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A086926 Product of Fibonacci and (shifted) triangular numbers.

%I A086926 #24 May 16 2025 07:27:36
%S A086926 0,0,1,6,18,50,120,273,588,1224,2475,4895,9504,18174,34307,64050,
%T A086926 118440,217192,395352,714951,1285350,2298660,4091241,7250221,12797568,
%U A086926 22507500,39452725,68942718,120132558,208776974,361937400,626015085,1080441264
%N A086926 Product of Fibonacci and (shifted) triangular numbers.
%H A086926 Colin Barker, <a href="/A086926/b086926.txt">Table of n, a(n) for n = 0..1000</a>
%H A086926 Yahia Djemmada, Abdelghani Mehdaoui, László Németh, and László Szalay, <a href="https://arxiv.org/abs/2407.04409">The Fibonacci-Fubini and Lucas-Fubini numbers</a>, arXiv:2407.04409 [math.CO], 2024. See p. 12.
%H A086926 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-5,0,3,1).
%F A086926 From _Franklin T. Adams-Watters_, Feb 03 2006: (Start)
%F A086926 a(n) = A000045(n)*A000217(n-1) = A000045(n)*n*(n-1)/2.
%F A086926 a(n) = (n/(n-2)*a(n-1) + n*(n-1))/((n-2)*(n-3))*a(n-2).
%F A086926 G.f.: x^2*(1+3x+x^3)/(1-x-x^2)^3. (End)
%F A086926 a(n) = Sum_{k=0..n-1} Sum_{i=0..n-1} i * C(n-k-1,k). - _Wesley Ivan Hurt_, Sep 19 2017
%F A086926 From _Colin Barker_, Sep 20 2017: (Start)
%F A086926 a(n) = ((-1)*(2^(-1-n)*((1-sqrt(5))^n - (1+sqrt(5))^n)*(-1+n)*n)) / sqrt(5).
%F A086926 a(n) = 3*a(n-1) - 5*a(n-3) + 3*a(n-5) + a(n-6) for n>5. (End)
%F A086926 a(n) = A001629(n-2)+3*A001629(n-3)+A001629(n-5). - _R. J. Mathar_, May 16 2025
%t A086926 Array[Fibonacci[#] PolygonalNumber[# - 1] &, 33, 0] (* or *)
%t A086926 LinearRecurrence[{3, 0, -5, 0, 3, 1}, {0, 0, 1, 6, 18, 50}, 33] (* or *)
%t A086926 CoefficientList[Series[x^2*(1 + 3 x + x^3)/(1 - x - x^2)^3, {x, 0, 32}], x] (* _Michael De Vlieger_, Dec 17 2017 *)
%o A086926 (MuPAD) numlib::fibonacci(n)*binomial(n,2) $ n = 0..35; // _Zerinvary Lajos_, May 09 2008
%o A086926 (PARI) concat(vector(2), Vec(x^2*(1 + 3*x + x^3) / (1 - x - x^2)^3 + O(x^40))) \\ _Colin Barker_, Sep 20 2017
%Y A086926 Cf. A000045, A000217, A045925.
%K A086926 nonn,easy
%O A086926 0,4
%A A086926 James FitzSimons (cherry(AT)getnet.net), Sep 20 2003
%E A086926 Definition and more terms from _Franklin T. Adams-Watters_, Feb 03 2006