A195339 -id:A195339 - cp's OEIS frontend

A195350 Expansion of (1 - 3x - x^2)/(1 - 4x + 2*x^3 + x^4).

1, 1, 3, 10, 37, 141, 541, 2080, 8001, 30781, 118423, 455610, 1752877, 6743881, 25945881, 99822160, 384048001, 1477556361, 5684635243, 21870622810, 84143330517, 323726495221, 1245480100021, 4791763116240, 18435456144001, 70927137880741

Offset: 0

Bruno Berselli, Sep 16 2011

Rewrite the Girard-Waring formulae to express the mean powers in terms of the mean symmetric functions of the data values; the results are polynomials in the mean symmetric polynomials, indexed by the power n. Then for 3 data points, the sum of the positive coefficients in the n-th such polynomial is a(n). a(n+1)/a(n) approaches 1/(2^(1/3)-1). See extended comment in A301417. - Gregory Gerard Wojnar, Mar 19 2018

Bruno Berselli, Table of n, a(n) for n = 0..1000
G. G. Wojnar, D. S. Wojnar, and L. Q. Brin, Universal peculiar linear mean relationships in all polynomials, arXiv:1706.08381 [math.GM], 2017. See Table GW. n=3 p. 22.
Index entries for linear recurrences with constant coefficients, signature (4,0,-2,-1).

Cf. A185962 (gives the coefficients of numerator and denominator of the g.f., row 4 and 5 of its triangular array). Sequences likewise related to A185962: A000012 (row 1 and 2), A001333 (row 2 and 3) and A006190 (row 3 and 4).

Cf. also A195339, A301417, A301420, A301421, A301424, A302764, A301483, A303647.

m:=26; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-3*x-x^2)/(1-4*x+2*x^3+x^4)));

[seq(coeftayl((1-3*x-x^2)/(1-4*x+2*x^3+x^4), x = 0, k), k=0..25)]; # Muniru A Asiru, Mar 20 2018

CoefficientList[Series[(1 - 3 x - x^2)/(1 - 4 x + 2 x^3 + x^4), {x, 0, 25}], x] (* Vincenzo Librandi, Mar 26 2013 *)

makelist(coeff(taylor((1-3*x-x^2)/(1-4*x+2*x^3+x^4), x, 0, n), x, n), n, 0, 25);

Vec((1-3*x-x^2)/(1-4*x+2*x^3+x^4)+O(x^26))

G.f.: (1-3*x-x^2)/((1-x)*(1-3*x-3*x^2-x^3)).
a(n) = 4*a(n-1) - 2*a(n-3) - a(n-4).
a(n) = A301483(n) - A303647(n-2) + A195339(n-4) (conjectured). - Gregory Gerard Wojnar, Apr 27 2018

A185962 Riordan array ((1-x)^2/(1-x+x^2), x(1-x)^2/(1-x+x^2)).

Original entry on oeis.org

1, -1, 1, -1, -2, 1, 0, -1, -3, 1, 1, 2, 0, -4, 1, 1, 3, 5, 2, -5, 1, 0, 0, 3, 8, 5, -6, 1, -1, -4, -6, -1, 10, 9, -7, 1, -1, -4, -10, -16, -10, 10, 14, -8, 1, 0, 1, 0, -10, -26, -24, 7, 20, -9, 1, 1, 6, 15, 20, 5, -30, -42, 0, 27, -10, 1

Offset: 0

Paul Barry, Feb 07 2011

Riordan array (g(x),xg(x)) where g(x)=(1-x)(1-x^2)(1-x^3)/(1-x^6).
Inverse is A185967. Row sums are A185963.
Diagonal sums are A185964. Central coefficients are A185965.
Subtriangle of the triangle given by (0, -1, 2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 19 2012

			Triangle begins:
   1;
  -1,  1;
  -1, -2,   1;
   0, -1,  -3,   1;
   1,  2,   0,  -4,   1;
   1,  3,   5,   2,  -5,   1;
   0,  0,   3,   8,   5,  -6,   1;
  -1, -4,  -6,  -1,  10,   9,  -7,  1;
  -1, -4, -10, -16, -10,  10,  14, -8,  1;
   0,  1,   0, -10, -26, -24,   7, 20, -9,   1;
   1,  6,  15,  20,   5, -30, -42,  0, 27, -10, 1;
  ...
From _Philippe Deléham_, Mar 19 2012: (Start)
(0, -1, 2, -1/2, 1/2, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins:
  1;
  0,  1;
  0, -1,  1;
  0, -1, -2,  1;
  0,  0, -1, -3,  1;
  0,  1,  2,  0, -4,  1;
  0,  1,  3,  5,  2, -5,  1;
  ... (End)

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

Cf. A195339, A195350.

