cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A053725 Number of n X n binary matrices of order dividing 3 (also number of solutions to X^3=I in GL(n,2)).

Original entry on oeis.org

1, 3, 57, 1233, 75393, 19109889, 6326835201, 6388287561729, 23576681450405889, 120906321631678693377, 1968421511613895105052673, 111055505036706392268074909697, 8965464105556083354144035638870017
Offset: 1

Views

Author

Vladeta Jovovic, Mar 23 2000

Keywords

References

  • V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Links

Crossrefs

Cf. A053722, A053846, A053856.
Cf. A053718, A053770, A053771, A053772, A053773, A053774, A053775, A053776, A053777.

Programs

  • PARI
    \\ See Morison theorem 2.6
\\ F(n,q,k) is number of solutions to X^k=I in GL(i, GF(q)) for i=1..n.
\\ q is power of prime and gcd(q, k) = 1.
B(n,q,e)={sum(m=0, n\e, x^(m*e)/prod(k=0, m-1, q^(m*e)-q^(k*e)))}
F(n,q,k)={if(gcd(q,k)<>1, error("no can do")); my(D=ffgen(q)^0); my(f=factor(D*(x^k-1))); my(p=prod(i=1, #f~, (B(n, q, poldegree(f[i,1])) + O(x*x^n))^f[i,2])); my(r=B(n,q,1)); vector(n, i, polcoeff(p, i)/polcoeff(r, i))}
F(10, 2, 3) \\ Andrew Howroyd, Jul 09 2018

A378666 Triangular array read by rows: T(n,k) is the number of n X n idempotent matrices over GF(3) having rank k, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 12, 1, 1, 117, 117, 1, 1, 1080, 10530, 1080, 1, 1, 9801, 882090, 882090, 9801, 1, 1, 88452, 72243171, 666860040, 72243171, 88452, 1, 1, 796797, 5873190687, 491992666011, 491992666011, 5873190687, 796797, 1, 1, 7173360, 476309310660, 360089838858960, 3267815287645062, 360089838858960, 476309310660, 7173360, 1
Offset: 0

Views

Author

Geoffrey Critzer, Dec 02 2024

Keywords

Comments

A matrix M is idempotent if M^2 = M.

Examples

			Triangle T(n,k) begins:
  1;
  1,    1;
  1,   12,      1;
  1,  117,    117,      1;
  1, 1080,  10530,   1080,    1;
  1, 9801, 882090, 882090, 9801, 1;
  ...

Crossrefs

Cf. A296548, A053846 (row sums).
Cf. A022167, A118180.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n=0, 1, b(n-1, k-1)+3^k*b(n-1, k)))
    end:
T:= (n,k)-> 3^(k*(n-k))*b(n, k):
seq(seq(T(n,k), k=0..n), n=0..8);  # Alois P. Heinz, Dec 02 2024
  • Mathematica
    nn = 8; \[Gamma][n_, q_] := Product[q^n - q^i, {i, 0, n - 1}]; B[n_, q_] := \[Gamma][n, q]/(q - 1)^n; \[Zeta][x_] := Sum[x^n/B[n, 3], {n, 0, nn}];Map[Select[#, # > 0 &] &, Table[B[n, 3], {n,0,nn}]*CoefficientList[Series[\[Zeta][x] \[Zeta][y x], {x, 0, nn}], {x, y}]] // Flatten

Formula

Sum_{n>=0} Sum_{k=0..n} T(n,k)*y^k*x^n/B(n) = e(x)*e(y*x) where e(x) = Sum_{n>=0} x^n/B(n) and B(n) = A053290(n)/2^n.
T(n,k) = A022167(n,k) * A118180(n,k). - Alois P. Heinz, Dec 02 2024

A381899 Irregular triangular array read by rows. T(n,k) is the number of length n words x on {0,1} such that I(x) + W(x)*(n-W(x)) = k, where I(x) is the number of inversions in x and W(x) is the number of 1's in x, n >= 0, 0 <= k <= floor(n^2/2).

