A330613 -id:A330613 - cp's OEIS frontend

A379555 Row products of the triangle A330613.

1, 1, 10, 312, 19800, 2112320, 339721200, 76736721600, 23161613673856, 9002333397657600, 4378779743334796800, 2605462349704907264000, 1861650500340096385920000, 1572904258939213777572323328, 1551130967414603865237595648000, 1765646154517023983944233381888000, 2297658475440816396798100972400640000

Offset: 0

Stefano Spezia, Dec 25 2024

nonn

Cf. A227514, A330613.

a[n_]:=(1-2n)^(n-1)(1-2n+2n^2)Pochhammer[-2(n-1)^2/(2n-1),n-1]; Array[a,17,0]

a(n) = (1 - 2*n)^(n-1)*(1 - 2*n + 2*n^2)*Pochhammer(-2*(n - 1)^2/(2*n - 1), n - 1).

a(n) ~ C*2^n*exp(3/(8*n)-n)*n^(2*n), where C = sqrt(2/e) = A227514.

A374708 Triangle T read by rows: T(n,k) = (n - k)n(4n^2 - 4nk + 2k^2 - 1 + (-1)^k)/4, with 0 <= k < n.

Original entry on oeis.org

1, 16, 4, 81, 36, 15, 256, 144, 80, 32, 625, 400, 255, 140, 65, 1296, 900, 624, 396, 240, 108, 2401, 1764, 1295, 896, 609, 364, 175, 4096, 3136, 2400, 1760, 1280, 864, 544, 256, 6561, 5184, 4095, 3132, 2385, 1728, 1215, 756, 369, 10000, 8100, 6560, 5180, 4080, 3100, 2320, 1620, 1040, 500

Offset: 1

Stefano Spezia, Jul 17 2024

T(n, k) is the k-th super- and subdiagonal sum of the Hankel matrix M(n) whose permanent is A374668(n).

			n\k|    0    1    2    3    4    5
---+------------------------------
1  |    1
2  |   16    4
3  |   81   36   15
4  |  256  144   80   32
5  |  625  400  255  140   65
6  | 1296  900  624  396  240  108
      ...
For n = 3 the matrix M is
  [ 1,  4, 15]
  [ 4, 15, 32]
  [15, 32, 65]
and therefore T(3, 0) = 1 + 15 + 65 = 81, T(3, 1) = 4 + 32 = 36, and T(3, 2) = 15.

Cf. A317614 (diagonal), A374668.
Cf. A333119, A330613.
Cf. A000583 (k=0), A035287 (k=1), A123865, A374709 (row sums).

T[n_,k_]:=(n-k)*n*(4*n^2 - 4*n*k+2*k^2-1+(-1)^k)/4; Table[T[n,k],{n,10},{k,0,n-1}]//Flatten

O.g.f.: x*(1 - 4*x^8*y^5 + x*(11 + 2*y) - x^7*y^4*(7 + 16*y) - x^2*(-11 + 6*y - 6*y^2) - x^5*y^2*(2 - 46*y - 3*y^2) - x^6*y^3*(-2 - 27*y + 4*y^2) - x^3*(-1 + 18*y + 38*y^2 - 2*y^3) - x^4*y*(2 + 14*y + 2*y^2 - y^3))/((1 - x)^5*(1 - x*y)^4*(1 + x*y)^2).

T(n,2) = A123865(n-1) for n > 1.

cp's OEIS Frontend

A379555 Row products of the triangle A330613.

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A374708 Triangle T read by rows: T(n,k) = (n - k)n(4n^2 - 4nk + 2k^2 - 1 + (-1)^k)/4, with 0 <= k < n.

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cp's OEIS Frontend

A379555 Row products of the triangle A330613.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A374708 Triangle T read by rows: T(n,k) = (n - k)*n*(4*n^2 - 4*n*k + 2*k^2 - 1 + (-1)^k)/4, with 0 <= k < n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A374708 Triangle T read by rows: T(n,k) = (n - k)n(4n^2 - 4nk + 2k^2 - 1 + (-1)^k)/4, with 0 <= k < n.