A228767 -id:A228767 - cp's OEIS frontend

A228827 Numerators of the first bisection of the inverse binomial transform of the rational sequence with e.g.f. (x/2)*(exp(-x)+1)/(exp(x)-1).

1, 25, 599, 4285, 15599, 169625, 33578309, 344155, 133697983, 941417335, 1729982389, 3184334285, 274574499509, 2625798955, 1611022490371, 123951819730625, 9814145542783, 3453861186955, -25128299959971711973, 2945661954537595, -260933954573210488051

Offset: 0

Paul Curtz & Michel Marcus, Sep 06 2013

The sequence to be transformed is A176328/A176591, its inverse binomial transform begins: 1, -2, 25/6, -9, 599/30, -45, 4285/42, -231, 15599/30, -1161, 169625/66, -5643, 33578309/2730, ...

It appears that a(n) - A000367(n) is a multiple of A002445(n), and the quotients are 0, 4, 20, 102, 520, 2570, 12300, ...

Cf. A228767 (other bisection).

fr(n) = {default(seriesprecision, n+1); egf = (x/2)*(exp(-x)+1)/(exp(x)-1);(n)!* polcoeff(egf, n);}
ibtfr(n) = sum(k = 0, n, (-1)^(n-k)*binomial(n, k) * fr(k));
lista(nn) = {forstep(n = 0, nn, 2, print1(numerator(ibtfr(n)), ", "););} \\ Michel Marcus, Sep 06 2013

A227978 a(0)=1, a(1)=2; for n>1, a(n) = n*(2^n+4)/4.

Original entry on oeis.org

1, 2, 4, 9, 20, 45, 102, 231, 520, 1161, 2570, 5643, 12300, 26637, 57358, 122895, 262160, 557073, 1179666, 2490387, 5242900, 11010069, 23068694, 48234519, 100663320, 209715225, 436207642, 905969691, 1879048220, 3892314141, 8053063710, 16642998303

Offset: 0

Paul Curtz, Oct 07 2013

The inverse binomial transform of A176328/A176591 (see Comments field in A228827) begins: 1, -2, 25/6, -9, 599/30, -45, 4285/42, -231, 15599/30, -1161, 169625/66, ... Consider these values without sign and the fractions rounded to the nearest integer, the sequence lists the resulting numbers.
Differences table of a(n):
  1, 2,  4,  9, 20,  45, 102, 231,  520, 1161, ...
  1, 2,  5, 11, 25,  57, 129, 289,  641, 1409, ... After 2: 2^m*(m+4)+1.
  1, 3,  6, 14, 32,  72, 160, 352,  768, 1664, ... A078836 (after 3).
  2, 3,  8, 18, 40,  88, 192, 416,  896, 1920, ... A129955.
  1, 5, 10, 22, 48, 104, 224, 480, 1024, 2176, ... A079861 (after 5).
  4, 5, 12, 26, 56, 120, 256, 544, 1152, 2432, ... After 5: 2^m*(m+12).
  1, 7, 14, 30, 64, 136, 288, 608, 1280, 2688, ... After 7: 2^m*(m+14).
  6, 7, 16, 34, 72, 152, 320, 672, 1408, 2944, ..., etc.
(n-1)*a(n)-n*a(n-1) = A001788(n-1) for n>1. [Bruno Berselli, Oct 11 2013]

Bruno Berselli, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Cf. A001788, A079862, A079863.

[1,2] cat [n*(2^n+4)/4: n in [2..40]]; // Bruno Berselli, Oct 11 2013

Join[{1, 2}, Table[n (2^n + 4)/4, {n, 2, 40}]] (* Bruno Berselli, Oct 11 2013 *)

a(n) = if (n == 0, 1, if (n == 1, 2, n*(2^n+4)/4)); \\ Michel Marcus, Oct 11 2013

a(2n+2) = A229135(n+1); a(2n-1) = -A228767(n) for n>0.
a(n) = 6*a(n-1) -13*a(n-2) +12*a(n-3) -4*a(n-4) for n>5.
G.f.: (1-4*x+5*x^2-x^3-2*x^4+2*x^5)/((1-x)^2*(1-2*x)^2). - Colin Barker, Oct 09 2013

More terms from Colin Barker, Oct 09 2013

A141519 Period 10: repeat [-1, 1, -3, 7, -5, 3, -7, 9, -9, 5].

Original entry on oeis.org

-1, 1, -3, 7, -5, 3, -7, 9, -9, 5, -1, 1, -3, 7, -5, 3, -7, 9, -9, 5, -1, 1, -3, 7, -5, 3, -7, 9, -9, 5, -1, 1, -3, 7, -5, 3, -7, 9, -9, 5, -1, 1, -3, 7, -5, 3, -7, 9, -9, 5, -1, 1, -3, 7, -5, 3, -7, 9, -9, 5, -1, 1, -3, 7, -5, 3, -7, 9, -9, 5, -1, 1, -3, 7, -5, 3, -7, 9, -9, 5, -1, 1, -3, 7, -5, 3, -7, 9, -9, 5, -1, 1, -3

Offset: 0

Paul Curtz, Aug 11 2008

It appears that abs(a(n)) = abs(A001469(n+1)) mod 10.

It also appears that abs(a(n)) = abs(A228767(n+4)) mod 10. - Michel Marcus, Sep 04 2013

Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,-1,-1,-1,-1,-1,-1).

Cf. A001469, A228767.

PadRight[{},120,{-1,1,-3,7,-5,3,-7,9,-9,5}] (* Harvey P. Dale, Mar 03 2023 *)

G.f.: ( -1-3*x^2+4*x^3+2*x^5-5*x^6-5*x^8-x^4+4*x^7 ) / ( (1+x)*(1+x+x^2+x^3+x^4)*(x^4-x^3+x^2-x+1) ). - R. J. Mathar, Oct 08 2011

a(n) = - Sum_{k=1..9} a(n-k). - Wesley Ivan Hurt, May 27 2021

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A228827 Numerators of the first bisection of the inverse binomial transform of the rational sequence with e.g.f. (x/2)*(exp(-x)+1)/(exp(x)-1).

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A227978 a(0)=1, a(1)=2; for n>1, a(n) = n*(2^n+4)/4.

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A141519 Period 10: repeat [-1, 1, -3, 7, -5, 3, -7, 9, -9, 5].

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