Original entry on oeis.org

1, 2, 2, 1, 1, 2, 0, 2, 2, 2, 2, 0, 0, 2, 3, 3, 4, 1, 1, 2, 0, 0, 0, 2, 2, 4, 4, 6, 4, 4, 2, 2, 2, 0, 0, 0, 0, 2, 2, 2, 4, 5, 7, 6, 9, 7, 7, 5, 4, 1, 1, 2, 0, 0, 0, 0, 0, 2, 2, 2, 2, 4, 4, 8, 6, 10, 12, 14, 12, 14, 10, 10, 6, 4, 2, 2
Offset: 0

Views

Author

Geoffrey Critzer, Mar 09 2025

Keywords

Comments

Sum_{k>=0} T(n,k)*2^k = A132186(n).
Sum_{k>=0} T(n,k)*3^k = A053846(n).
Sum_{k>=0} T(n,k)*q^k = the number of idempotent n X n matrices over GF(q).
It appears that if n is even the n-th row converges to 2,0,0,...,21,13,9,5,4,1,1 which is A226622 reversed, and if n is odd the sequence is twice A226635.
From Alois P. Heinz, Mar 09 2025: (Start)
Sum_{k>=0} k * T(n,k) = 3*A001788(n-1) for n>=1.
Sum_{k>=0} (-1)^k * T(n,k) = A060546(n). (End)

Examples

			Triangle T(n,k) begins:
  1;
  2;
  2, 1, 1;
  2, 0, 2, 2, 2;
  2, 0, 0, 2, 3, 3, 4, 1, 1;
  2, 0, 0, 0, 2, 2, 4, 4, 6, 4, 4, 2, 2;
  ...
T(4,5) = 3 because we have: {0, 1, 0, 0}, {0, 1, 0, 1}, {1, 1, 0, 1}.

Links

Crossrefs

Row sums give A000079.
Cf. A001788, A060546, A226635, A226622, A132186, A053846, A083906.

Programs

  • Maple
    b:= proc(i, j) option remember; expand(`if`(i+j=0, 1,
     `if`(i=0, 0, b(i-1, j))+`if`(j=0, 0, b(i, j-1)*z^i)))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(
         expand(add(b(n-j, j)*z^(j*(n-j)), j=0..n))):
seq(T(n), n=0..10);  # Alois P. Heinz, Mar 09 2025
  • Mathematica
    nn = 7; B[n_] := FunctionExpand[QFactorial[n, q]]*q^Binomial[n, 2];e[z_] := Sum[z^n/B[n], {n, 0, nn}]; Map[CoefficientList[#, q] &, Table[B[n], {n, 0, nn}] CoefficientList[Series[e[z]^2, {z, 0, nn}],z]]

Formula

Sum_{n>=0} Sum_{k>=0} T(n,k)*q^k*x^n/(n_q!*q^binomial(n,2)) = e(x)^2 where e(x) = Sum_{n>=0} x^n/(n_q!*q^binomial(n,2)) where n_q! = Product{i=1..n} (q^n-1)/(q-1).

A053848 Number of n X n matrices over GF(3) of order dividing 4 (i.e., number of solutions of X^4=I in GL(n,3)).

Original entry on oeis.org

2, 20, 1640, 901424, 1333386848, 9762556479680, 552311189659544960, 178664076508334107310336, 177339376736100844376103875072, 928195308949863753322027981355832320, 37279296908672313898400994512625009709475840
Offset: 1

Views

Author

Vladeta Jovovic, Mar 28 2000

Keywords

References

  • V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Links

Crossrefs

Cf. A053718, A053725, A053846.

Programs

Extensions

a(10)-a(11) from Andrew Howroyd, Jul 09 2018

A053849 Number of n X n matrices over GF(3) of order dividing 5 (i.e., number of solutions of X^5=I in GL(n,3)).

Original entry on oeis.org

1, 1, 1, 303265, 2972290465, 21908753010145, 149203663865159905, 46239110653796245209025, 2921531591617287960635324005825, 141479937596692631672320398001559452225, 6324451981634068043025384996830468552444687425
Offset: 1

Views

Author

Vladeta Jovovic, Mar 28 2000

Keywords

Links

Crossrefs

Cf. A053725, A053770, A053846.

