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A084851
Binomial transform of binomial(n+2,2).
6
1, 4, 13, 38, 104, 272, 688, 1696, 4096, 9728, 22784, 52736, 120832, 274432, 618496, 1384448, 3080192, 6815744, 15007744, 32899072, 71827456, 156237824, 338690048, 731906048, 1577058304, 3388997632, 7264534528, 15535702016, 33151778816
OFFSET
0,2
COMMENTS
Essentially the same as A049611.
FORMULA
G.f.: (1 - x)^2/(1 - 2*x)^3.
a(n) = (n^2 + 7*n + 8)*2^(n - 3).
a(n) = Sum_{k=0..n} C(n, k)*C(k+2, 2).
a(n) = A049611(n+1).
EXAMPLE
From Bruno Berselli, Jul 17 2018: (Start)
Let the triangle:
1
3, 4
6, 9, 13
10, 16, 25, 38
15, 25, 41, 66, 104
21, 36, 61, 102, 168, 272
28, 49, 85, 146, 248, 416, 688
36, 64, 113, 198, 344, 592, 1008, 1696, etc.
where the first column is A000217 (without 0). The other terms are calculated with the recurrence T(r, c) = T(r-1, c-1) + T(r, c-1).
The sequence is the right side of the triangle.
(End)
MAPLE
a := n -> hypergeom([-n, 3], [1], -1);
seq(round(evalf(a(n), 32)), n=0..31); # Peter Luschny, Aug 02 2014
MATHEMATICA
CoefficientList[ Series[(1 - x)^2/(1 - 2 x)^3, {x, 0, 28}], x] (* Robert G. Wilson v, Jun 28 2005 *)
LinearRecurrence[{6, -12, 8}, {1, 4, 13}, 30] (* Harvey P. Dale, Aug 05 2019 *)
PROG
(Magma) [(n^2+7*n+8)*2^(n-3): n in [0..40]]; // Vincenzo Librandi, Aug 03 2014
CROSSREFS
Cf. A000217, A049611, A058396 (first differences).
Sequence in context: A089092 A181527 A049611 * A094706 A325927 A056014
KEYWORD
nonn,easy
AUTHOR
Paul Barry, Jun 09 2003
STATUS
approved