96073, 3111, 22, 1, 1, 25, 1452, 95214, 2459994, 9283247, 1863594, 12789, 30, 1, 1, 30, 2775, 383714, 30170387, 537001197, 833593500, 42430061, 53836, 42, 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

EXAMPLE

Array begins:

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

n\k| 0  1    2       3         4            5              6

---|------------------------------------------------------------

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

2  | 1  2    4       6         9           12             16 ...

3  | 1  3   14      44       129          316            714 ...

4  | 1  5   53     458      3411        19865          95214 ...

5  | 1  7  198    5929    145168      2459994       30170387 ...

6  | 1 11  782   96073   9283247    537001197    19578605324 ...

7  | 1 15 3111 1863594 833593500 189076534322 23361610029905 ...

...

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];

K[q_List, t_, k_] := SeriesCoefficient[1/Product[g = GCD[t, q[[j]]]; (1 - x^(q[[j]]/g))^g, {j, 1, Length[q]}], {x, 0, k}];

RowSumMats[n_, m_, k_] := Module[{s = 0}, Do[s += permcount[q]* SeriesCoefficient[Exp[Sum[K[q, t, k]/t*x^t, {t, 1, n}]], {x, 0, n}], {q, IntegerPartitions[m]}]; s/m!];

Table[RowSumMats[n-k, n-k, k], {n, 1, 11}, {k, n-1, 0, -1}] // Flatten (* Jean-Fran├žois Alcover, Sep 12 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}

K(q, t, k)={polcoeff(1/prod(j=1, #q, my(g=gcd(t, q[j])); (1 - x^(q[j]/g) + O(x*x^k))^g), k)}

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

for(n=1, 8, for(k=0, 6, print1(RowSumMats(n, n, k), ", ")); print)

CROSSREFS

Rows 2..6 are A002620(n+2), A058389, A058390, A058391, A058392.

Cf. A304942, A306017.

Sequence in context: A055080 A034367 A034371 * A101321 A210764 A091186

Adjacent sequences:  A318948 A318949 A318950 * A318952 A318953 A318954

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Sep 05 2018

STATUS

approved

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Last modified December 9 19:51 EST 2019. Contains 329879 sequences. (Running on oeis4.)