CoefficientList[CoefficientList[Series[1/(1 - y*x + x/(1 - x)^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Jul 23 2017 *)

T(n,k) = Sum_{i=0..(2*k+2)} C(2*k+2,i)*Sum_{j=0..(n-k-i)} C(k+j,j)*C(j,n-k-i-j)*(-1)^(n-k-j).
G.f.: 1/(1-y*x+x/(1-x)^2). - Philippe Deléham, Feb 07 2012
T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k) - 2*T(n-2,k-1) + T(n-3,k-1), T(0,0) = T(1,1) = T(2,2) = 1, T(1,0) = T(2,0) = -1, T(2,1) = -2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham , Nov 11 2013

A301483 a(n) = floor(a(n-1)/(2^(1/3)-1)) with a(1)=1.

Original entry on oeis.org

1, 3, 11, 42, 161, 619, 2381, 9160, 35241, 135583, 521631, 2006882, 7721121, 29705639, 114287161, 439699520, 1691665681, 6508382763, 25039844851, 96336348522, 370636962881, 1425959779059, 5486126574341, 21106896023080, 81205027571321, 312421897357543

Offset: 1

Gregory Gerard Wojnar, Mar 22 2018

nonn

a(n+1)/a(n) approaches 1/(2^(1/3)-1).

Cf. A024537, A195350 (also has 1/(2^(1/3)-1) ratio), A303647.

[n le 1 select 1 else Floor(Self(n-1)/(2^(1/3)-1)): n in [1..30]]; // Vincenzo Librandi, Apr 04 2018

a:=proc(n) option remember;
   if n<1 then 0  else if n=1 then 1 else floor(a(n-1)/(2^(1/3)-1))
end if end if end proc:
seq(a(n), n=1..25);

RecurrenceTable[{a[1]==1, a[n]==Floor[a[n-1]/(2^(1/3)-1)]}, a, {n, 30}] (* Vincenzo Librandi, Apr 04 2018 *)

a=vector(50); a[1]=1; for(n=2, #a, a[n]=a[n-1]\(2^(1/3)-1)); a \\ Altug Alkan, Mar 22 2018

Conjectures from Colin Barker, Apr 01 2018: (Start)
G.f.: x*(1 - x - x^2) / ((1 - x)*(1 - 3*x - 3*x^2 - x^3)).
a(n) = 4*a(n-1) - 2*a(n-3) - a(n-4) for n>4.
(End)
a(n) = A195350(n) + A303647(n-2) - A195339(n-4) (conjectured).

A303647 a(n) = ceiling(a(n-1)/(2^(1/3)-1)+1), a(1)=1.

Original entry on oeis.org

1, 5, 21, 82, 317, 1221, 4699, 18080, 69561, 267625, 1029641, 3961362, 15240637, 58635641, 225590199, 867918160, 3339160721, 12846826845, 49425880861, 190157283842, 731596320957, 2814686695261, 10829006332499, 41662675404240, 160289731905481, 616686228261665

Offset: 1

Gregory Gerard Wojnar, Apr 27 2018

nonn

Cf. A195350, A301483, A195339.

a := proc(n) option remember;
       if n<1 then 0 else
       if n=1 then 1 else ceil(a(n-1)/(2^(1/3)-1)+1)
     end if end if end proc;
seq(a(n), n=0..10);

Nest[Append[#, Ceiling[#[[-1]]/(2^(1/3) - 1) + 1]] &, {1}, 25] (* or *)
Rest@ CoefficientList[Series[x (1 + x + x^2)/((1 - x) (1 - 3 x - 3 x^2 - x^3)), {x, 0, 25}], x] (* Michael De Vlieger, Apr 28 2018 *)

a(n) = if (n==1, 1, ceil(a(n-1)/(2^(1/3)-1)+1)); \\ Michel Marcus, Apr 28 2018

a(n) = A301483(n+2) - A195350(n+2) + A195339(n-2) (conjectured).
Conjectures from Colin Barker, Apr 28 2018: (Start)
G.f.: x*(1 + x + x^2) / ((1 - x)*(1 - 3*x - 3*x^2 - x^3)).
a(n) = 4*a(n-1) - 2*a(n-3) - a(n-4) for n>4.
(End)

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A195350 Expansion of (1 - 3*x - x^2)/(1 - 4*x + 2*x^3 + x^4).

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A185962 Riordan array ((1-x)^2/(1-x+x^2), x(1-x)^2/(1-x+x^2)).

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A301483 a(n) = floor(a(n-1)/(2^(1/3)-1)) with a(1)=1.

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A303647 a(n) = ceiling(a(n-1)/(2^(1/3)-1)+1), a(1)=1.

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A195350 Expansion of (1 - 3x - x^2)/(1 - 4x + 2*x^3 + x^4).