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 A034007 First differences of A045891. 17
 1, 0, 2, 4, 9, 20, 44, 96, 208, 448, 960, 2048, 4352, 9216, 19456, 40960, 86016, 180224, 376832, 786432, 1638400, 3407872, 7077888, 14680064, 30408704, 62914560, 130023424, 268435456, 553648128, 1140850688, 2348810240 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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 Number of ordered pairs of (possibly empty) ordered partitions, each not beginning with 1. - Christian G. Bower, Jan 23 2004 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 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 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 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-mk) - 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 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Frank Ellermann, Illustration of binomial transforms. Milan Janjic, Two Enumerative Functions. Milan Janjic and Boris Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013. Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5. Index entries for linear recurrences with constant coefficients, signature (4,-4). FORMULA Sum_{ k = 0..n } (k+4)*binomial(n,k) gives 4, 9, 20, 44, 96, 208, 448, 960, 2048, 4352, ... - N. J. A. Sloane, Jan 30 2008 a(n) = (n+5)*2^(n-4), n >= 3; a(0)=1, a(1)=0, a(2)=2. G.f.: ((1-x)^2/(1-2*x))^2. a(n) = Sum_{k=0..n} (k+1)*C(n-3,n-k). - Peter Luschny, Apr 20 2015 From Amiram Eldar, Jan 13 2021: (Start) Sum_{n>=2} 1/a(n) = 512*log(2) - 74327/210. Sum_{n>=2} (-1)^n/a(n) = 14579/70 - 512*log(3/2). (End) MATHEMATICA Join[{1, 0, 2, a=4}, Table[a=(2*(n+7)*a)/(n+6), {n, 2, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 15 2011 *) PROG (PARI) a(n)=if(n<3, [1, 0, 2][n+1], (n+5)*2^(n-4)) \\ Charles R Greathouse IV, Jun 01 2011 CROSSREFS Cf. A045891. Convolution of A034008 with itself. Columns of A091613 converge to this sequence. Sequence in context: A123720 A179744 A266930 * A109975 A129891 A130587 Adjacent sequences:  A034004 A034005 A034006 * A034008 A034009 A034010 KEYWORD easy,nonn AUTHOR STATUS approved

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Last modified June 22 09:57 EDT 2021. Contains 345375 sequences. (Running on oeis4.)