A171186 -id:A171186 - 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.

A171185 G.f.: exp( Sum_{n>=1} (x^n/n)*[Sum_{k=0..[n/2]} A034807(n,k)^3] ), where A034807 is a triangle of Lucas polynomials.

Original entry on oeis.org

1, 1, 5, 14, 40, 126, 408, 1332, 4473, 15377, 53627, 189724, 680475, 2467975, 9038578, 33399571, 124400702, 466619283, 1761467038, 6688059913, 25527326897, 97901917060, 377123873505, 1458573962761, 5662223702216, 22056563938599

Offset: 0

Paul D. Hanna, Dec 14 2009

nonn

			G.f.: A(x) = 1 + x + 5*x^2 + 14*x^3 + 40*x^4 + 126*x^5 + 408*x^6 +...
log(A(x)) = x + 9*x^2/2 + 28*x^3/3 + 73*x^4/4 + 251*x^5/5 + 954*x^6/6 +...+ A171215(n)*x^n/n +...

Cf. A171215, A093128, A171186.

{a(n)=polcoeff(exp(sum(m=1,n,(x^m/m)*sum(k=0, m\2, (binomial(m-k, k)+binomial(m-k-1, k-1))^3))+x*O(x^n)),n)}

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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