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

0,3

Comments

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).

Formula

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

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

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

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

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

a(n)= Sum_{k=0..n} Sum_{j=0..k} (-1)^j*n*C(n-j,k-j).

a(n)= Sum_{k=0..n} ((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)) with C(n,k) the binomial coefficient and 2F1( ) the hypergeometric function.

E.g.f.: x * (4*exp(2*x) - exp(-x)) / 3. - Amiram Eldar, Dec 30 2025

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 *)

Prog

(Magma) [n*(2^(n + 1) + (-1)^n)/3: n in [0..35]]; // Vincenzo Librandi, Oct 21 2012
(Python)
def A193449(n): return (((1<<n+1)|1)//3)*n # Chai Wah Wu, Apr 18 2025

Crossrefs

Second column of A124860.

Row sums of A193450 or A193451.

Cf. A001045.

Sequence in context: A272289 A272320 A137806 * A378796 A197160 A182420

Adjacent sequences: A193446 A193447 A193448 * A193450 A193451 A193452

Keyword

nonn,easy

Author

Olivier Gérard, Jul 26 2011

Status

approved