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A158920 Binomial transform of A008805 (triangular numbers with repeats).

%I A158920 #27 Feb 20 2025 05:03:17
%S A158920 1,2,6,16,41,102,248,592,1392,3232,7424,16896,38144,85504,190464,
%T A158920 421888,929792,2039808,4456448,9699328,21037056,45481984,98041856,
%U A158920 210763776,451936256,966787072,2063597568,4395630592,9344909312,19830669312
%N A158920 Binomial transform of A008805 (triangular numbers with repeats).
%H A158920 Harvey P. Dale, <a href="/A158920/b158920.txt">Table of n, a(n) for n = 1..1000</a>
%H A158920 Milan Janjic and Boris Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
%H A158920 Milan Janjic and Boris Petkovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Janjic/janjic45.html">A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers</a>, J. Int. Seq. 17 (2014) # 14.3.5.
%H A158920 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12,8).
%F A158920 A007318 * (1, 1, 3, 3, 6, 6, 10, 10, 15, 15, ...) = binomial transform of triangular numbers A000217 with repeats.
%F A158920 From _R. J. Mathar_, Apr 02 2009: (Start)
%F A158920 G.f.: x*(x-1)^4/(1-2*x)^3.
%F A158920 a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3), n > 5. (End)
%F A158920 32*a(n) = 2^(n+1) + 3*A001787(n+1) + A001788(n+1), n>=3. - _R. J. Mathar_, Feb 25 2023
%e A158920 a(4) = 16 = (1, 3, 3, 1) dot (1, 1, 3, 3) = (1 + 3 + 9 + 3).
%p A158920 A000217 := proc(n) n*(n+1)/2 ; end: A008805 := proc(n) A000217( 1+floor(n/2) ) ; end: L := [seq(A008805(n), n=0..100)] ; read("transforms"); BINOMIAL(L) ; # _R. J. Mathar_, Apr 02 2009
%t A158920 Join[{1,2},LinearRecurrence[{6,-12,8},{6,16,41},30]] (* _Harvey P. Dale_, Feb 25 2012 *)
%Y A158920 Cf. A008805, A045891.
%K A158920 nonn,easy
%O A158920 1,2
%A A158920 _Gary W. Adamson_, Mar 30 2009