A007466 Exponential-convolution of natural numbers with themselves.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

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

a(n+1) is the number of strings of length n defined on {0,1,2,3} that contain at most one 2, at most one 3, and have no restriction on the number of 0s and 1s. For example, for n=2, a(3)=14 since from the 16 strings of length 2 we exclude 22 and 33. - Enrique Navarrete, May 03 2025

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

Crossrefs

Cf. A002061, A007318, A141611, A228643.

Programs

Haskell

a007466 n = a228643 n n  -- Reinhard Zumkeller, Aug 29 2013

Magma

[2^(n-1)*(n+(n-1)*(n-2)/4) : n in [1..30]]; // Wesley Ivan Hurt, Jul 11 2014

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 + (n-1)*(n-2)/4), {n, 30}] (* Wesley Ivan Hurt, Jul 11 2014 *)

SageMath

def A007466(n): return 2^(n-3)*(n^2+n+2)
[A007466(n) for n in range(1,31)] # G. C. Greubel, Sep 22 2024

Formula

E.g.f.: (Sum_{n >= 1} n*x^(n-1)/(n-1)!)^2.

a(n) = 2^(n-1)*n + 2^(n-3)*(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.: x*(1 - 2*x + 2*x^2)/(1-2*x)^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

a(n) = A228643(n, n). - Reinhard Zumkeller, Aug 29 2013

a(n) = Sum_{k=0..n-1} A141611(n-1, k). - G. C. Greubel, Sep 22 2024