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A295288
Binomial transform of the centered triangular numbers A005448.
2
1, 5, 19, 62, 184, 512, 1360, 3488, 8704, 21248, 50944, 120320, 280576, 647168, 1478656, 3350528, 7536640, 16842752, 37421056, 82706432, 181927936, 398458880, 869269504, 1889533952, 4093640704, 8841592832, 19042140160, 40902852608
OFFSET
0,2
COMMENTS
The sequence is column 3 of triangle in A207630.
First difference is given by A055818(n+3,3) for n > 0.
REFERENCES
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.
LINKS
C. Corsani, D. Merlini, and R. Sprugnoli, Left-inversion of combinatorial sums, Discrete Mathematics, 180 (1998) 107-122.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv:1406.3081 [math.CO], 2014.
FORMULA
G.f.: (1 - x + x^2)/(1 - 2*x)^3.
a(n+3) = 8*a(n) - 12*a(n+1) + 6*a(n+2).
a(n+1) = 2*a(n) + 3*(n + 2)*2^(n-1).
a(n+1) = 2*a(n) + 3*A001792(n) = 2*a(n) + A001787(n+2) - A001792(n).
a(n) = (3*n^2 + 9*n + 8)*2^(n - 3).
a(n) = (1/8)*A077588(n+2)*A000079(n).
EXAMPLE
a(0) = (3*0^2 + 9*0 + 8)*2^(-3) = 8/8 = 1.
MAPLE
A:=n->(3*n^2+9*n+8)*2^(n-3); seq(A(n), n=0..70);
MATHEMATICA
Table[(3 n^2 + 9 n + 8) 2^(n-3), {n, 0, 70}]
LinearRecurrence[{6, -12, 8}, {1, 5, 19}, 50] (* G. C. Greubel, Oct 17 2018 *)
PROG
(Maxima) makelist((3*n^2 + 9*n + 8)*2^(n - 3), n, 0, 70);
(PARI) a(n) = (3*n^2 + 9*n + 8)*2^(n - 3) \\ Felix Fröhlich, Nov 19 2017
(Magma) I:=[1, 5, 19]; [n le 3 select I[n] else 6*Self(n-1) -12*Self(n-2) +8*Self(n-3): n in [1..40]]; // G. C. Greubel, Oct 17 2018
KEYWORD
nonn,easy
AUTHOR
STATUS
approved