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A007466 Exponential-convolution of natural numbers with themselves.
(Formerly M3478)
8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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
This is triangle A228643: a(n) = A228643(n,n). - Reinhard Zumkeller, Aug 29 2013
From David Callan, Jul 11 2014: (Start)
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
Binomial transform of A002061. - Jules Beauchamp, Jan 04 2022
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
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]
Hyeong-Kwan Ju and Seunghyun Seo, Enumeration of 0/1-matrices avoiding some 2x2 matrices, arXiv:1107.1299 [math.CO], 2011.
Hyeong-Kwan Ju and Seunghyun Seo, Enumeration of (0,1)-matrices avoiding some 2 X 2 matrices, Discrete Math., 312 (2012), 2473-2481.
N. J. A. Sloane, Transforms
FORMULA
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.: (1+x)^2*exp(2*x). - Vladeta Jovovic, Sep 09 2003
G.f.: -(2*x^3-2*x^2+x)/(2*x-1)^3. - Vladimir Kruchinin, Sep 28 2011
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
MAPLE
A007466:=n->2^(n-1)*n+1/4*2^(n-1)*(n-1)*(n-2): seq(A007466(n), n=1..30);
MATHEMATICA
Table[2^(n - 1)*n + 1/4*2^(n - 1)*(n - 1)*(n - 2), {n, 30}] (* Wesley Ivan Hurt, Jul 11 2014 *)
PROG
(Haskell)
a007466 n = a228643 n n -- Reinhard Zumkeller, Aug 29 2013
(Magma) [2^(n-1)*n+1/4*2^(n-1)*(n-1)*(n-2) : n in [1..30]]; // Wesley Ivan Hurt, Jul 11 2014
CROSSREFS
Sequence in context: A084613 A099063 A057223 * A062109 A118042 A006645
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)