A133138 -id:A133138

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A133140

a(0) = 2, a(n) = 2^n + 2 for n>=1.

+10
3

2, 4, 6, 10, 18, 34, 66, 130, 258, 514, 1026, 2050, 4098, 8194, 16386, 32770, 65538, 131074, 262146, 524290, 1048578, 2097154, 4194306, 8388610, 16777218, 33554434, 67108866, 134217730, 268435458, 536870914, 1073741826, 2147483650, 4294967298

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

Sum of the coefficients of the polynomial Q(n,x)=(1+x)[(1+x)^(n-1)+x^(n-1)], Q(0,x)=2.

Also row sums of A133138. - R. J. Mathar, Jun 12 2008

LINKS

Table of n, a(n) for n=0..32.

Index entries for linear recurrences with constant coefficients, signature (3, -2).

FORMULA

a(n)=2*A094373(n). a(n)=A052548(n), n>0. - R. J. Mathar, Jun 12 2008

MATHEMATICA

Table[If[n < 2, 2, 1] + Total[Table[Coefficient[Expand[(1 + x) ((1 + x)^(n - 1) + x^(n - 1))], x^m], {m, 1, n}]], {n, 0, 22}] (* Mats Granvik, May 19 2012 *)

Join[{2}, 2^Range[40]+2] (* Harvey P. Dale, Aug 16 2014 *)

KEYWORD

nonn,easy

AUTHOR

Paul Curtz, Sep 21 2007

EXTENSIONS

Edited by R. J. Mathar, Jun 12 2008

STATUS

approved

A233508

Numerators of the triangle of polynomial coefficients P(0,x)=1, 2*P(n)=(1+x)*((1+x)^(n-1)+x^(n-1)). Of the first array of A133135.

+10
2

1, 1, 1, 1, 3, 1, 1, 3, 2, 1, 1, 2, 3, 5, 1, 1, 5, 5, 5, 3, 1, 1, 3, 15, 10, 15, 7, 1, 1, 7, 21, 35, 35, 21, 4, 1, 1, 4, 14, 28, 35, 28, 14, 9, 1, 1, 9, 18, 42, 63, 63, 42, 18, 5, 1, 1, 5, 45, 60, 105, 126, 105, 60, 45, 11, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

Discovered via Euler polynomials A060096(n)/A060097(n).

The fractional sequence is 1, 1, 1, 1/2, 3/2, 1, 1/2, 3/2, 2, 1, 1/2, 2, 3, 5/2, 1,... =a(n)/b(n). There is a correspondant sequence for Bernoulli polynomials (*).

LINKS

Table of n, a(n) for n=0..65.

FORMULA

a(n) = reduced A133138(n)/A007395.

EXAMPLE

1, 1,

1, 3, 1,

1, 3, 2, 1,

1, 2, 3, 5, 1,

1, 5, 5, 5, 3, 1, etc.

MATHEMATICA

p[n_] := (1+x)*((1+x)^(n-1)+x^(n-1))/2; t[n_, k_] := Coefficient[p[n], x, k] // Numerator; Table[t[n, k], {n, 0, 10 }, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 16 2013 *)

CROSSREFS

Cf. (*) A193815.

KEYWORD

nonn,tabl,frac

AUTHOR

Paul Curtz, Dec 11 2013

STATUS

approved

A233808

Autosequence preceding A198631(n)/A006519(n+1). Numerators.

+10
2

0, 0, 1, 3, 3, 5, 5, 7, 7, -3, -3, 121, 121, -1261, -1261, 20583, 20583, -888403, -888403, 24729925, 24729925, -862992399, -862992399, 36913939769, 36913939769, -1899853421885, -1899853421885

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

The fractions are g(n)=0, 0, 1, 3/2, 3/2, 5/4, 5/4, 7/4, 7/4, -3/8, -3/8, 121/8, 121/8, -1261/8, -1261/8, 20583/8, 20583/8, -888403/16, -888403/16,... . The denominators are 1, 1, followed by A053644(n+1).

g(n+2) - g(n+1) = A198631(n)/A006519(n+1).

The corresponding fractions to g(n) are f(n) in A165142(n).

g(n) differences table:

0, 0, 1, 3/2, 3/2, 5/4,

0, 1, 1/2, 0, -1/4, 0,

1, -1/2, -1/2, -1/4, 1/4, 1/2, Euler twin numbers (new),

-3/2, 0, 1/4, 1/2, 1/4, -1,

3/2, 1/4, 1/4, -1/4, -5/4, -5/8,

-5/4, 0, -1/2, -1, 5/8, 13/2, etc.

Like A198631(n)/A006519(n+1),g(n) is an autosequence of the second kind.

If we proceed, here for Euler polynomials, like in A233565 for Bernoulli polynomials, we obtain

1) A133138(n)/A007395(n) (unreduced form) or

2) A233508(n)/A232628(n) (reduced form),the first array in A133135.

The Bernoulli's corresponding fractions to 1) are A193815(n)/(A003056(n) with 1 instead of 0).

LINKS

Table of n, a(n) for n=0..26.

FORMULA

a(n) = 0, 0, followed by (-1)^n *A141424(n).

MATHEMATICA

max = 27; p[0] = 1; p[n_] := (1 + x)*((1 + x)^(n - 1) + x^(n - 1))/2; t = Table[Coefficient[p[n], x, k], {n, 0, max + 2}, {k, 0, max + 2}]; a[n_] := (-1)^n*Inverse[t][[n, 2]] // Numerator; a[0] = 0; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Jan 11 2016 *)

CROSSREFS

Cf. A051716/A051717, Bernoulli twin numbers.

KEYWORD

sign,frac

AUTHOR

Paul Curtz, Dec 16 2013

STATUS

approved

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