A035045 -id:A035045 - cp's OEIS frontend

A215919 a(n) = -3*a(n-1) + a(n-3), with a(0)=0, a(1)=-3, a(2)=12.

0, -3, 12, -36, 105, -303, 873, -2514, 7239, -20844, 60018, -172815, 497601, -1432785, 4125540, -11879019, 34204272, -98487276, 283582809, -816544155, 2351145189, -6769852758, 19493014119, -56127897168, 161613838746, -465348502119, 1339917609189, -3858138988821

Offset: 0

Roman Witula, Aug 27 2012

The Berndt-type sequence number 10 for the argument 2Pi/9 defined by the first trigonometric relation from the section "Formula" below. The sequence a(n) is connected with sequences A215917 and A215885 - see the respective formula.

We have A035045(n)=abs(a(n+1)/3) for every n=0,1,...,5 and A035045(7) + a(7)/3 = 1, A035045(8) - a(8)/3 = 10, A035045(9) + a(9)/3 = 63, and A035045(10) - a(10)/3 = 320 - all these four results-numbers are in A069269.

			We have a(2)=-4*a(1), a(3)=-3*a(2), a(6)/a(3) = -24.25, and a(9) = 579*a(3).

D. Chmiela and R. Witula, Two parametric quasi-Fibonacci numbers of the ninth order, (submitted, 2012).
R. Witula, Ramanujan type formulas for arguments 2Pi/7 and 2Pi/9, Demonstratio Math. (in press, 2012).

Index entries for linear recurrences with constant coefficients, signature (-3,0,1).

Cf. A215917, A215885, A215664.

LinearRecurrence[{-3, 0, 1}, {0, -3, 12}, 50]

a(n) = c(1)*(-c(2))^(-n) + c(2)*(-c(4))^(-n) + c(4)*(-c(1))^(-n), where c(j) := 2*cos(2*Pi*j/9).
a(n) = A215917(n+1) + A215917(n) - 2*(-1)^n*A215885(n).
G.f.: -3*x*(1-x)/(1+3*x-x^3).

A115990 Riordan array (1/sqrt(1-2x-3x^2), (1-2x-3x^2)/(2(1-3x)) - sqrt(1-2x-3x^2)/2).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 7, 5, 3, 1, 19, 13, 8, 4, 1, 51, 35, 22, 12, 5, 1, 141, 96, 61, 35, 17, 6, 1, 393, 267, 171, 101, 53, 23, 7, 1, 1107, 750, 483, 291, 160, 77, 30, 8, 1, 3139, 2123, 1373, 839, 476, 244, 108, 38, 9, 1, 8953, 6046, 3923, 2423, 1406, 752, 360, 147, 47, 10

Offset: 0

Paul Barry, Feb 10 2006

First column is central trinomial coefficients A002426. Second column is number of directed animals of size n+1, A005773(n+1). Row sums are A005717 (number of horizontal steps in all Motzkin paths of length n). First column has e.g.f. exp(x) I_0(2x). Row sums have e.g.f. dif(exp(x) I_1(2x),x).

Riordan array (1/sqrt(1-2*x-3*x^2), (1+x-sqrt(1-2*x-3*x^2))/2).

			Triangle begins
    1;
    1,  1;
    3,  2,  1;
    7,  5,  3,  1;
   19, 13,  8,  4,  1;
   51, 35, 22, 12,  5,  1;
  141, 96, 61, 35, 17,  6,  1;

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A115991, A005773 (k=1), A025566 (k=2), A035045 (k=3), A152948 (diag. n=k+2), .

Flat(List([0..10], n-> List([0..n], k-> Sum([0..n], j-> Binomial(n-k, j-k)*Binomial(j, n-j)) ))); # G. C. Greubel, May 09 2019

[[(&+[Binomial(n-k, j-k)*Binomial(j, n-j): j in [0..n]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 09 2019

A115990 := proc(n,k)
    add(binomial(n-k,j-k)*binomial(j,n-j),j=0..n) ;
end proc:
seq(seq(A115990(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Jun 25 2023

Table[Sum[ Binomial[n-k, j-k]*Binomial[j, n-j], {j, 0, n}], {n, 0, 10}, {k, 0, n} ] // Flatten (* G. C. Greubel, Mar 07 2017 *)

{T(n, k) = sum(j=0, n, binomial(n-k, j-k)*binomial(j, n-j))}; \\ G. C. Greubel, May 09 2019

[[sum(binomial(n-k, j-k)*binomial(j, n-j) for j in (0..n)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 09 2019

Number triangle T(n,k) = Sum_{j=0..n} C(n-k,j-k)*C(j,n-j).

cp's OEIS Frontend

A215919 a(n) = -3*a(n-1) + a(n-3), with a(0)=0, a(1)=-3, a(2)=12.

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A115990 Riordan array (1/sqrt(1-2x-3x^2), (1-2x-3x^2)/(2(1-3x)) - sqrt(1-2x-3x^2)/2).

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cp's OEIS Frontend

A215919 a(n) = -3*a(n-1) + a(n-3), with a(0)=0, a(1)=-3, a(2)=12.

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Mathematica

Formula

A115990 Riordan array (1/sqrt(1-2*x-3*x^2), (1-2*x-3*x^2)/(2*(1-3*x)) - sqrt(1-2*x-3*x^2)/2).

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Author

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Examples

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Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A115990 Riordan array (1/sqrt(1-2x-3x^2), (1-2x-3x^2)/(2(1-3x)) - sqrt(1-2x-3x^2)/2).