A213569 Principal diagonal of the convolution array A213568.

1, 7, 25, 71, 181, 435, 1009, 2287, 5101, 11243, 24553, 53223, 114661, 245731, 524257, 1114079, 2359261, 4980699, 10485721, 22020055, 46137301, 96468947, 201326545, 419430351, 872415181, 1811939275, 3758096329, 7784628167

Offset

1,2

Comments

Create a triangle having first column T(n,1) = 2*n-1 for n = 1,2,3... The remaining terms are set to T(r,c) = T(r,c-1) + T(r-1,c-1). The sum of the terms in row n is a(n). The first five rows of the triangle are 1; 3,4; 5,8,12; 7,12,20,32; 9,16,28,48,80. - J. M. Bergot, Jan 17 2013

Starting at n=1, a(n) = (n+1)*2^n - 2*n - 1. A001787(n) = n*2^n. - J. M. Bergot, Jan 27 2013

Links

Clark Kimberling, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Formula

a(n) = 6*a(n-1) - 13*a(n-2) + 12*a(n-3) - 4*a(n-4).

G.f.: x*(1 + x - 4*x^2)/( (1-2*x)^2*(1-x)^2 ).

a(n) = A001787(n+1)- 2*n - 1. - J. M. Bergot, Jan 22 2013

a(n) = Sum_{k=1..n} Sum_{i=0..n} (n-i) * C(k,i). - Wesley Ivan Hurt, Sep 19 2017

Maple

f:= gfun:-rectoproc({a(n) = 6*a(n-1) - 13*a(n-2) + 12*a(n-3) - 4*a(n-4),
  a(1)=1, a(2)=7, a(3)=25, a(4)=71}, a(n), remember):
map(f, [$1..30]); # Robert Israel, Sep 19 2017

Mathematica

(* First program *)
b[n_]:= 2^(n-1); c[n_]:= n;
t[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n-k+1, k], {n, 12}, {k, n, 1, -1}]]
r[n_]:= Table[t[n, k], {k, 1, 60}]  (* A213568 *)
d = Table[t[n, n], {n, 1, 40}] (* A213569 *)
s[n_]:= Sum[t[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A047520 *)
(* Additional programs *)
LinearRecurrence[{6, -13, 12, -4}, {1, 7, 25, 71}, 30] (* Harvey P. Dale, Jan 06 2015 *)
Table[2^n*(n+1) -(2*n+1), {n, 30}] (* G. C. Greubel, Jul 25 2019 *)

Prog

(PARI) my(x='x+O('x^30)); Vec(x*(1+x-4*x^2)/((1-2*x)^2*(1-x)^2)) \\ Altug Alkan, Sep 19 2017
(PARI) vector(30, n, 2^n*(n+1) -(2*n+1)) \\ G. C. Greubel, Jul 25 2019
(Magma) [2^n*(n+1) -(2*n+1): n in [1..30]]; // G. C. Greubel, Jul 25 2019
(Sage) [2^n*(n+1) -(2*n+1) for n in (1..30)] # G. C. Greubel, Jul 25 2019
(GAP) List([1..30], n-> 2^n*(n+1) -(2*n+1)); # G. C. Greubel, Jul 25 2019

Crossrefs

Cf. A001787, A213568, A213500.

Sequence in context: A155245 A155291 A155221 * A299269 A048477 A294837

Adjacent sequences: A213566 A213567 A213568 * A213570 A213571 A213572

Keyword

nonn,easy

Author

Clark Kimberling, Jun 18 2012

Status

approved