A112477 -id:A112477 - cp's OEIS frontend

A112478 Expansion of (1 + x + sqrt(1 + 6*x + x^2))/2.

1, 2, -2, 6, -22, 90, -394, 1806, -8558, 41586, -206098, 1037718, -5293446, 27297738, -142078746, 745387038, -3937603038, 20927156706, -111818026018, 600318853926, -3236724317174, 17518619320890, -95149655201962, 518431875418926, -2832923350929742, 15521467648875090

Offset: 0

Paul Barry, Sep 07 2005

This is the A-sequence for the Delannoy triangle A008288. See the W. Lang link under A006232 for Sheffer a- and z-sequences where also Riordan A- and Z-sequences are explained. O.g.f. A(y) = y/Finv(y) = 2*y/(-(1 + y) + sqrt(y^2 + 6*y + 1)) = ((1 + y) + sqrt(1 + 6*y + y^2))/2 with Finv the inverse function of F(x) = x*(1 + x)/(1 - x). The o.g.f. of the Z-sequence is 1.

			G.f. = 1 + 2*x - 2*x^2 + 6*x^3 - 22*x^4 + 90*x^5 - 394*x^6 + 1806*x^7 + ...

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

A minor variation of A006318. See A085403 for yet another version.
Row sums of number triangle A112477.
Cf. A364393, A364407, A364408, A364409, A366266, A366267, A366268.
Cf. A366325.

CoefficientList[Series[(1+x+Sqrt[1+6*x+x^2])/2, {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)

{a(n) = polcoeff((1 + x + sqrt(1 + 6*x + x^2 + x*O(x^n)))/2, n)}; /* Michael Somos, Jul 07 2020 */

G.f.: (1 + x + sqrt(1 + 6*x + x^2))/2. - Sergei N. Gladkovskii, Jan 04 2012
G.F.: G(0) where G(k)= 1 + x + x/G(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Jan 04 2012
D-finite with recurrence: n*a(n) + 3*(2*n-3)*a(n-1) + (n-3)*a(n-2) = 0. - R. J. Mathar, Nov 24 2012
a(n) ~ (-1)^(n+1) * sqrt(3*sqrt(2) - 4) * (3 + 2*sqrt(2))^n / (2 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 12 2014
0 = a(n)*(a(n+1) + 15*a(n+2) + 4*a(n+3)) + a(n+1)*(-3*a(n+1) + 34*a(n+3) + 15*a(n+3)) + a(n+2)*(-3*a(n+2) + a(n+3)) for all integer n > 0. - Michael Somos, Jul 07 2020
From Seiichi Manyama, Oct 08 2023: (Start)
G.f. satisfies A(x) = 1 + x + x/A(x).
a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(2*k-1,k) * binomial(n+k-2,n-k)/(2*k-1). (End)

A112475 Riordan array (1/(1+x),x(1+x)/(1-x)).

Original entry on oeis.org

1, -1, 1, 1, 1, 1, -1, 1, 3, 1, 1, 1, 5, 5, 1, -1, 1, 7, 13, 7, 1, 1, 1, 9, 25, 25, 9, 1, -1, 1, 11, 41, 63, 41, 11, 1, 1, 1, 13, 61, 129, 129, 61, 13, 1, -1, 1, 15, 85, 231, 321, 231, 85, 15, 1, 1, 1, 17, 113, 377, 681, 681, 377, 113, 17, 1

Offset: 0

Paul Barry, Sep 07 2005

Equivalent to Delannoy triangle A008288 with prepended column 1,-1,1,-1,... Row sums are A111954. Diagonal sums are A112476. Inverse is A112477.

			Triangle starts:
   1;
  -1, 1;
   1, 1, 1;
  -1, 1, 3,  1;
   1, 1, 5,  5,  1;
  -1, 1, 7, 13,  7, 1;
   1, 1, 9, 25, 25, 9, 1;
  ...

Cf. A008288, A111954, A112476, A112477.

T[n_,k_]:=SeriesCoefficient[(x(1+x)/(1-x))^k/(1+x),{x,0,n}]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten (* Stefano Spezia, May 26 2024 *)

T(n,k) = Sum{j=0..n-k} C(k-1,j)*C(n-j-1,n-k-j).

cp's OEIS Frontend

A112478 Expansion of (1 + x + sqrt(1 + 6*x + x^2))/2.

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