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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.

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A374668 a(n) is the permanent of the n-th order Hankel matrix M whose generic element is given by M(i,j) = A317614(i+j-1) with i,j = 1, ..., n.

Original entry on oeis.org

1, 1, 31, 10254, 12238276, 41596930860, 309346186680924, 4522151204857137264, 116038936382978521700928, 4918677318766771695942334272, 323424014903141386787887115413440, 31725978444319999354629697685162941056, 4460612938377274751881312432310360618154240
Offset: 0

Views

Author

Stefano Spezia, Jul 15 2024

Keywords

Comments

The Hankel transform of A317614 has the following polynomial as g.f. 1 + x - x^2 - 624*x^3 - 9216*x^4 + 138240*x^5 - 331776*x^6: the matrices are singular for n > 6.

Examples

			a(7) = 4522151204857137264:
  [  1,   4,  15,  32,  65, 108,  175]
  [  4,  15,  32,  65, 108, 175,  256]
  [ 15,  32,  65, 108, 175, 256,  369]
  [ 32,  65, 108, 175, 256, 369,  500]
  [ 65, 108, 175, 256, 369, 500,  671]
  [108, 175, 256, 369, 500, 671,  864]
  [175, 256, 369, 500, 671, 864, 1105]
which is the singular matrix M of minimal order.

Links

Crossrefs

Cf. A317614.
Cf. A000583 (trace of M), A006010 (sum of the 1st row or column of M), A035287 (super- or subdiagonal sum of M), A346174, A374708 (k-th super- or subdiagonal sum of M).

Programs

  • Mathematica
    A317614[n_]:=(1/2)*(n^3 + n*Mod[n,2]); a[n_]:=Permanent[Table[A317614[i+j-1], {i, n}, {j, n}]]; Join[{1}, Array[a, 12]]

A374709 a(n) = n*(6*n^4 + 8*n^3 + 1 - (-1)^n)/16.

Original entry on oeis.org

0, 1, 20, 132, 512, 1485, 3564, 7504, 14336, 25425, 42500, 67716, 103680, 153517, 220892, 310080, 425984, 574209, 761076, 993700, 1280000, 1628781, 2049740, 2553552, 3151872, 3857425, 4684004, 5646564, 6761216, 8045325, 9517500, 11197696, 13107200, 15268737, 17706452
Offset: 0

Views

Author

Stefano Spezia, Jul 17 2024

Keywords

Links

Crossrefs

Row sums of A374708.
Cf. A000583, A016789, A229147.

Programs

  • Mathematica
    LinearRecurrence[{4,-4,-4,10,-4,-4,4,-1},{0,1,20,132,512,1485,3564,7504},35]

Formula

O.g.f.: x*(1 + 16*x + 56*x^2 + 68*x^3 + 35*x^4 + 4*x^5)/((1 - x)^6*(1 + x)^2).
a(n) = 4*a(n-1) - 4*a(n-2) - 4*a(n-3) + 10*a(n-4) - 4*a(n-5) - 4*a(n-6) + 4*a(n-7) - a(n-8) for n > 7.
E.g.f.: x*((8 + 73*x + 99*x^2 + 34*x^3 + 3*x^4)*cosh(x) + (7 + 73*x + 99*x^2 + 34*x^3 + 3*x^4)*sinh(x))/8.
a(2*n) = 4*A229147(n) = 4*A000583(n)*A016789(n).
Showing 1-2 of 2 results.

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

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