A184883 -id:A184883 - cp's OEIS frontend

A184884 Diagonal sums of number triangle A184883.

1, 1, 2, 6, 11, 27, 60, 132, 301, 669, 1502, 3370, 7543, 16919, 37912, 84968, 190457, 426841, 956698, 2144238, 4805827, 10771315, 24141588, 54108332, 121272549, 271806901, 609198390, 1365390546, 3060236911, 6858880431, 15372743856, 34454786384, 77223188593, 173079605553, 387921692082, 869445237846

Offset: 0

Paul Barry, Jan 24 2011

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

Cf. A183883.

A184883:= func< n,k | (&+[Binomial(k,j)*Binomial(2*(n-k), j)*2^j: j in [0..k]]) >;
A184884:= func< n | (&+[A184883(n, j): j in [0..Floor(n/2)]]) >;
[A184884(n): n in [0..40]]; // G. C. Greubel, Nov 19 2021

LinearRecurrence[{1,2,2,-1,1}, {1,1,2,6,11}, 45] (* G. C. Greubel, Nov 19 2021 *)

def A184883(n,k): return simplify( hypergeometric([-k, 2*(k-n)], [1], 2) )
def A184884(n): return sum( A184883(n, j) for j in (0..n//2) )
[A184884(n) for n in (0..40)] # G. C. Greubel, Nov 19 2021

G.f.: (1-x^2)/(1-x-2*x^2-2*x^3+x^4-x^5).
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..k} C(2*n-4*k,j)*C(k,j)*2^j.
a(n) = Sum_{k=0..floor(n/2)} Hypergeometric2F1([-k, 2*(k-n)], [1], 2). - G. C. Greubel, Nov 19 2021

A099463 Bisection of tribonacci numbers.

Original entry on oeis.org

0, 1, 2, 7, 24, 81, 274, 927, 3136, 10609, 35890, 121415, 410744, 1389537, 4700770, 15902591, 53798080, 181997601, 615693474, 2082876103, 7046319384, 23837527729, 80641778674, 272809183135, 922906855808, 3122171529233

Offset: 0

Paul Barry, Oct 16 2004

Binomial transform of A099462.
From Paul Barry, Feb 07 2006: (Start)
a(n+1) gives row sums of number triangle A114123 or A184883.
Partial sums are A113300. (End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 24.
Meng-Han Wu, Henryk A. Witek, and Rafał Podeszwa, Clar Covers and Zhang-Zhang Polynomials of Zigzag and Armchair Carbon Nanotubes, MATCH Commun. Math. Comput. Chem. (2025) Vol. 93, 415-462. See p. 437.
Index entries for linear recurrences with constant coefficients, signature (3,1,1).

Cf. A000073, A058265, A099462, A113300, A114123, A184883.

[n le 3 select (n-1) else 3*Self(n-1) +Self(n-2) +Self(n-3): n in [1..31]]; // G. C. Greubel, Nov 20 2021

LinearRecurrence[{3,1,1},{0,1,2},30] (* or *) Join[{0},Mean/@ Partition[ LinearRecurrence[ {1,1,1},{1,1,1},60],2]] (* Harvey P. Dale, Apr 02 2012 *)

def A184883(n, k): return simplify( hypergeometric([-k, 2*(k-n)], [1], 2) )
def A099463(n): return sum( A184883(n, k) for k in (0..n) )
[0]+[A099463(n-1) for n in (1..40)] # G. C. Greubel, Nov 20 2021

G.f.: x*(1-x)/(1-3*x-x^2-x^3).
a(n) = Sum_{k=0..n} binomial(n, k)*Sum_{j=0..floor((k-1)/2)} binomial(j, k-2*j-1)*4^j.
From Paul Barry, Feb 07 2006: (Start)
a(n) = 3*a(n-1) + a(n-2) + a(n-3).
a(n) = Sum_{k=0..n} Sum_{j=0..n} C(2*k, n-k-j)*C(n-k, j)*2^(n-k-j). (End)
a(n)/a(n-1) tends to 3.38297576..., the square of the tribonacci constant A058265. - Gary W. Adamson, Feb 28 2006
If p[1]=2, p[2]=3, p[i]=4, (i>2), and if A is Hessenberg matrix of order n defined by: A[i,j] = p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n+1) = det A. - Milan Janjic, May 02 2010

A114123 Riordan array (1/(1-x), x*(1+x)^2/(1-x)^2).

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 13, 9, 1, 1, 25, 41, 13, 1, 1, 41, 129, 85, 17, 1, 1, 61, 321, 377, 145, 21, 1, 1, 85, 681, 1289, 833, 221, 25, 1, 1, 113, 1289, 3653, 3649, 1561, 313, 29, 1, 1, 145, 2241, 8989, 13073, 8361, 2625, 421, 33, 1, 1, 181, 3649, 19825, 40081, 36365, 16641, 4089, 545, 37, 1

Offset: 0

Paul Barry, Feb 07 2006, Oct 22 2006

Row sums are A099463(n+1). Diagonal sums are A116404.

Triangle formed of even-numbered columns of the Delannoy triangle A008288. - Philippe Deléham, Mar 11 2013

			Triangle begins
  1;
  1,  1;
  1,  5,   1;
  1, 13,   9,   1;
  1, 25,  41,  13,   1;
  1, 41, 129,  85,  17,  1;
  1, 61, 321, 377, 145, 21, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A008288, A099463 (row sums), A116404 (diagonal sums), A184883.

T:= func< n, k | (&+[Binomial(2*k, j)*Binomial(n-k, j)*2^j: j in [0..n-k]]) >;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2021

T := (n,k) -> hypergeom([-2*k, k-n], [1], 2);
seq(seq(round(evalf(T(n,k),99)),k=0..n),n=0..9); # Peter Luschny, Sep 16 2014

T[n_, k_] := Hypergeometric2F1[-2k, k-n, 1, 2];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* Jean-François Alcover, Jun 13 2019 *)

def A114123(n,k): return round( hypergeometric([-2*k, k-n], [1], 2) )
flatten([[A114123(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Nov 20 2021

T(n, k) = Sum_{j=0..n} C(2*k,n-k-j)*C(n-k,j)*2^(n-k-j).
T(n, k) = Sum_{j=0..n-k} C(2*k,j)*C(n-k,j)*2^j.
Sum_{k=0..n} T(n, k) = A099463(n+1).
Sum_{k=0..floor(n/2)} T(n, k) = A116404(n).
T(n, k) = hypergeom([-2*k, k-n], [1], 2). - Peter Luschny, Sep 16 2014
T(n, n-k) = A184883(n, k). - G. C. Greubel, Nov 20 2021

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A184884 Diagonal sums of number triangle A184883.

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A099463 Bisection of tribonacci numbers.

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A114123 Riordan array (1/(1-x), x*(1+x)^2/(1-x)^2).

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