cp's OEIS Frontend

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

%I A193449 #29 Apr 22 2025 03:39:49
%S A193449 0,1,6,15,44,105,258,595,1368,3069,6830,15015,32772,70993,152922,
%T A193449 327675,699056,1485477,3145734,6640975,13981020,29360121,61516466,
%U A193449 128625315,268435464,559240525,1163220318,2415919095,5010795188,10379504289,21474836490,44381328715
%N A193449 Products of the Jacobsthal numbers and the integers: a(n) =  n * A001045(n+1).
%C A193449 a(n) = n * A001045(n+1).
%C A193449 This sequence is the sum of several triangles of integers (see formula section)
%H A193449 Vincenzo Librandi, <a href="/A193449/b193449.txt">Table of n, a(n) for n = 0..1000</a>
%H A193449 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,-4,-4).
%F A193449 G.f.:  x*(1 + 4*x)/( 2*x^2+x-1)^2
%F A193449 a(n) = n*(2^(n + 1) + (-1)^n)/3
%F A193449 a(n)= sum( sum( (-1)^(j+k)*(j+k)*C(n-k+j,j), j=0..k), k=0..n)
%F A193449 a(n)= sum( n*C(n, k)*2F1( (1, -k); -n )(-1), k=0..n)
%F A193449 a(n)= sum( sum( (-1)^j*n*C(n-j,k-j), j=0..k), k=0..n)
%F A193449 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.
%t A193449 Table[Sum[n Binomial[n, k] HypergeometricPFQ[{1, -k}, {-n}, -1], {k, 0, n}], {n, 0, 35}]
%t A193449 CoefficientList[Series[(x*(1 + 4*x))/(2*x^2 + x - 1)^2, {x, 0, 100}], x] (* _Vincenzo Librandi_, Oct 21 2012 *)
%o A193449 (Magma) [n*(2^(n + 1) + (-1)^n)/3: n in [0..35]]; // _Vincenzo Librandi_, Oct 21 2012
%o A193449 (Python)
%o A193449 def A193449(n): return (((1<<n+1)|1)//3)*n # _Chai Wah Wu_, Apr 18 2025
%Y A193449 Cf. A001045, Equals second column of A124860, equals sum of A193450 or A193451.
%K A193449 nonn,easy
%O A193449 0,3
%A A193449 _Olivier Gérard_, Jul 26 2011