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A073372
Second convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.
7
1, 3, 12, 34, 99, 261, 678, 1692, 4149, 9959, 23568, 55014, 127031, 290457, 658602, 1482240, 3314025, 7365915, 16285300, 35832810, 78500811, 171293293, 372412782, 806963364, 1743173469, 3754782351, 8066319768, 17285917742, 36957928479, 78847115649
OFFSET
0,2
FORMULA
a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A001045(k+1) and c(k) = A073371(k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+2, 2) * binomial(n-k, k) * 2^k.
a(n) = ((30+9*n)*(n+1)*U(n+1) + 2*(33+9*n)*(n+2)*U(n))/162 with U(n) = A001045(n+1), n>=0.
G.f.: 1/(1 - (1+2*x)*x)^3.
E.g.f.: (1/162)*(32*(4 + 9*x + 3*x^2)*exp(2*x) + (34 - 24*x + 3*x^2)*exp(-x)). - G. C. Greubel, Sep 28 2022
MATHEMATICA
CoefficientList[Series[-(-1+x+2x^2)^(-3), {x, 0, 78}], x] (* or *) Table[(-3*(-1)^n*n^2+3*2^(n+2)*n^2-15*(-1)^n*n+9*2^(n+2)*n-16*(-1)^n+2^(n+4))/162, {n, 42}] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2012 *)
PROG
(Magma) [(2^(n+3)*(16+15*n+3*n^2) +(-1)^n*(34+21*n+3*n^2))/162: n in [0..40]]; // G. C. Greubel, Sep 28 2022
(SageMath)
def A073372(n): return (2^(n+3)*(16+15*n+3*n^2) +(-1)^n*(34+21*n+3*n^2))/162
[A073372(n) for n in range(40)] # G. C. Greubel, Sep 28 2022
CROSSREFS
Third (m=2) column of triangle A073370.
Sequence in context: A060298 A304975 A226546 * A305023 A026573 A326660
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Aug 02 2002
STATUS
approved