A007107 -id:A007107 - cp's OEIS frontend

A321711 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

1, 1, 0, 3, 0, 0, 11, 9, 0, 1, 53, 120, 60, 40, 9, 309, 1410, 1800, 1590, 885, 216, 2119, 16560, 39960, 55120, 52065, 29016, 7570, 16687, 202755, 801780, 1696555, 2433165, 2300403, 1326850, 357435, 148329, 2624496, 15606360, 48387024, 99650670, 141429456, 135382464, 79738800, 22040361, 1468457, 36080100, 304274880, 1323453180, 3760709526, 7493549868, 10570597800, 10199809980, 6103007505, 1721632024

Offset: 0

Gheorghe Coserea, Nov 27 2018

			For n=3 we have s1 = z1 + z2 + z3, s2 = z1^2 + z2^2 + z3^2, s12 = z1*z2 + z1*z3 + z2*z3, f1 = z1^2 + z2^2 + z3^2 + t*z2*z3 + z1*(z2 + z3), f2 = z1^2 + z2^2 + z3^2 + t*z1*z3 + z2*(z1 + z3), f3 = z1^2 + z2^2 + z3^2 + t*z1*z2 + z3*(z1 + z2), [(z1*z2*z3)^2] f1*f2*f3 = 11 + 9*t + t^3, therefore P_3(t) = 11 + 9*t + t^3.
A(x;t) = 1 + x + 3*x^2 + (11 + 9*t + t^3)*x^3 + (53 + 120*t + 60*t^2 + 40*t^3 + 9*t^4)*x^4 + ...
Triangle starts:
n\k [0]    [1]     [2]     [3]      [4]      [5]      [6]      [7]
[0] 1;
[1] 1;     0;
[2] 3;     0;      0;
[3] 11,    9,      0,      1;
[4] 53,    120,    60,     40,      9;
[5] 309,   1410,   1800,   1590,    885,     216;
[6] 2119,  16560,  39960,  55120,   52065,   29016,   7570;
[7] 16687, 202755, 801780, 1696555, 2433165, 2300403, 1326850, 357435;
[8] ...

Gheorghe Coserea, Rows n = 0..13, flattened
Shmuel Friedland, Giorgio Ottaviani, The number of singular vector tuples and uniqueness of best rank one approximation of tensors, arXiv:1210.8316 [math.AG], 2013.

Cf. A000255, A000681, A007107, A274308, A284989.

P(n, t='t) = {
  my(z=vector(n, k, eval(Str("z", k))),
     s1=sum(k=1, #z, z[k]), s2=sum(k=1, #z, z[k]^2), s12=(s1^2 - s2)/2,
     f=vector(n, k, s2 + t*(s12 - z[k]*(s1 - z[k])) + z[k]*(s1 - z[k])), g=1);
  for (i=1, n, g *= f[i]; for(j=1, n, g=substpol(g, z[j]^3, 0)));
  for (k=1, n, g=polcoef(g, 2, z[k]));
  g;
};
seq(N) = concat([[1], [1, 0], [3, 0, 0]], apply(n->Vecrev(P(n,'t)), [3..N]));
concat(seq(9))

Let z1..zn be n variables and s1 = Sum_{k=1..n} zk, s2 = Sum_{k=1..n} zk^2, s12 = (s1^2 - s2)/2, fk = s2 + t*(s12 - zk*(s1 - zk)) + zk*(s1 - zk) for k=1..n; we define P_n(t) = [(z1..zn)^2] Product_{k=1..n} fk.
A000255(n) = T(n,0).
A007107(n) = T(n,n).
A000681(n) = Sum_{k=0..n} T(n,k).
A274308(n) = Sum_{k=0..n} T(n,k)*2^k.

A364068 Triangle T(n,k) read by rows: Number of traceless binary n X n matrices with all row and column sums equal to k, 1<=k<=n.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 9, 9, 1, 0, 44, 216, 44, 1, 0, 265, 7570, 7570, 265, 1, 0, 1854, 357435, 1975560, 357435, 1854, 1, 0, 14833, 22040361, 749649145, 749649145, 22040361, 14833, 1, 0, 133496, 1721632024

Offset: 1

R. J. Mathar, Jul 04 2023

			    0
    1        0
    2        1         0
    9        9         1      0
   44      216        44      1    0
  265     7570      7570    265    1 0
 1854   357435   1975560 357435 1854 1 0
14833 22040361 749649145

Cf. A000166 (k=1), A007107 (k=2), A284989 (see 1st col), A284990 (see 1st col, k=3), A007105 (k=3?), A284991 (see 1st col, k=4), A008300 (any trace)

T(n,n)=0. (k=n would require a 1 on the diagonal)
T(n,n-1)=1. (1 at all entries but the diagonal)
T(n,n-k) = T(n,k-1). (Flip entries 0<->1 and erase diagonal) - R. J. Mathar, Jul 26 2023

A174584 Let J_n be n X n matrix which contains 1's only, I=I_n be the n X n identity matrix and P=P_n be the incidence matrix of the cycle (1,2,3,...,n). Then a(n) is the number of (0,1) n X n matrices A<=J_n-I-P-P^2-P^3 with exactly two 1's in every row and column.

Original entry on oeis.org

0, 1, 31, 3114, 381022

Offset: 5

Vladimir Shevelev, Mar 23 2010

V. S. Shevelev, Development of the rook technique for calculating the cyclic indicators of (0,1)-matrices, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 21-28 (in Russian).
S. E. Grigorchuk, V. S. Shevelev, An algorithm of computing the cyclic indicator of couples discordant permutations with restricted position, Izvestia Vuzov of the North-Caucasus region, Nature sciences 3 (1997), 5-13 (in Russian).

A001499 A007107 A082491 A000186 A174564 A174580 A174581 A174582

A174585 Let J_n be n X n matrix which contains 1's only, I=I_n be the n X n identity matrix and P=P_n be the incidence matrix of the cycle (1,2,3,...,n). Then a(n) is the number of (0,1,2) n X n matrices A<=2(J_n-I-P-P^2-P^3) with exactly one 1 and one 2 in every row and column.

Original entry on oeis.org

0, 2, 132, 9800, 1309928

Offset: 5

Vladimir Shevelev, Mar 23 2010

V. S. Shevelev, Development of the rook technique for calculating the cyclic indicators of (0,1)-matrices, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 21-28 (in Russian).
S. E. Grigorchuk, V. S. Shevelev, An algorithm of computing the cyclic indicator of couples discordant permutations with restricted position, Izvestia Vuzov of the North-Caucasus region, Nature sciences 3 (1997), 5-13 (in Russian).

A001499 A007107 A082491 A000186 A174564 A174580 A174581 A174582 A174584

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A321711 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

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A364068 Triangle T(n,k) read by rows: Number of traceless binary n X n matrices with all row and column sums equal to k, 1<=k<=n.

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Formula

A174584 Let J_n be n X n matrix which contains 1's only, I=I_n be the n X n identity matrix and P=P_n be the incidence matrix of the cycle (1,2,3,...,n). Then a(n) is the number of (0,1) n X n matrices A<=J_n-I-P-P^2-P^3 with exactly two 1's in every row and column.

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Author

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Crossrefs

A174585 Let J_n be n X n matrix which contains 1's only, I=I_n be the n X n identity matrix and P=P_n be the incidence matrix of the cycle (1,2,3,...,n). Then a(n) is the number of (0,1,2) n X n matrices A<=2(J_n-I-P-P^2-P^3) with exactly one 1 and one 2 in every row and column.

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Author

Keywords

References

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