A330287 -id:A330287 - cp's OEIS frontend

A238261 Decimal expansion of a constant related to A187235.

4, 9, 1, 0, 8, 1, 4, 9, 6, 4, 5, 6, 8, 2, 5, 5, 8, 9, 8, 7, 5, 1, 5, 3, 4, 8, 0, 5, 2, 4, 0, 3, 5, 2, 1, 9, 7, 8, 9, 8, 7, 0, 5, 2, 8, 1, 7, 6, 7, 8, 4, 7, 1, 7, 6, 1, 3, 9, 4, 1, 1, 2, 0, 2, 2, 5, 6, 4, 1, 7, 8, 7, 7, 8, 7, 9, 9, 4, 7, 9, 7, 2, 9, 5, 1, 8, 1, 9, 7, 4, 1, 5, 3, 5, 5, 4, 4, 6, 1, 4, 2, 5, 0, 5, 3

Offset: 1

Vaclav Kotesovec, Feb 21 2014

			4.9108149645682558987515348...

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p.261.

Cf. A187235, A238262, A002465, A129256, A342111, A330287.

RealDigits[N[-(2*LambertW[-1,-1/2/Sqrt[E]])^2/(1+2*LambertW[-1,-1/2/Sqrt[E]]), 105]][[1]]

Equals lim n->infinity (A187235(n)/(n-1)!)^(1/n).

Equals -(2*LambertW(-1,-exp(-1/2)/2))^2 / (1 + 2*LambertW(-1,-exp(-1/2)/2)).

A330613 Triangle read by rows: T(n, k) = 1 + k - 2n - 2kn + 2n^2, with 0 <= k < n.

Original entry on oeis.org

1, 5, 2, 13, 8, 3, 25, 18, 11, 4, 41, 32, 23, 14, 5, 61, 50, 39, 28, 17, 6, 85, 72, 59, 46, 33, 20, 7, 113, 98, 83, 68, 53, 38, 23, 8, 145, 128, 111, 94, 77, 60, 43, 26, 9, 181, 162, 143, 124, 105, 86, 67, 48, 29, 10, 221, 200, 179, 158, 137, 116, 95, 74, 53, 32, 11

Offset: 1

Stefano Spezia, Dec 20 2019

T(n, k) is the k-th super- and subdiagonal sum of the matrix M(n) whose permanent is A330287(n).

			n\k|   0   1   2   3   4   5
---+------------------------
1  |   1
2  |   5   2
3  |  13   8   3
4  |  25  18  11   4
5  |  41  32  23  14   5
6  |  61  50  39  28  17   6
...
For n = 3 the matrix M is
      1, 2, 3
      2, 4, 6
      3, 6, 8
and therefore T(3, 0) = 1 + 4 + 8 = 13, T(3, 1) = 2 + 6 = 8 and T(3, 2) = 3.

Stefano Spezia, First 150 rows of the triangle, flattened

Cf. A005408, A330287.

Cf. A000027: diagonal; A001105: 2nd column; A001844: 1st column; A016789: 1st subdiagonal; A016885: 2nd subdiagonal; A017029: 3rd subdiagonal; A017221: 4th subdiagonal; A017461: 5th subdiagonal; A081436: row sums; A132209: 3rd column; A164284: 7th subdiagonal; A269044: 6th subdiagonal.

Flatten[Table[1+k-2n-2k*n+2n^2,{n,1,11},{k,0,n-1}]] (* or *)
r[n_] := Table[SeriesCoefficient[(1-x*(2-5x+2(1+x)y))/((1-x)^3*(1-y)^2), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 0, n-1}]; Flatten[Array[r, 11]] (* or *)
r[n_] := Table[SeriesCoefficient[Exp[x+y]*(1+2x(x-y)+y), {x, 0, i}, {y, 0, j}]*i!*j!, {i, n, n}, {j, 0, n-1}]; Flatten[Array[r, 11]]

O.g.f.: (1 - x*(2 - 5*x + 2*(1 + x)*y))/((1 - x)^3*(1 - y)^2).
E.g.f.: exp(x+y)*(1 + 2*x*(x - y) + y).
T(n, k) = A001844(n-1) - k*A005408(n-1), with 0 <= k < n. [Typo corrected by Stefano Spezia, Feb 14 2020]

A330700 a(n) = (n - 1)n(2n^2 + 4n - 1)/6.

Original entry on oeis.org

0, 0, 5, 29, 94, 230, 475, 875, 1484, 2364, 3585, 5225, 7370, 10114, 13559, 17815, 23000, 29240, 36669, 45429, 55670, 67550, 81235, 96899, 114724, 134900, 157625, 183105, 211554, 243194, 278255, 316975, 359600, 406384, 457589, 513485, 574350, 640470, 712139, 789659

Offset: 0

Stefano Spezia, Dec 26 2019

Conjectures: (Start)
For n > 1, a(n) is the absolute value of the trace of the 2nd exterior power of an n X n square matrix M(n) defined as M[i,j,n] = i*j if i < 3 or j < 3 and M[i,j,n] = 2*(i + j) - 4 otherwise (see A330287). Equivalently, a(n) is the absolute value of the coefficient of the term [x^(n-2)] in the characteristic polynomial of M(n), or the absolute value of the sum of all principal minors of M(n) of size 2.
For k > 2, the trace of the k-th exterior power of M(n) is equal to zero.
(End)

Wikipedia, Characteristic polynomial
Wikipedia, Exterior algebra
Wikipedia, Harmonic number
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A001105 (super- and subdiagonal sum of M(n)), A001844 (trace of M(n)), A005843 (antitrace of M(n)), A268581, A319840, A322844, A330287 (permanent of M(n)).

I:=[0, 0, 5, 29, 94]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]];

Mathematica
```
Table[(n-1)n(2n^2+4n-1)/6,{n,0,39}]
```

my(x='x + O('x^39)); concat([0, 0], Vec(serlaplace((1/6)*exp(x)*x^2*(15+14*x+2*x^2))))

(x^2*(5+4*x-x^2)/(1-x)^5).series(x, 40).coefficients(x, sparse=False)

O.g.f.: x^2*(5 + 4*x - x^2)/(1 - x)^5.
E.g.f.: exp(x)*x^2*(15 + 14*x + 2*x^2)/6.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4.
a(n) = A000217(n-1)*A268581(n-1)/3 for n > 0.
Sum_{k>=2} 1/a(k) = (1/5)*((18 + 7*sqrt(6))*H(2-sqrt(3/2)) + (18 - 7*sqrt(6))*H(2+sqrt(3/2)) - 30) = 0.254905801002729039998040617... where H(x) = Integral_{t=0..1} (1 - t^x)/(1 - t) dt is the function that interpolates the harmonic numbers.

cp's OEIS Frontend

A238261 Decimal expansion of a constant related to A187235.

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A330613 Triangle read by rows: T(n, k) = 1 + k - 2n - 2kn + 2n^2, with 0 <= k < n.

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A330700 a(n) = (n - 1)n(2n^2 + 4n - 1)/6.

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cp's OEIS Frontend

A238261 Decimal expansion of a constant related to A187235.

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A330613 Triangle read by rows: T(n, k) = 1 + k - 2*n - 2*k*n + 2*n^2, with 0 <= k < n.

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A330700 a(n) = (n - 1)*n*(2*n^2 + 4*n - 1)/6.

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Magma

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PARI

Sage

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A330613 Triangle read by rows: T(n, k) = 1 + k - 2n - 2kn + 2n^2, with 0 <= k < n.

A330700 a(n) = (n - 1)n(2n^2 + 4n - 1)/6.