A135852 -id:A135852 - cp's OEIS frontend

A099035 a(n) = (n+1)*2^(n-1) - 1.

1, 5, 15, 39, 95, 223, 511, 1151, 2559, 5631, 12287, 26623, 57343, 122879, 262143, 557055, 1179647, 2490367, 5242879, 11010047, 23068671, 48234495, 100663295, 209715199, 436207615, 905969663, 1879048191, 3892314111, 8053063679

Offset: 1

Ralf Stephan, Sep 28 2004

Row sums of triangle A135852. - Gary W. Adamson, Dec 01 2007

Binomial transform of [1, 4, 6, 8, 10, 12, 14, 16, ...]. Equals A128064 * A000225, (A000225 starting 1, 3, 7, 15, ...). - Gary W. Adamson, Dec 28 2007

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

First differences of A066524.

Cf. A135852, A128064, A000225.

[(n+1)*2^(n-1) -1: n in [1..30]]; // G. C. Greubel, Dec 31 2017

Table[(n + 1)*2^(n - 1) - 1, {n,1,30}] (* G. C. Greubel, Dec 31 2017 *)
LinearRecurrence[{5,-8,4},{1,5,15},30] (* Harvey P. Dale, Dec 28 2022 *)

a(n)=(n+1)*2^(n-1)-1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = A057711(n+1) - 1 = A058966(n+3)/2 = (A087323(n)-1)/2 = (A074494(n+1)-2)/3 = (A003261(n+1)-3)/4 = A036289(n+1)/4 - 1, n>0.
a(n) = A131056(n+1) - 2. - Juri-Stepan Gerasimov, Oct 02 2011
From Colin Barker, Mar 23 2012: (Start)
a(n) = 5*a(n-1) - 8*a(n-2) + 4*a(n-3).
G.f.: x*(1-2*x^2)/((1-x)*(1-2*x)^2). (End)
E.g.f.: ((2*x+1)*exp(2*x) - 2*exp(x) + 1)/2. - G. C. Greubel, Dec 31 2017

A135853 A103516 * A007318 as an infinite lower triangular matrix.

Original entry on oeis.org

1, 4, 2, 6, 6, 3, 8, 12, 12, 4, 10, 20, 30, 20, 5, 12, 30, 60, 60, 30, 6, 14, 42, 105, 140, 105, 42, 7, 16, 56, 168, 280, 280, 168, 56, 8, 18, 72, 252, 504, 630, 504, 252, 72, 9, 20, 90, 360, 840, 1260, 1260, 840, 360, 90, 10

Offset: 0

Gary W. Adamson, Dec 01 2007

			First few rows of the triangle are:
   1;
   4,   2;
   6,   6,  3;
   8,  12,  12,   4;
  10,  20,  30,  20,   5;
  12,  30,  60,  60,  30,   6;
  14,  42, 105, 140, 105,  42,   7;
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows

Cf. A007318, A103516.
Cf. A103517 (1st column), A135854 (row sums).
Cf. A135852 (= A007318 * A103516).

T[n_, k_]:= If[k==n, n+1, If[k==0, 2*(n+1), (k+1)*Binomial[n+1, k+1]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//flatten (* G. C. Greubel, Dec 07 2016 *)

def A135853(n,k):
    if (n==0): return 1
    elif (k==0): return 2*(n+1)
    else: return (k+1)*binomial(n+1, k+1)
flatten([[A135853(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 06 2022

T(n, k) = (A103516 * A007318)(n, k).
Sum_{k=0..n} T(n, k) = A135854(n).
T(n, k) = (k+1)*binomial(n+1, k+1), with T(n, n) = n+1, T(n, 0) = 2*(n+1). - G. C. Greubel, Dec 07 2016
T(n, 0) = A103517(n). - G. C. Greubel, Feb 06 2022

cp's OEIS Frontend

A099035 a(n) = (n+1)*2^(n-1) - 1.

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