A165201 -id:A165201 - cp's OEIS frontend

A236830 Riordan array (1/(1-xC(x)^3), xC(x)), C(x) the g.f. of A000108.

1, 1, 1, 4, 2, 1, 16, 7, 3, 1, 65, 27, 11, 4, 1, 267, 108, 43, 16, 5, 1, 1105, 440, 173, 65, 22, 6, 1, 4597, 1812, 707, 267, 94, 29, 7, 1, 19196, 7514, 2917, 1105, 398, 131, 37, 8, 1, 80380, 31307, 12111, 4597, 1680, 575, 177, 46, 9, 1, 337284, 130883, 50503, 19196, 7085, 2488, 808, 233, 56, 10, 1

Offset: 0

Philippe Deléham, Feb 01 2014

T(n+3,n) = A011826(n+5).

			Triangle begins:
      1;
      1,    1;
      4,    2,    1;
     16,    7,    3,    1;
     65,   27,   11,    4,   1;
    267,  108,   43,   16,   5,   1;
   1105,  440,  173,   65,  22,   6,  1;
   4597, 1812,  707,  267,  94,  29,  7, 1;
  19196, 7514, 2917, 1105, 398, 131, 37, 8, 1;
Production matrix is:
   1  1
   3  1   1
   6  1   1   1
  10  1   1   1   1
  15  1   1   1   1   1
  21  1   1   1   1   1   1
  28  1   1   1   1   1   1   1
  36  1   1   1   1   1   1   1   1
  45  1   1   1   1   1   1   1   1   1
  55  1   1   1   1   1   1   1   1   1   1
  66  1   1   1   1   1   1   1   1   1   1   1
  78  1   1   1   1   1   1   1   1   1   1   1   1
  91  1   1   1   1   1   1   1   1   1   1   1   1   1
  ...

G. C. Greubel, Rows n= 0..100 of triangle, flattened

Cf. (columns): A165201, A026726, A026671, A026674, A026672, A026675, A026673, A026843, A026842 or A026846 or A026849, A026844, A026841 or A026848.

Flat(List([0..12], n-> List([0..n], k-> Sum([0..n-k], j-> Binomial(2*n-k-j-2, n-k-j)*Fibonacci(2*j-1) )))); # G. C. Greubel, Jul 18 2019

[(&+[Binomial(2*n-k-j-2, n-k-j)*Fibonacci(2*j-1): j in [0..n-k]]): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 18 2019

A236830 := (n,k) -> add(combinat:-fibonacci(2*i-1)*binomial(2*n-2-k-i,n-k-i), i = 0..n-k): seq(seq(A236830(n, k), k = 0..n), n = 0..10); # Peter Bala, Feb 18 2018

(* The function RiordanArray is defined in A256893. *)
c[x_] := (1 - Sqrt[1 - 4 x])/(2 x);
RiordanArray[1/(1 - # c[#]^3)&, # c[#]&, 11] // Flatten (* Jean-François Alcover, Jul 16 2019 *)
Table[Sum[Binomial[2*n-k-j-2, n-k-j]*Fibonacci[2*j-1], {j,0,n-k}], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jul 18 2019 *)

T(n,k) = sum(j=0,n-k, binomial(2*n-k-j-2, n-k-j)*fibonacci(2*j -1));
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jul 18 2019

[[sum( binomial(2*n-k-j-2, n-k-j)*fibonacci(2*j -1) for j in (0..n-k) ) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jul 18 2019

Sum_{k=0..n} T(n,k) = A026726(n).
G.f.: 1/((x^2*C(x)^4-x*C(x))*y-x*C(x)^3+1), where C(x) the g.f. of A000108. - Vladimir Kruchinin, Apr 22 2015
From Peter Bala, Feb 18 2018: (Start)
T(n,k) = Sum_{i = 0..n-k} Fibonacci(2*i-1)*binomial(2*n-2-k-i,n-k-i).
The n-th row polynomial of row reverse triangle is the n-th degree Taylor polynomial of the rational function (1 - 3*x + 2*x^2)/(1 - 3*x + x^2) * 1/(1 - x)^n about 0. For example, for n = 4, (1 - 3*x + 2*x^2)/(1 - 3*x + x^2) * 1/(1 - x)^4 = 1 + 4*x + 11*x^2 + 27*x^3 + 65*x^4 + O(x^5), giving row 4 as (65, 27, 11, 4, 1). (End)

A165202 Expansion of (1+x)/(1 - x + x^2)^2.

Original entry on oeis.org

1, 3, 3, -1, -6, -6, 1, 9, 9, -1, -12, -12, 1, 15, 15, -1, -18, -18, 1, 21, 21, -1, -24, -24, 1, 27, 27, -1, -30, -30, 1, 33, 33, -1, -36, -36, 1, 39, 39, -1, -42, -42, 1, 45, 45, -1, -48, -48, 1, 51, 51, -1, -54, -54, 1, 57, 57, -1, -60, -60, 1

Offset: 0

Paul Barry, Sep 07 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-3,2,-1).

Cf. A100050 (first differences).

