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

Author

Clark Kimberling, Jun 18 2012

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A001787, A213568, A213500.

Programs

GAP

List([1..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

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 *)

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

Sage

[2^n*(n+1) -(2*n+1) for n in (1..30)] # G. C. Greubel, Jul 25 2019

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