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A007466
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Exponential-convolution of natural numbers with themselves.
(Formerly M3478)
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8
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1, 4, 14, 44, 128, 352, 928, 2368, 5888, 14336, 34304, 80896, 188416, 434176, 991232, 2244608, 5046272, 11272192, 25034752, 55312384, 121634816, 266338304, 580911104, 1262485504, 2734686208, 5905580032, 12717129728
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OFFSET
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1,2
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COMMENTS
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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
With offset 0, a(n) is the number of 2 X n 0-1 matrices that do not contain
1 1 0 0
0 0 or 1 1, as a 2 X 2 submatrix,
See Ju and Seo link, Theorem 3.2. (End)
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
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
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FORMULA
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E.g.f.: (Sum_{n >= 1} n*x^(n-1)/(n-1)!)^2.
a(n) = 2^(n-1)*n+1/4*2^(n-1)*(n-1)*(n-2).
a(n) = Sum_{k=0..(n+2)} C(n+2, k) * floor(k/2)^2. - Paul Barry, Mar 06 2003
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
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MAPLE
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A007466:=n->2^(n-1)*n+1/4*2^(n-1)*(n-1)*(n-2): seq(A007466(n), n=1..30);
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MATHEMATICA
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Table[2^(n - 1)*n + 1/4*2^(n - 1)*(n - 1)*(n - 2), {n, 30}] (* Wesley Ivan Hurt, Jul 11 2014 *)
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PROG
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(Haskell)
(Magma) [2^(n-1)*n+1/4*2^(n-1)*(n-1)*(n-2) : n in [1..30]]; // Wesley Ivan Hurt, Jul 11 2014
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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