A193449 Products of the Jacobsthal numbers and the integers: a(n) = n * A001045(n+1).

0, 1, 6, 15, 44, 105, 258, 595, 1368, 3069, 6830, 15015, 32772, 70993, 152922, 327675, 699056, 1485477, 3145734, 6640975, 13981020, 29360121, 61516466, 128625315, 268435464, 559240525, 1163220318, 2415919095, 5010795188, 10379504289, 21474836490, 44381328715

Offset: 0

Author

Olivier Gérard, Jul 26 2011

Keywords

• nonn
• easy

Comments

a(n) = n * A001045(n+1).

This sequence is the sum of several triangles of integers (see formula section)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,3,-4,-4).

Crossrefs

Cf. A001045, Equals second column of A124860, equals sum of A193450 or A193451.

Programs

Magma

[n*(2^(n + 1) + (-1)^n)/3: n in [0..35]]; // Vincenzo Librandi, Oct 21 2012

Mathematica

Table[Sum[n Binomial[n, k] HypergeometricPFQ[{1, -k}, {-n}, -1], {k, 0, n}], {n, 0, 35}]
CoefficientList[Series[(x*(1 + 4*x))/(2*x^2 + x - 1)^2, {x, 0, 100}], x] (* Vincenzo Librandi, Oct 21 2012 *)

Python

def A193449(n): return (((1<Chai Wah Wu, Apr 18 2025

Formula

G.f.: x*(1 + 4*x)/( 2*x^2+x-1)^2

a(n) = n*(2^(n + 1) + (-1)^n)/3

a(n)= sum( sum( (-1)^(j+k)*(j+k)*C(n-k+j,j), j=0..k), k=0..n)

a(n)= sum( n*C(n, k)*2F1( (1, -k); -n )(-1), k=0..n)

a(n)= sum( sum( (-1)^j*n*C(n-j,k-j), j=0..k), k=0..n)

a(n)= sum( (1+2*k)*C(n+1, k+1)*2F1( (1, n+2); k+2 )(-1) - C(n+2, k+2) 2F1( (2, n+3); k+3 )(-1) - (-1)^(k) * 2^(k-n-2) * (n-3*k+1) , k=0..n) with C(n,k) the binomial coefficient and 2F1( ) the hypergeometric function.