A067961 -id:A067961 - cp's OEIS frontend

A171187 a(n) = Sum_{k=0..[n/2]} A034807(n,k)^n, where A034807 is a triangle of Lucas polynomials.

1, 1, 5, 28, 273, 6251, 578162, 107060591, 29911744769, 27309372325966, 100510174785157275, 579282314757603925315, 5692451844585536053973346, 272831740026972379247127727751, 36494329378701187545939734030067963

Offset: 0

Paul D. Hanna, Dec 13 2009

nonn

			The n-th term equals the sum of the n-th powers of the n-th row of triangle A034807:
a(0) = 2^0 = 1;
a(1) = 1^1 = 1;
a(2) = 1^2 + 2^2 = 5;
a(3) = 1^3 + 3^3 = 28;
a(4) = 1^4 + 4^4 + 2^4 = 273;
a(5) = 1^5 + 5^5 + 5^5 = 6251;
a(6) = 1^6 + 6^6 + 9^6 + 2^6 = 578162;
a(7) = 1^7 + 7^7 + 14^7 + 7^7 = 107060591; ...

Cf. A171186, A034807, A067961.

{a(n)=sum(k=0, n\2, (binomial(n-k, k)+binomial(n-k-1, k-1))^n)}

a(n) = Sum_{k=0..[n/2]} [C(n-k,k) + C(n-k-1,k-1)]^n.

Ignoring the zeroth term, equals the logarithmic derivative of A171186.

A197990 Number of binary arrangements of total n 1's, without adjacent 1's on n X n torus connected n-s.

Original entry on oeis.org

1, 1, 4, 27, 664, 19375, 712536, 31474709, 1623421808, 95752130751, 6356272757680, 468976366239799, 38071162011854412, 3372179632719015287, 323631920261745650114, 33452466695808298399785, 3705187274710433648959456, 437779689881887196512539391

Offset: 0

Vaclav Kotesovec, Oct 20 2011

Andrew Howroyd, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 408.

Cf. A067961, A197989.

permopak[part_,k_]:=(hist=ConstantArray[0,k];
Do[hist[[part[[t]]]]++,{t,1,Length[part]}];
(Length[part])!/Product[(hist[[t]])!,{t,1,k}]);
waz1t[k_,n_]:=(If[n-k+1Harvey P. Dale, Nov 24 2016 *)

a(n) = if(n<=1, 1, n*binomial(n^2-n-1,n-1) + n*(-1)^n) \\ Andrew Howroyd, Mar 27 2023

a(n) = n*binomial(n^2-n-1,n-1) + n*(-1)^n, n > 1. - Vaclav Kotesovec, Oct 20 2011

a(0)=1 prepended by Andrew Howroyd, Mar 27 2023

cp's OEIS Frontend

A171187 a(n) = Sum_{k=0..[n/2]} A034807(n,k)^n, where A034807 is a triangle of Lucas polynomials.

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