A130103 Expansion of e.g.f. e^(2x)-(1+x)*e^x+x.

0, 1, 1, 4, 11, 26, 57, 120, 247, 502, 1013, 2036, 4083, 8178, 16369, 32752, 65519, 131054, 262125, 524268, 1048555, 2097130, 4194281, 8388584, 16777191, 33554406, 67108837, 134217700, 268435427, 536870882, 1073741793, 2147483616

Offset

0,4

Comments

a(n) = A130102(n+1)/2.

Partial sums are A130104.

Essentially the same as the Euler numbers A000295.

a(n) = Sum_{i=1..n} i*2^(n-i) - Ctibor O. Zizka, Feb 23 2008

Links

Seiichi Manyama, Table of n, a(n) for n = 0..3321

Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, Chip-Firing on Infinite k-ary Trees, arXiv:2501.06675 [math.CO], 2025. See p. 18.

Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order"

Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Formula

G.f.: x(1-3x+5x^2-2x^3)/((1-x)^2*(1-2x)).

E.g.f.: e^(2x)-(1+x)*e^x+x.

a(n) = 2^n-n-1+C(1,n)-C(0,n).

Example

G.f. = x + x^2 + 4*x^3 + 11*x^4 + 26*x^5 + 57*x^6 + 120*x^7 + 247*x^8 + ...

Maple

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]+n od: seq(a[n], n=0..30); # Zerinvary Lajos, Feb 22 2008

Mathematica

Join[{0, 1}, LinearRecurrence[{4, -5, 2}, {1, 4, 11}, 40]] (* Harvey P. Dale, May 16 2014 *)
a[ n_] := If[ n < 2, Boole[n == 1], 2^n - (1 + n)]; (* Michael Somos, Aug 17 2015 *)

Prog

(PARI) {a(n) = if( n<2, n==1, 2^n - (1+n))}; /* Michael Somos, Aug 17 2015 */

Crossrefs

Cf. A000295.

Sequence in context: A030196 A248425 A356621 * A000295 A125128 A034334

Adjacent sequences: A130100 A130101 A130102 * A130104 A130105 A130106

Keyword

easy,nonn

Author

Paul Barry, May 07 2007

Status

approved