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A007466 Exponential-convolution of natural numbers with themselves.
(Formerly M3478)
6
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). - Aug 29 2013

With offset 0, a(n) is the number of 2 X n 0-1 matrices that do not contain, as a 2 X 2 submatrix,

  1 1 or 0 0

  0 0 .. 1 1.

(See Ju and Seo link, Theorem 3.2.) - David Callan, Jul 11 2014

a(n) = 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

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

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 arXiv version]

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

Index entries for linear recurrences with constant coefficients, signature (6,-12,8)

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

Adjacent sequences:  A007463 A007464 A007465 * A007467 A007468 A007469

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified March 21 07:23 EDT 2019. Contains 321367 sequences. (Running on oeis4.)