A054975 -id:A054975 - cp's OEIS frontend

A321615 Triangle read by rows: T(n,k) is the number of k X k integer matrices with sum of elements n, with no zero rows or columns, up to row and column permutation.

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 6, 3, 1, 0, 1, 9, 13, 3, 1, 0, 1, 17, 38, 20, 3, 1, 0, 1, 23, 97, 82, 23, 3, 1, 0, 1, 36, 217, 311, 126, 24, 3, 1, 0, 1, 46, 453, 968, 624, 151, 24, 3, 1, 0, 1, 65, 868, 2825, 2637, 933, 162, 24, 3, 1, 0, 1, 80, 1585, 7394, 10098, 4942, 1132, 165, 24, 3, 1

Offset: 0

Andrew Howroyd, Nov 14 2018

Also the number of non-isomorphic multiset partitions of weight n with k parts and k vertices, where the weight of a multiset partition is the sum of sizes of its parts. - Gus Wiseman, Nov 18 2018

			Triangle begins:
    1
    0  1
    0  1    1
    0  1    2    1
    0  1    6    3    1
    0  1    9   13    3    1
    0  1   17   38   20    3    1
    0  1   23   97   82   23    3    1
    0  1   36  217  311  126   24    3    1
    0  1   46  453  968  624  151   24    3    1
    0  1   65  868 2825 2637  933  162   24    3    1

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Columns k=0..3 are A000007, A000012, A054974, A054975.
Row sums are A319616.
Cf. A007716, A048291, A054976, A057149, A057150, A104601, A120732, A318795, A320808.

(* See A318795 for M[m, n, k]. *)
T[n_, k_] := M[k, k, n] - 2 M[k, k-1, n] + M[k-1, k-1, n];
Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 24 2018, from PARI *)

\\ See A318795 for M.
T(n, k) = if(k==0, n==0, M(k, k, n) - 2*M(k, k-1, n) + M(k-1, k-1, n));

\\ See A340652 for G.
T(n)={[Vecrev(p) | p<-Vec(1 + sum(k=1, n, y^k*(polcoef(G(k, n, n, y), k, y) - polcoef(G(k-1, n, n, y), k, y))))]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 16 2024

Column k=0 inserted by Andrew Howroyd, Jan 17 2024

A055007 Number of nonnegative integer 4 X 4 matrices with no zero rows or columns and with sum of elements equal to n.

Original entry on oeis.org

1, 0, 0, 0, 24, 528, 4648, 26224, 112666, 401424, 1246000, 3476368, 8905432, 21266208, 47875272, 102482048, 210000931, 414160240, 789572072, 1460372624, 2628456428, 4615495808, 7924479264, 13328517504, 21997272036, 35674700896, 56926058920, 89477437120

Offset: 0

Vladeta Jovovic, May 30 2000

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1).

Cf. A054688, A054974, A054975, A052366.

Number of nonnegative integer p X q matrices with no zero rows or columns and with sum of elements equal to n is Sum_{k=0...q} (-1)^k*C(q, k)*m(p, q-k, n) where m(p, q, n)=Sum_{k=0..p} (-1)^k*C(p, k)*C((p-k)*q+n-1, n).
For p = q = 4 we get a(n) = (1/15!)*(n^15 + 120*n^14 + 6580*n^13 + 218400*n^12 + 4637542*n^11 + 61261200*n^10 + 423591740*n^9 + 164392800*n^8 - 17247717487*n^7 - 47940252360*n^6 + 346941238280*n^5 + 557885764800*n^4 - 4897231459056*n^3 + 8643549191040*n^2 - 5894285241600*n + 1307674368000).
G.f.: -(16*x^15 -192*x^14 +1040*x^13 -3356*x^12 +7200*x^11 -10952*x^10 +12544*x^9 -11712*x^8 +9664*x^7 -7088*x^6 +4224*x^5 -1844*x^4 +560*x^3 -120*x^2 +16*x -1) / (x -1)^16. - Colin Barker, Jul 11 2013

More terms from James Sellers, May 31 2000

A055005 Number of nonnegative integer 3 X 3 matrices with no zero rows or columns and with sum of elements equal to n.

Original entry on oeis.org

1, 0, 0, 6, 63, 306, 1038, 2844, 6750, 14437, 28521, 52911, 93258, 157509, 256581, 405171, 622719, 934542, 1373158, 1979820, 2806281, 3916812, 5390496, 7323822, 9833604, 13060251, 17171415, 22366045, 28878876, 36985383, 47007231

Offset: 0

Vladeta Jovovic, May 30 2000

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A054688, A054974, A054975.

a(n)=([0,1,0,0,0,0,0,0,0; 0,0,1,0,0,0,0,0,0; 0,0,0,1,0,0,0,0,0; 0,0,0,0,1,0,0,0,0; 0,0,0,0,0,1,0,0,0; 0,0,0,0,0,0,1,0,0; 0,0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,0,1; 1,-9,36,-84,126,-126,84,-36,9]^n*[1;0;0;6;63;306;1038;2844;6750])[1,1] \\ Charles R Greathouse IV, Aug 14 2023

Number of nonnegative integer p X q matrices with no zero rows or columns and with sum of elements equal to n is Sum_{k=0...q} (-1)^k*C(q, k)*m(p, q-k, n) where m(p, q, n)=Sum_{k=0..p} (-1)^k*C(p, k)*C((p-k)*q+n-1, n).
For p=q=3 we get a(n)=C(n + 8, 8) - 6*C(n + 5, 5) + 9*C(n + 3, 3) + 6*C(n + 2, 2) - 18*C(n + 1, 1) + 9=(1/8!)*(n^8 + 36*n^7 + 546*n^6 + 2520*n^5 - 7791*n^4 - 43596*n^3 + 148364*n^2 - 140400*n + 40320).
G.f.: -(9*x^8-54*x^7+132*x^6-171*x^5+135*x^4-78*x^3+36*x^2-9*x+1) / (x-1)^9. - Colin Barker, Jul 13 2013

cp's OEIS Frontend

A321615 Triangle read by rows: T(n,k) is the number of k X k integer matrices with sum of elements n, with no zero rows or columns, up to row and column permutation.

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A055007 Number of nonnegative integer 4 X 4 matrices with no zero rows or columns and with sum of elements equal to n.

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Extensions

A055005 Number of nonnegative integer 3 X 3 matrices with no zero rows or columns and with sum of elements equal to n.

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Author

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Crossrefs

Programs

PARI

Formula