Hankel transform of A165201.

a:=[1,3,3,-1];; for n in [5..70] do a[n]:=2*a[n-1]-3*a[n-2]+ 2*a[n-3] -a[n-4]; od; a; # G. C. Greubel, Jul 18 2019

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1+x)/(1-x+x^2)^2 )); // G. C. Greubel, Jul 18 2019

LinearRecurrence[{2,-3,2,-1}, {1,3,3,-1}, 70] (* G. C. Greubel, Jul 18 2019 *)
(-1)^Quotient[#-1,3]{1,1+#,#}[[Mod[#,3,1]]]&/@Range[0, 10] (* Federico Provvedi, Jul 18 2021 *)

my(x='x+O('x^70)); Vec((1+x)/(1-x+x^2)^2) \\ G. C. Greubel, Jul 18 2019

((1+x)/(1-x+x^2)^2).series(x, 70).coefficients(x, sparse=False) # G. C. Greubel, Jul 18 2019

a(n) = cos(Pi*n/3) + sin(Pi*n/3)*(2n/3 + 1)*sqrt(3).

a(n) = A099254(n) + A099254(n-1). - R. J. Mathar, May 02 2013

A108080 a(n) = Sum_{i=0..n} binomial(2*n+i,n-i).

Original entry on oeis.org

1, 3, 12, 50, 211, 895, 3805, 16193, 68940, 293526, 1249622, 5318976, 22634700, 96296410, 409573584, 1741574006, 7403616923, 31466106703, 133704121665, 568008916093, 2412570019447, 10245302874071, 43500597657111, 184670002546295, 783850164628721, 3326671128027805, 14116630429874265

Offset: 0

Ralf Stephan, Jun 03 2005

nonn

Apparently a bisection of A026847.

Row sums of A159965. - Paul Barry, Apr 28 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000108, A026847, A159965, A165201.

CoefficientList[Series[x/(x*Sqrt[1-4*x]-(1-2*x-(1-3*x)*(1-Sqrt[1-4*x])/(2*x))), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *)

x='x+O('x^66); Vec(x/(x*sqrt(1-4*x)-(1-2*x-(1-3*x)*(1-sqrt(1-4*x))/(2*x)))) \\ Joerg Arndt, May 15 2013

From Paul Barry, Apr 28 2009: (Start)
G.f.: x/(x*sqrt(1-4x)-(1-2x-(1-3x)*c(x))), c(x) the g.f. of A000108.
a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n+k,j-k)*C(n,j). (End)
From Paul Barry, Sep 07 2009: (Start)
G.f.: (1/sqrt(1-4x))*(1/(1-xc(x)^3)), c(x) the g.f. of A000108.
a(n) = Sum_{k=0..n} C(2n,n-k)*F(k+1) = Sum_{k=0..n} C(2n,k)*F(n-k+1).
a(n) = Sum_{k=0..n} C(2k,k) * A165201(n-k). (End)
From Vaclav Kotesovec, Oct 24 2012: (Start)
Recurrence: n*(17*n-93)*a(n) = 4*(34*n^2 - 189*n + 98)*a(n-1) - 5*(51*n^2 - 271*n + 252)*a(n-2) - 4*(17*n^2 - 184*n + 406)*a(n-3) + 44*(2*n-7) * a(n-4).
a(n) ~ 1/2*(1+1/sqrt(5))*(sqrt(5)+2)^n. (End)
a(n) = binomial(2*n, n)*hypergeom([1, -n, 1+2*n], [(1+n)/2, 1+n/2], -1/4). - Stefano Spezia, Jun 17 2025

A165203 Expansion of (1+x)c(x)^3/(1-xc(x)^3), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 5, 20, 81, 332, 1372, 5702, 23793, 99576, 417664, 1754866, 7383204, 31096466, 131084954, 552969854, 2334012425, 9856336324, 41639407776, 175971686398, 743888534968, 3145439344550, 13302946909338, 56272308538682

Offset: 0

Paul Barry, Sep 07 2009

Hankel transform is A165204.

Iain Fox, Table of n, a(n) for n = 0..1595 (first 201 terms from Vincenzo Librandi)

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1+x)*(1-Sqrt(1-4*x))^3/(x*(8*x^2 - (1-Sqrt(1-4*x))^3)) )); // G. C. Greubel, Jul 18 2019

Rest[CoefficientList[Series[(1+x)*((1-x)*Sqrt[1-4*x]+5*x-1)/(2*(1-4*x-x^2)), {x, 0, 30}], x]] (* Vaclav Kotesovec, Feb 01 2014 *)

first(n) = x='x+O('x^(n+1)); Vec((1+x)*((1-x)*sqrt(1-4*x)+5*x-1)/(2*(1-4*x-x^2))) \\ Iain Fox, Feb 27 2018

((1+x)*(1-sqrt(1-4*x))^3/(x*(8*x^2 - (1-sqrt(1-4*x))^3)) ).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jul 18 2019

G.f. (for offset 1): (1+x)*((1-x)*sqrt(1-4*x)+5*x-1)/(2*(1-4*x-x^2)).
a(n) = (A165201(n) - 0^n) + A165201(n+1).
Conjecture: (n+1)*(5*n-31)*a(n) +(5*n^2+74*n+62)*a(n-1) +(-285*n^2+ 1072*n-757)*a(n-2) +(695*n^2-3674*n+4206)*a(n-3) +2*(45*n-74)*(2*n-7)*a(n-4)=0. - R. J. Mathar, Dec 11 2011
a(n) ~ (18/sqrt(5)-8) * (2+sqrt(5))^(n+2). - Vaclav Kotesovec, Feb 01 2014

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A236830 Riordan array (1/(1-x*C(x)^3), x*C(x)), C(x) the g.f. of A000108.

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A165202 Expansion of (1+x)/(1 - x + x^2)^2.

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A108080 a(n) = Sum_{i=0..n} binomial(2*n+i,n-i).

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A165203 Expansion of (1+x)*c(x)^3/(1-x*c(x)^3), c(x) the g.f. of A000108.

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A236830 Riordan array (1/(1-xC(x)^3), xC(x)), C(x) the g.f. of A000108.

A165203 Expansion of (1+x)c(x)^3/(1-xc(x)^3), c(x) the g.f. of A000108.