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%I #51 Jan 01 2024 02:19:11
%S 0,1,4,11,25,51,97,176,309,530,894,1490,2462,4043,6610,10773,17519,
%T 28445,46135,74770,121115,196116,317484,513876,831660,1345861,2177872,
%U 3524111,5702389,9226935,14929789
%N Apply partial sum operator thrice to Fibonacci numbers.
%C With offset 4, number of 132-avoiding two-stack sortable permutations which contain exactly one subsequence of type 51234.
%H G. C. Greubel, <a href="/A014162/b014162.txt">Table of n, a(n) for n = 0..1000</a>
%H Hung Viet Chu, <a href="https://arxiv.org/abs/2106.03659">Partial Sums of the Fibonacci Sequence</a>, arXiv:2106.03659 [math.CO], 2021.
%H Ligia Loretta Cristea, Ivica Martinjak, and Igor Urbiha, <a href="http://arxiv.org/abs/1606.06228">Hyperfibonacci Sequences and Polytopic Numbers</a>, arXiv:1606.06228 [math.CO], 2016.
%H E. S. Egge and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0205206">132-avoiding Two-stack Sortable Permutations, Fibonacci Numbers, and Pell Numbers</a>, arXiv:math/0205206 [math.CO], 2002.
%H T. Langley, J. Liese, and J. Remmel, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Langley/langley2.html">Generating Functions for Wilf Equivalence Under Generalized Factor Order </a>, J. Int. Seq. 14 (2011) # 11.4.2.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,1,2,-1).
%F a(n) = Sum_{k=0..n} A000045(n-k)*k*(k+1)/2. - _Benoit Cloitre_, Jan 06 2003
%F G.f.: x/((1-x)^3*(1-x-x^2)).
%F From _Paul Barry_, Oct 07 2004: (Start)
%F a(n-2) = Sum_{k=0..floor(n/2)} binomial(n-k, k+3).
%F a(n-2) = Sum_{k=0..n} binomial(k, n-k+3). (End)
%F Convolution of A000045 and A000217 (Fibonacci and triangular numbers). - _Ross La Haye_, Nov 08 2004
%F a(n) = Fibonacci(n+6) - (n^2 + 7*n + 16)/2.
%p with(combinat); seq(fibonacci(n+6)-(n^2+7*n+16)*(1/2), n = 0..40); # _G. C. Greubel_, Sep 05 2019
%t Nest[Accumulate,Fibonacci[Range[0,30]],3] (* or *) LinearRecurrence[{4,-5,1,2,-1},{0,1,4,11,25},40] (* _Harvey P. Dale_, Aug 19 2017 *)
%o (PARI) a(n)=fibonacci(n+6)-n*(n+7)/2-8 \\ _Charles R Greathouse IV_, Jun 11 2015
%o (Magma) [Fibonacci(n+6) - (n^2 + 7*n + 16)/2: n in [0..40]]; // _G. C. Greubel_, Sep 05 2019
%o (Sage) [fibonacci(n+6) - (n^2 + 7*n + 16)/2 for n in (0..40)] # _G. C. Greubel_, Sep 05 2019
%o (GAP) List([0..40], n-> Fibonacci(n+6) - (n^2 + 7*n + 16)/2); # _G. C. Greubel_, Sep 05 2019
%Y Cf. A000045, A001924, A228074.
%Y Right-hand column 6 of triangle A011794.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
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