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A034007 First differences of A045891.

%I A034007 #81 Jan 02 2025 02:54:53
%S A034007 1,0,2,4,9,20,44,96,208,448,960,2048,4352,9216,19456,40960,86016,
%T A034007 180224,376832,786432,1638400,3407872,7077888,14680064,30408704,
%U A034007 62914560,130023424,268435456,553648128,1140850688,2348810240
%N A034007 First differences of A045891.
%C A034007 Let M_n be the n X n matrix m_(i,j) = 4 + abs(i-j) then det(M_n) = (-1)^(n+1)*a(n+2). - _Benoit Cloitre_, May 28 2002
%C A034007 Number of ordered pairs of (possibly empty) ordered partitions, each not beginning with 1. - _Christian G. Bower_, Jan 23 2004
%C A034007 If X_1, X_2, ..., X_n are 2-blocks of a (2n+4)-set X then, for n>=1, a(n+3) is the number of (n+1)-subsets of X intersecting each X_i, (i=1,2,...,n). - _Milan Janjic_, Nov 18 2007
%C A034007 Except for an initial 1, this is the p-INVERT of (1,1,1,1,1,...) for p(S) = (1 - S^2)^2; see A291000. - _Clark Kimberling_, Aug 24 2017
%C A034007 Conjecture 1: For compositions of n+k-1, a(n) is the number of runs of 1 of length k. Example: Among the compositions of 4+2-1 = 5, there are a(4) = 4 runs of two 1's: 3,[1,1]; [1,1],3; 1,2,[1,1] and [1,1],2,1. - _Gregory L. Simay_, Feb 18 2018
%C A034007 Conjecture 2: More generally, let R(n,m,k) = the number of runs of k m's in all compositions of n. Then R(n,m,k) = A045623(n-m*k) - 2*A045623(n-m*(k+1)) + A045623(n-m*(k+2)). For example, R(7,1,1) = A045623(6) - 2*A045623(5) + A045623(4) = 144 - 2*64 + 28 = 44 = a(7). - _Gregory L. Simay_, Feb 20 2018
%H A034007 Vincenzo Librandi, <a href="/A034007/b034007.txt">Table of n, a(n) for n = 0..1000</a>
%H A034007 Frank Ellermann, <a href="/A001792/a001792.txt">Illustration of binomial transforms</a>.
%H A034007 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>.
%H A034007 Milan Janjic and Boris Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv:1301.4550 [math.CO], 2013.
%H A034007 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 A034007 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4).
%F A034007 a(n) = Sum_{k = 0..n-3} (k+4)*binomial(n-3,k) for n >= 3. - _N. J. A. Sloane_, Jan 30 2008
%F A034007 a(n) = (n+5)*2^(n-4), n >= 3; a(0)=1, a(1)=0, a(2)=2.
%F A034007 G.f.: ((1-x)^2/(1-2*x))^2.
%F A034007 a(n) = Sum_{k=0..n} (k+1)*C(n-3,n-k). - _Peter Luschny_, Apr 20 2015
%F A034007 From _Amiram Eldar_, Jan 13 2021: (Start)
%F A034007 Sum_{n>=2} 1/a(n) = 512*log(2) - 74327/210.
%F A034007 Sum_{n>=2} (-1)^n/a(n) = 14579/70 - 512*log(3/2). (End)
%F A034007 E.g.f.: (1/16)*(11 - 12*x + 2*x^2 + (5+2*x)*exp(2*x)). - _G. C. Greubel_, Sep 27 2022
%t A034007 Join[{1,0,2,a=4},Table[a=(2*(n+7)*a)/(n+6),{n,2,100}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 15 2011 *)
%o A034007 (PARI) a(n)=if(n<3,[1,0,2][n+1],(n+5)*2^(n-4)) \\ _Charles R Greathouse IV_, Jun 01 2011
%o A034007 (Magma) [1,0,2] cat [(n+5)*2^(n-4): n in [3..30]]; // _G. C. Greubel_, Sep 27 2022
%o A034007 (SageMath) [1,0,2]+[(n+5)*2^(n-4) for n in range(3,30)] # _G. C. Greubel_, Sep 27 2022
%Y A034007 Cf. A045623, A045891, A291000.
%Y A034007 Convolution of A034008 with itself.
%Y A034007 Columns of A091613 converge to this sequence.
%K A034007 easy,nonn
%O A034007 0,3
%A A034007 _Wolfdieter Lang_