A146559 -id:A146559 - cp's OEIS frontend

A106664 Expansion of g.f.: (1-3x+x^2)/((1-x)(1+x)(1-2x+2*x^2)).

-1, 1, 2, 5, 4, 1, -8, -15, -16, 1, 32, 65, 64, 1, -128, -255, -256, 1, 512, 1025, 1024, 1, -2048, -4095, -4096, 1, 8192, 16385, 16384, 1, -32768, -65535, -65536, 1, 131072, 262145, 262144, 1, -524288, -1048575, -1048576, 1, 2097152, 4194305, 4194304, 1, -8388608, -16777215, -16777216, 1, 33554432

Offset: 0

Creighton Dement, May 13 2005

Superseeker finds that a(n+2) - a(n) = A090131(n+1) (or with different signs, see A078069).

Floretion Algebra Multiplication Program, FAMP Code: 2ibaseiseq[ + .5'i + .5i' - .5'ii' + .5'jj' + .5'kk' + .5e]

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,-2,2).

Cf. A000035, A010673, A078069, A090131, A099087, A146559.

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!(  (1-3*x+x^2)/((1-x^2)*(1-2*x+2*x^2)) )); // G. C. Greubel, Sep 08 2021

CoefficientList[Series[(1-3x+x^2)/((1-x)(1+x)(1-2x+2x^2)),{x,0,60}],x] (* Harvey P. Dale, Mar 20 2013 *)

def A106664_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( sinh(x) -exp(x)*(cos(x)-sin(x)) ).egf_to_ogf().list()
A106664_list(50) # G. C. Greubel, Sep 08 2021

a(n) = (1/2)*(A010673(n) - A099087(n+2)).
a(n) = (1/2)*(1 - (-1)^n - (1-i)^(n+1) - (1+i)^(n+1)), with i=sqrt(-1). - Ralf Stephan, Nov 16 2010
From G. C. Greubel, Sep 08 2021: (Start)
a(n) = (1-(-1)^n)/2 - 2^((n+1)/2)*cos((n+1)*Pi/4).
a(n) = A000035(n) - A146559(n).
E.g.f.: sinh(x) - exp(x)*(cos(x) - sin(x)). (End)

A141665 A signed half of Pascal's triangle A007318: p(x,n) = (1+I*x)^n; t(n,m) = real part of coefficients(p(x,n)).

Original entry on oeis.org

1, 1, 0, 1, 0, -1, 1, 0, -3, 0, 1, 0, -6, 0, 1, 1, 0, -10, 0, 5, 0, 1, 0, -15, 0, 15, 0, -1, 1, 0, -21, 0, 35, 0, -7, 0, 1, 0, -28, 0, 70, 0, -28, 0, 1, 1, 0, -36, 0, 126, 0, -84, 0, 9, 0, 1, 0, -45, 0, 210, 0, -210, 0, 45, 0, -1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 05 2008

Polynomials like these are seen in complex dynamics.
This method symmetrically breaks up Pascal's triangle A007318 into two parts as polynomial coefficient vectors. See the examples for the s(n,m) = imaginary part of coefficients(p(x,n)).
From Johannes W. Meijer, Mar 10 2012: (Start)
The row sums equal A146559 and the two antidiagonal sums lead to A104862 (minus a(0)) and A110161 (minus a(0)).
The mirror of this triangle (for the absolute values of the coefficients) is A119467. (End)

			s(n,m) = imaginary part of coefficients(p(x,n))
  {0},
  {0,   1},
  {0,   2,   0},
  {0,   3,   0,   -1},
  {0,   4,   0,   -4,   0},
  {0,   5,   0,  -10,   0,   1},
  {0,   6,   0,  -20,   0,   6,   0},
  {0,   7,   0,  -35,   0,  21,   0,   -1},
  {0,   8,   0,  -56,   0,  56,   0,   -8,   0},
  {0,   9,   0,  -84,   0, 126,   0,  -36,   0,   1},
  {0,  10,   0, -120,   0, 252,   0, -120,   0,  10,   0}

G. C. Greubel, Rows n=0..100 of triangle, flattened
Clark Kimberling, Polynomials associated with reciprocation, JIS 12 (2009) 09.3.4, section 5.

Cf. A007318, A146559, A104862, A110161, A119467.

