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%I M3478 #75 Feb 18 2022 22:56:00
%S 1,4,14,44,128,352,928,2368,5888,14336,34304,80896,188416,434176,
%T 991232,2244608,5046272,11272192,25034752,55312384,121634816,
%U 266338304,580911104,1262485504,2734686208,5905580032,12717129728
%N Exponential-convolution of natural numbers with themselves.
%C Define a triangle T by T(n,1) = n*(n-1)+1 and T(r,c) = T(r,c-1) + T(r-1,c-1), then a(n) = T(n,n). - _J. M. Bergot_, Mar 03 2013
%C This is triangle A228643: a(n) = A228643(n,n). - _Reinhard Zumkeller_, Aug 29 2013
%C From _David Callan_, Jul 11 2014: (Start)
%C With offset 0, a(n) is the number of 2 X n 0-1 matrices that do not contain
%C 1 1 0 0
%C 0 0 or 1 1, as a 2 X 2 submatrix,
%C See Ju and Seo link, Theorem 3.2. (End)
%C a(n) is the sum of all ways of adding the k-tuples of the terms in the (n-1)-st row of Pascal's triangle A007318. For n=4 take row 3 of A007318: 1,3,3,1, giving (1)+(3)+(3)+(1)=8; (1+3)+(3+3)+(3+1)=14; (1+3+3)+(3+3+1)=14; (1+3+3+1)=8. The sum of these four terms is 8+14+14+8=44. - _J. M. Bergot_, Jun 17 2017
%C Binomial transform of A002061. - _Jules Beauchamp_, Jan 04 2022
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Reinhard Zumkeller, <a href="/A007466/b007466.txt">Table of n, a(n) for n = 1..1000</a>
%H M. Bernstein and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002.
%H M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H Hyeong-Kwan Ju and Seunghyun Seo, <a href="http://arxiv.org/abs/1107.1299">Enumeration of 0/1-matrices avoiding some 2x2 matrices</a>, arXiv:1107.1299 [math.CO], 2011.
%H Hyeong-Kwan Ju and Seunghyun Seo, <a href="http://dx.doi.org/10.1016/j.disc.2012.04.019">Enumeration of (0,1)-matrices avoiding some 2 X 2 matrices</a>, Discrete Math., 312 (2012), 2473-2481.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12,8)
%F E.g.f.: (Sum_{n >= 1} n*x^(n-1)/(n-1)!)^2.
%F a(n) = 2^(n-1)*n+1/4*2^(n-1)*(n-1)*(n-2).
%F a(n) = Sum_{k=0..(n+2)} C(n+2, k) * floor(k/2)^2. - _Paul Barry_, Mar 06 2003
%F E.g.f.: (1+x)^2*exp(2*x). - _Vladeta Jovovic_, Sep 09 2003
%F G.f.: -(2*x^3-2*x^2+x)/(2*x-1)^3. - _Vladimir Kruchinin_, Sep 28 2011
%F E.g.f.: U(0) where U(k)= 1 + 2*x/( 1 - x/(2 + x - 4/( 2 + x*(k+1)/U(k+1)))) ; (continued fraction, 3rd kind, 4-step). - _Sergei N. Gladkovskii_, Oct 28 2012
%p A007466:=n->2^(n-1)*n+1/4*2^(n-1)*(n-1)*(n-2): seq(A007466(n), n=1..30);
%t Table[2^(n - 1)*n + 1/4*2^(n - 1)*(n - 1)*(n - 2), {n, 30}] (* _Wesley Ivan Hurt_, Jul 11 2014 *)
%o (Haskell)
%o a007466 n = a228643 n n -- _Reinhard Zumkeller_, Aug 29 2013
%o (Magma) [2^(n-1)*n+1/4*2^(n-1)*(n-1)*(n-2) : n in [1..30]]; // _Wesley Ivan Hurt_, Jul 11 2014
%Y Cf. A002061, A007318, A228643.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
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