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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 (list; graph; refs; listen; history; text; internal format)
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

G. C. Greubel, Table of n, a(n) for n = 0..1000

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv:1406.3081 [math.CO], 2014.

C. Corsani, D. Merlini, R. Sprugnoli, Left-inversion of combinatorial sums, Discrete Mathematics, 180 (1998) 107-122.

Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

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

CROSSREFS

Cf. A000079, A001787, A001792, A005448, A055818, A077588, A207629, A207630.

Sequence in context: A212339 A072111 A173627 * A141397 A305779 A143131

Adjacent sequences:  A295285 A295286 A295287 * A295289 A295290 A295291

KEYWORD

nonn,easy

AUTHOR

Franck Maminirina Ramaharo, Nov 19 2017

STATUS

approved

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Last modified May 25 11:45 EDT 2019. Contains 323549 sequences. (Running on oeis4.)