Programs

Extensions

a(10)-a(11) from Andrew Howroyd, Jul 09 2018

A053852 Number of n X n matrices over GF(3) of order dividing 7 (i.e., number of solutions of X^7=I in GL(n,3)).

Original entry on oeis.org

1, 1, 1, 1, 1, 115562653240321, 92079975413927255041, 55043567702937434517811201, 30375957967569132050957664153601, 16344540545803963971405840043681904641, 8722002954856094967866703998081059674593281, 1007937807674669630303410866111304336524953920798402867201
Offset: 1

Views

Author

Vladeta Jovovic, Mar 28 2000

Keywords

References

  • V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Links

Crossrefs

Cf. A053725, A053772, A053846.

Programs

Extensions

a(10)-a(12) from Andrew Howroyd, Jul 09 2018

A053853 Number of n X n matrices over GF(3) of order dividing 8 (i.e., number of solutions of X^8=I in GL(n,3)).

Original entry on oeis.org

2, 32, 4448, 3816128, 26288771456, 1354765603506176, 413011432853757876224, 1232292753203369699693195264, 12961108500525078696110888464867328, 1011066029229309888379062265909092037296128, 580367747201355872811056515502067372016693810429952
Offset: 1

Views

Author

Vladeta Jovovic, Mar 28 2000

Keywords

References

  • V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Links

Crossrefs

Cf. A053725, A053773, A053846.

Programs

Extensions

a(10)-a(11) from Andrew Howroyd, Jul 09 2018

A053855 Number of n X n matrices over GF(3) of order dividing 10 (i.e., number of solutions of X^10=I in GL(n,3)).

Original entry on oeis.org

2, 14, 236, 619220, 11890945640, 613445895807320, 70424130340088781680, 325784192369064628390662800, 38844516111082042308571222950605600, 13159967487181842256922128281356518089759200, 9916640076650391701813048608335186503022719034948800
Offset: 1

Views

Author

Vladeta Jovovic, Mar 28 2000

Keywords

References

  • V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Links

Crossrefs

Cf. A053725, A053775, A053846.

Programs

Extensions

a(10)-a(11) from Andrew Howroyd, Jul 09 2018

A297892 Triangle read by rows. T(n,k) is the number of n X n diagonalizable matrices over GF(3) that have rank k, 0 <= k <= n, n >= 0.

Original entry on oeis.org

1, 1, 2, 1, 24, 14, 1, 234, 1638, 236, 1, 2160, 147420, 254880, 12692, 1, 19602, 12349260, 208173240, 124394292, 1783784, 1, 176904, 1011404394, 157378969440, 916910326332, 157779262368, 811523288
Offset: 0

Views

Author

Geoffrey Critzer, Jan 07 2018

Keywords

Examples

			Triangle begins
  1;
  1,     2;
  1,    24,       14;
  1,   234,     1638,       236;
  1,  2160,   147420,    254880,     12692;
  1, 19602, 12349260, 208173240, 124394292, 1783784;

Links

Crossrefs

Cf. A296548, A053846 (main diagonal), A290516 (row sums).

Programs

  • Mathematica
    nn = 5; g[n_] := (q - 1)^n  q^Binomial[n, 2] FunctionExpand[QFactorial[n, q]] /. q -> 3;G[n] := Sum[u z^r/g[r], {r, 0, nn}]; Grid[Map[Select[#, # > 0 &] &,Table[g[n], {n, 0, nn}] CoefficientList[Series[Sum[(u z)^r/g[r] , {r, 0, nn}]^2 Sum[
       z^r/g[r], {r, 0, nn}], {z, 0, nn}], {z, u}]]]

Formula

T(n,k)/A053290(n) is the coefficient of y^k*x^n in the expansion of Sum_{n>=0} x^n\A053290(n) * (Sum_{n>=0} y*x^n\A053290(n))^2.
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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