From Johannes W. Meijer, Mar 10 2012: (Start)
nmax:=10: for n from 0 to nmax do p(x,n) := (1+I*x)^n: for m from 0 to n do t(n,m) := Re(coeff(p(x,n), x, m)) od: od: seq(seq(t(n,m), m=0..n), n=0..nmax);
nmax:=10: for n from 0 to nmax do for m from 0 to n do A119467(n,m) := binomial(n,m) * (1+(-1)^(n-m))/2: if (m mod 4 = 2) then x(n,m):= -1 else x(n,m):= 1 end if: od: od: for n from 0 to nmax do for m from 0 to n do t(n,m) := A119467(n,n-m)*x(n,m) od: od: seq(seq(t(n,m), m=0..n), n=0..nmax); # (End)

p[x_, n_] := If[n == 0, 1, Product[(1 + I*x), {i, 1, n}]]; Table[Expand[p[x, n]], {n, 0, 10}]; Table[Im[CoefficientList[p[x, n], x]], {n, 0, 10}]; Flatten[%] Table[Re[CoefficientList[p[x, n], x]], {n, 0, 10}]; Flatten[%]

p(x,n) = (1+I*x)^n
t(n,m) = real part of coefficients(p(x,n))
s(n,m) = imaginary part of coefficients(p(x,n))

Edited and information added by Johannes W. Meijer, Mar 10 2012

A158499 Expansion of (1 + sqrt(1-4x))/(2-4x).

Original entry on oeis.org

1, 1, 1, 0, -5, -24, -90, -312, -1053, -3536, -11934, -40664, -140114, -488240, -1719380, -6113200, -21921245, -79200160, -288045110, -1053728920, -3874721030, -14313562480, -53093391980, -197669347600, -738398308850, -2766700765024

Offset: 0

Paul Barry, Mar 20 2009

Hankel transform is A056594 with g.f. 1/(1+x^2).
Row sums of the Riordan array (sqrt(1-4*x)/(1-2*x), x*c(x)^2), c(x) the g.f. of A000108.
The inverse Catalan transform yields A146559. - R. J. Mathar, Mar 20 2009

Matthew House, Table of n, a(n) for n = 0..1669
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.

Cf. A000108, A056594, A146559, A158495.

[n le 2 select 1 else (6*(n-2)*Self(n-1) - 4*(2*n-5)*Self(n-2))/(n-1): n in [1..30]]; // G. C. Greubel, Mar 17 2025

CoefficientList[ Series[(1 + Sqrt[1 - 4x])/(2 - 4x), {x, 0, 26}], x] (* Robert G. Wilson v, Nov 08 2015 *)

my(x='x+O('x^33)); Vec(((1-4*x)+sqrt(1-4*x))/(2*(1-2*x)*sqrt(1-4*x))) \\ Altug Alkan, Nov 08 2015

@CachedFunction
def a(n): # a = A158499
    if n<2: return 1
    else: return (6*(n-1)*a(n-1) - 4*(2*n-3)*a(n-2))/n
[a(n) for n in range(41)] # G. C. Greubel, Mar 17 2025

a(n) = Sum_{k=0..n} binomial(2*k,k)*A158495(n-k).

n*a(n) + 6*(1-n)*a(n-1) + 4*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011

Name edited by Matthew House, Nov 08 2015

A221131 Table, T, read by antidiagonals where T(-j,k) = ((1+sqrt(j))^k + (1-sqrt(j))^k)/2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -2, 1, 1, 1, -2, -5, -4, 1, 1, 1, -3, -8, -7, -4, 1, 1, 1, -4, -11, -8, 1, 0, 1, 1, 1, -5, -14, -7, 16, 23, 8, 1, 1, 1, -6, -17, -4, 41, 64, 43, 16, 1, 1, 1, -7, -20, 1, 76, 117, 64, 17, 16, 1, 1, 1, -8, -23, 8, 121, 176, 29, -128, -95, 0, 1

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com) and Robert G. Wilson v, Jan 02 2013

