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