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%I #95 Sep 27 2023 03:57:48
%S 1,1,3,7,16,36,80,176,384,832,1792,3840,8192,17408,36864,77824,163840,
%T 344064,720896,1507328,3145728,6553600,13631488,28311552,58720256,
%U 121634816,251658240,520093696,1073741824,2214592512,4563402752
%N First differences of A045623.
%C Let M_n be the n X n matrix m_(i,j) = 3 + abs(i-j), then det(M_n) =(-1)^(n+1)*a(n+1). - _Benoit Cloitre_, May 28 2002
%C If X_1, X_2, ..., X_n are 2-blocks of a (2n+3)-set X then, for n>=1, a(n+2) is the number of (n+1)-subsets of X intersecting each X_i, (i=1..n). - _Milan Janjic_, Nov 18 2007
%C Equals row sums of triangle A152194. - _Gary W. Adamson_, Nov 28 2008
%C An elephant sequence, see A175655. For the central square 16 A[5] vectors, with decimal values between 19 and 400, lead to this sequence (without the first leading 1). For the corner squares these vectors lead to the companion sequence A045623. - _Johannes W. Meijer_, Aug 15 2010
%C a(n) is the total number of runs of 1 in the compositions of n+1. For example, a(3) = A045623(3) - A045623(2) = 12 - 5 = 7 runs of only 1 in the compositions of 4, enumerated "()" as follows: 3,(1); (1),3; 2,(1,1);(1),2,(1); (1,1),2; (1,1,1,1). More generally, the total number of runs of only part k in the compositions of n+k is A045623(n) - A045623(n-k). - _Gregory L. Simay_, May 02 2017
%C This is essentially the p-INVERT of (1,1,1,1,1,...) for p(S) = 1 - S - S^2 + S^3; see A291000. - _Clark Kimberling_, Aug 24 2017
%H Vincenzo Librandi, <a href="/A045891/b045891.txt">Table of n, a(n) for n = 0..1000</a>
%H Paul Barry, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Barry/barry84.html">A Catalan Transform and Related Transformations on Integer Sequences</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%H Frank Ellermann, <a href="/A001792/a001792.txt">Illustration of binomial transforms</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H Milan Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv 1301.4550 [math.CO], 2013.
%H Milan Janjic and B. 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 Thomas Selig and Haoyue Zhu, <a href="https://arxiv.org/abs/2309.11788">New combinatorial perspectives on MVP parking functions and their outcome map</a>, arXiv:2309.11788 [math.CO], 2023. See p. 29.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4).
%F a(n) = Sum_{k=0..n-2} (k+3)*binomial(n-2,k) for n >= 2. - _N. J. A. Sloane_, Jan 30 2008
%F a(n) = (n+4)*2^(n-3), n >= 2, with a(0) = a(1) = 1.
%F G.f.: (1-x)^3/(1-2*x)^2.
%F Equals binomial transform of A027656.
%F Starting 1, 3, 7, 16, ... this is ((n+5)*2^n - 0^n)/4, the binomial transform of (1, 2, 2, 3, 3, ...). - _Paul Barry_, May 20 2003
%F From _Paul Barry_, Nov 29 2004: (Start)
%F a(n) = ((n+4)*2^(n-1) + 3*C(0, n) - C(1, n))/4;
%F a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k)*(k+1). (End)
%F a(n) = A045623(n-1) + 2^(n-2) = A034007(n+1) - 2^(n-2) for n>=2. - _Philippe Deléham_, Apr 20 2009
%F G.f.: 1 + Q(0)*x/(1-x)^2, where Q(k)= 1 + (k+1)*x/(1 - x - x*(1-x)/(x + (k+1)*(1-x)/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Apr 25 2013
%F a(n) = Sum_{k=0..n} (k+1)*C(n-2,n-k). _Peter Luschny_, Apr 20 2015
%F From _Amiram Eldar_, Jan 13 2021: (Start)
%F Sum_{n>=0} 1/a(n) = 128*log(2) - 1292/15.
%F Sum_{n>=0} (-1)^n/a(n) = 782/15 - 128*log(3/2). (End)
%F E.g.f.: (2 - x + exp(2*x)*(2 + x))/4. - _Stefano Spezia_, Mar 26 2022
%e G.f. = 1 + x + 3*x^2 + 7*x^3 + 16*x^4 + 36*x^5 + 80*x^6 + ... - _Michael Somos_, Mar 26 2022
%t Join[{1,1,a=3,b=7},Table[c=4*b-4*a;a=b;b=c,{n,100}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 15 2011 *)
%t Table[ If[n<2, 1, 2^(n-3)*(n+4)], {n, 0, 30}] (* _Jean-François Alcover_, Sep 12 2012 *)
%t LinearRecurrence[{4,-4},{1,1,3,7},40] (* _Harvey P. Dale_, May 03 2019 *)
%o (PARI) v=[1,1,3,7];for(i=1,99,v=concat(v,4*(v[#v]-v[#v-1])));v \\ _Charles R Greathouse IV_, Jun 01 2011
%o (Magma) [1,1] cat [(n+4)*2^(n-3): n in [2..40]]; // _G. C. Greubel_, Sep 27 2022
%o (SageMath) [1,1]+[(n+4)*2^(n-3) for n in range(2,40)] # _G. C. Greubel_, Sep 27 2022
%Y Cf. A027656, A034007, A045623, A152194, A175655.
%K easy,nonn,nice
%O 0,3
%A _Wolfdieter Lang_
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