.j\k.........0..1...2....3...4....5....6......7.......8......9......10
.0: A000012..1..1...1....1...1....1....1......1.......1......1.......1
-1: A146559..1..1...0...-2..-4...-4....0......8......16.....16.......0
-2: A087455..1..1..-1...-5..-7....1...23.....43......17....-95....-241
-3: A138230..1..1..-2...-8..-8...16...64.....64....-128...-512....-512
-4: A006495..1..1..-3..-11..-7...41..117.....29....-527..-1199.....237
-5: A138229..1..1..-4..-14..-4...76..176...-104...-1264..-1904....3776
-6: A090592..1..1..-5..-17...1..121..235...-377...-2399..-2159...12475
-7: A090590..1..1..-6..-20...8..176..288...-832...-3968..-1280...29184
-8: A025172..1..1..-7..-23..17..241..329..-1511...-5983...1633...57113
-9: A120743..1..1..-8..-26..28..316..352..-2456...-8432...7696...99712
-10: ........1..1..-9..-29..41..401..351..-3709..-11279..18241..160551

Cf. A084097,
Rows: A000012, A146559, A087455, A138230, A006495, A138229, A090592, A090590, A025172, A120743.
Columns: A000012, A000012, A024000, 1-3n, A028884, A134593.

T[j_, k_] := Expand[((1 + Sqrt[j])^k + (1 - Sqrt[j])^k)/2]; Table[ T[ -j + k, k], {j, 0, 11}, {k, 0, j}] // Flatten

A305708 Expansion of e.g.f. exp(cos(x)/exp(x) - 1).

Original entry on oeis.org

1, -1, 1, 1, -11, 43, -83, -275, 3833, -21561, 51369, 375593, -5860147, 40452371, -101676235, -1409619211, 23912208945, -189650997937, 454996127889, 11250036170129, -204691511497499, 1799897065507003, -3741969787709699, -164548323889940675, 3183842522596250537, -30356999697044585833

Offset: 0

Ilya Gutkovskiy, Jun 08 2018

sign

			exp(cos(x)/exp(x) - 1) = 1 - x + x^2/2! + x^3/3! - 11*x^4/4! + 43*x^5/5! - 83*x^6/6! - 275*x^7/7! + ...

Cf. A004211, A009116, A009216, A009235, A009255, A009276, A146559, A155585.

a:=series(exp(cos(x)/exp(x)-1),x=0,26): seq(n!*coeff(a,x,n),n=0..25); # Paolo P. Lava, Mar 26 2019

nmax = 25; CoefficientList[Series[Exp[Cos[x]/Exp[x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Re[(-1 - I)^k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 25}]

A370359 Imaginary part of (n + n*i)^n where i = sqrt(-1).

Original entry on oeis.org

0, 1, 8, 54, 0, -12500, -373248, -6588344, 0, 6198727824, 320000000000, 9129973459552, 0, -19384006821904192, -1422336873671426048, -56050417968750000000, 0, 211773507042902211629312, 20145360934551827238617088, 1012950863698080557631477248, 0, -5982809106827246101894271407104

Offset: 0

Chai Wah Wu, Feb 16 2024

sign

Cf. A009545, A121625, A146559.

a(n) = imag((n + n*I)^n); \\ Michel Marcus, Feb 16 2024

def A370359(n): return n**n*((0, 1, 2, 2)[n&3]<<((n>>1)&-2))*(-1 if n&4 else 1)

a(n) = n^n*A009545(n) = n^n*Sum_{j=0..floor((n-1)/2)} binomial(n,2*j+1)*(-1)^j.
a(n) = 0 if and only if n == 0 mod 4.
a(4n) = 0.
a(4n+1) = (4n+1)^(4n+1)*(-4)^n.
a(4n+2) = 2*(4n+2)^(4n+2)*(-4)^n.
a(4n+3) = 2*(4n+3)^(4n+3)*(-4)^n.

cp's OEIS Frontend

A106664 Expansion of g.f.: (1-3*x+x^2)/((1-x)*(1+x)*(1-2*x+2*x^2)).

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A141665 A signed half of Pascal's triangle A007318: p(x,n) = (1+I*x)^n; t(n,m) = real part of coefficients(p(x,n)).

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A158499 Expansion of (1 + sqrt(1-4*x))/(2-4*x).

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A221131 Table, T, read by antidiagonals where T(-j,k) = ((1+sqrt(j))^k + (1-sqrt(j))^k)/2.

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Mathematica

A305708 Expansion of e.g.f. exp(cos(x)/exp(x) - 1).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

A370359 Imaginary part of (n + n*i)^n where i = sqrt(-1).

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Author

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PARI

Python

Formula

A106664 Expansion of g.f.: (1-3x+x^2)/((1-x)(1+x)(1-2x+2*x^2)).

A158499 Expansion of (1 + sqrt(1-4x))/(2-4x).