A104495 -id:A104495 - cp's OEIS frontend

A099602 Triangle, read by rows, such that row n equals the inverse binomial transform of column n of the triangle of trinomial coefficients (A027907), omitting leading zeros.

1, 1, 1, 1, 2, 1, 2, 5, 4, 1, 1, 5, 8, 5, 1, 3, 13, 22, 18, 7, 1, 1, 9, 26, 35, 24, 8, 1, 4, 26, 70, 101, 84, 40, 10, 1, 1, 14, 61, 131, 160, 116, 49, 11, 1, 5, 45, 171, 363, 476, 400, 215, 71, 13, 1, 1, 20, 120, 363, 654, 752, 565, 275, 83, 14, 1, 6, 71, 356, 1017, 1856, 2282, 1932

Offset: 0

Paul D. Hanna, Oct 25 2004

Row sums form A099603, where A099603(n) = Fibonacci(n+1)*2^floor((n+1)/2). Central coefficients of even-indexed rows form A082759, where A082759(n) = Sum_{k=0..n} binomial(n,k)*trinomial(n,k). Antidiagonal sums form A099604.

Matrix inverse equals triangle A104495, which is generated from self-convolutions of the Catalan sequence (A000108).

			Rows begin:
  1;
  1,  1;
  1,  2,   1;
  2,  5,   4,   1;
  1,  5,   8,   5,   1;
  3, 13,  22,  18,   7,   1;
  1,  9,  26,  35,  24,   8,   1;
  4, 26,  70, 101,  84,  40,  10,   1;
  1, 14,  61, 131, 160, 116,  49,  11,  1;
  5, 45, 171, 363, 476, 400, 215,  71, 13,  1;
  1, 20, 120, 363, 654, 752, 565, 275, 83, 14, 1;
  ...
The binomial transform of row 2 = column 2 of A027907: BINOMIAL[1,2,1] = [1,3,6,10,15,21,28,36,45,55,...].
The binomial transform of row 3 = column 3 of A027907: BINOMIAL[2,5,4,1] = [2,7,16,30,50,77,112,156,210,...].
The binomial transform of row 4 = column 4 of A027907: BINOMIAL[1,5,8,5,1] = [1,6,19,45,90,161,266,414,615,...].
The binomial transform of row 5 = column 5 of A027907: BINOMIAL[3,13,22,18,7,1] = [3,16,51,126,266,504,882,1452,...].

Cf. A027907, A082759, A099603, A099604.

Cf. A104495.

{T(n,k)=polcoeff(polcoeff((1+(y+1)*x-(y+1)*x^2)/(1-(y+1)*(y+2)*x^2+(y+1)^2*x^4)+x*O(x^n),n,x)+y*O(y^k),k,y)}

{T(n,k)=(matrix(n+1,n+1,i,j,if(i>=j,polcoeff(polcoeff( (1+x*y/(1+x))/(1+x-y^2*(1-(1+4*x+O(x^i))^(1/2))^2/4+O(y^j)),i-1,x),j-1,y)))^-1)[n+1,k+1]}

G.f.: (1 + (y+1)*x - (y+1)*x^2)/(1 - (y+1)*(y+2)*x^2 + (y+1)^2*x^4).

A104496 Expansion of 2(2x+1)/((x+1)(sqrt(4x+1)+1)).

Original entry on oeis.org

1, 0, 0, -1, 5, -19, 67, -232, 804, -2806, 9878, -35072, 125512, -452388, 1641028, -5986993, 21954973, -80884423, 299233543, -1111219333, 4140813373, -15478839553, 58028869153, -218123355523, 821908275547, -3104046382351, 11747506651599, -44546351423299, 169227201341651

Offset: 0

Paul D. Hanna, Mar 11 2005

Previous name was: Row sums of triangle A104495. A104495 equals the matrix inverse of triangle A099602, where row n of A099602 equals the inverse Binomial transform of column n of the triangle of trinomial coefficients (A027907).

Absolute row sums of triangle A104495 forms A014137 (partial sums of Catalan numbers).

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A104495, A099602, A027907, A000108.

gf := (2*(2*x+1))/((x+1)*(sqrt(4*x+1)+1)): ser := series(gf,x,30):
seq(coeff(ser,x,n),n=0..28); # Peter Luschny, Apr 25 2016

CoefficientList[Series[(1+2*x)/(1+x)/(1+x - (1-(1+4*x)^(1/2))^2/4), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 06 2014 *)

{a(n)=local(X=x+x*O(x^n));polcoeff( (1+2*X)/(1+X)/(1+X-(1-(1+4*X)^(1/2))^2/4),n,x)}

from itertools import accumulate
def A104496_list(size):
    if size < 1: return []
    L, accu = [1], [1]
    for n in range(size-1):
        accu = list(accumulate(accu + [-accu[0]]))
        L.append(-(-1)**n*accu[-1])
    return L
print(A104496_list(29)) # Peter Luschny, Apr 25 2016

G.f.: A(x) = (1 + 2*x)/(1+x)/(1+x - x^2*Catalan(-x)^2), where Catalan(x)=(1-(1-4*x)^(1/2))/(2*x) (cf. A000108).
a(n) ~ (-1)^n * 2^(2*n+1) / (3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 06 2014
D-finite with recurrence: (n+1)*a(n) +(7*n-3)*a(n-1) +2*(7*n-12)*a(n-2) +4*(2*n-5)*a(n-3)=0. - R. J. Mathar, Jan 23 2020

New name using the g.f. of the author by Peter Luschny, Apr 25 2016

cp's OEIS Frontend

A099602 Triangle, read by rows, such that row n equals the inverse binomial transform of column n of the triangle of trinomial coefficients (A027907), omitting leading zeros.

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A104496 Expansion of 2(2x+1)/((x+1)(sqrt(4x+1)+1)).

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cp's OEIS Frontend

A099602 Triangle, read by rows, such that row n equals the inverse binomial transform of column n of the triangle of trinomial coefficients (A027907), omitting leading zeros.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

PARI

Formula

A104496 Expansion of 2*(2*x+1)/((x+1)*(sqrt(4*x+1)+1)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A104496 Expansion of 2(2x+1)/((x+1)(sqrt(4x+1)+1)).