A114123 -id:A114123 - cp's OEIS frontend

A099463 Bisection of tribonacci numbers.

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

A116404 Expansion of (1-x)/((1-x)^2 - x^2*(1+x)^2).

Original entry on oeis.org

1, 1, 2, 6, 15, 35, 84, 204, 493, 1189, 2870, 6930, 16731, 40391, 97512, 235416, 568345, 1372105, 3312554, 7997214, 19306983, 46611179, 112529340, 271669860, 655869061, 1583407981, 3822685022, 9228778026, 22280241075, 53789260175

Offset: 0

Paul Barry, Feb 07 2006

Diagonal sums of number triangle A114123.
Binomial transform of A114122.
Congruent to 1,1,0,0,1,1,... modulo 2.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Vincent Vatter, Growth rates of permutation classes: from countable to uncountable, arXiv:1605.04297 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (2,0,2,1).

Cf. A000129, A056594, A114122, A114123.

I:=[1,1,2,6]; [n le 4 select I[n] else 2*Self(n-1)+2*Self(n-3)+Self(n-4): n in [1..36]]; // Vincenzo Librandi, Aug 19 2017

LinearRecurrence[{2,0,2,1}, {1,1,2,6}, 35] (* Emanuele Munarini, Apr 27 2017 *)
CoefficientList[Series[(1-x)/((1-x)^2 -x^2(1+x)^2), {x, 0, 35}], x] (* Vincenzo Librandi, Aug 19 2017 *)

Vec((1-x)/((1-x)^2-x^2*(1+x)^2) + O(x^40)) \\ Michel Marcus, Aug 19 2017

def A116404(n): return sum( round( hypergeometric([-n+2*k, -2*k], [1], 2) ) for k in (0..n//2) )
[A116404(n) for n in (0..35)] # G. C. Greubel, Nov 20 2021

G.f.: (1-x)/(1 - 2*x - 2*x^3 - x^4).
a(n) = 2*a(n-1) + 2*a(n-3) + a(n-4).
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} C(2*k,n-2*k-j)*C(n-2*k,j)*2^(n-2*k-j).
2*a(n) = A056594(n) + A000129(n+1). - R. J. Mathar, Oct 25 2011
a(n) = Sum_{k=0..floor(n/2)} hypergeometric2F1([-2*k, -n+2*k], [1], 2). - G. C. Greubel, Nov 20 2021

A184883 Number triangle T(n,k) = [k<=n]*Hypergeometric2F1([-k, 2k-2n], [1], 2).

Original entry on oeis.org

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

Offset: 0

Paul Barry, Jan 24 2011

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

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

Cf. A099463 (row sums), A114123, A184884 (diagonal sums).

T:= func< n,k | (&+[Binomial(k,j)*Binomial(2*(n-k), j)*2^j: j in [0..k]]) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 19 2021

A184883[n_, k_]:= Hypergeometric2F1[-k, 2*(k-n), 1, 2];
Table[A184883[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 19 2021 *)

def A184883(n,k): return simplify( hypergeometric([-k, 2*(k-n)], [1], 2) )
flatten([[A184883(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Nov 19 2021

T(n,k) = [k<=n]*Sum_{j=0..k} C(2*n-2*k,j)*C(k,j)*2^j.
T(n, n-k) = A114123(n, k).
Sum_{k=0..n} T(n, k) = A099463(n+1).
Sum_{k=0..floor(n/2)} T(n, k) = A184884(n).
T(n, k) = Hypergeometric2F1([-k, 2*(k-n)], [1], 2). - G. C. Greubel, Nov 19 2021

A216182 Riordan array ((1+x)/(1-x)^2, x(1+x)^2/(1-x)^2).

Original entry on oeis.org

1, 3, 1, 5, 7, 1, 7, 25, 11, 1, 9, 63, 61, 15, 1, 11, 129, 231, 113, 19, 1, 13, 231, 681, 575, 181, 23, 1, 15, 377, 1683, 2241, 1159, 265, 27, 1, 17, 575, 3653, 7183, 5641, 2047, 365, 31, 1, 19, 833, 7183, 19825, 22363, 11969, 3303, 481, 35, 1

Offset: 0

Philippe Deléham, Mar 11 2013

Triangle formed of odd-numbered columns of the Delannoy triangle A008288.

			Triangle begins
   1;
   3,   1;
   5,   7,    1;
   7,  25,   11,    1;
   9,  63,   61,   15,    1;
  11, 129,  231,  113,   19,    1;
  13, 231,  681,  575,  181,   23,   1;
  15, 377, 1683, 2241, 1159,  265,  27,  1;
  17, 575, 3653, 7183, 5641, 2047, 365, 31, 1;
  ...

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

Cf. (columns:) A005408, A001845, A001847, A001849, A008419.
Cf. Diagonals: A000012, A004767, A060820.
Cf. A008288 (Delannoy triangle), A114123 (even-numbered columns of A008288).

A216182[n_, k_]:= Hypergeometric2F1[-n +k, -2*k-1, 1, 2];
Table[A216182[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 19 2021 *)

def A216182(n,k): return simplify( hypergeometric([-n+k, -2*k-1], [1], 2) )
flatten([[A216182(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Nov 19 2021

T(2n, n) = A108448(n+1).
Sum_{k=0..n} T(n,k) = A073717(n+1).
From G. C. Greubel, Nov 19 2021: (Start)
T(n, k) = A008288(n+k+1, 2*k+1).
T(n, k) = hypergeometric([-n+k, -2*k-1], [1], 2). (End)

A112743 An aerated Delannoy triangle.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 1, 0, 5, 0, 1, 0, 5, 0, 7, 0, 1, 1, 0, 13, 0, 9, 0, 1, 0, 7, 0, 25, 0, 11, 0, 1, 1, 0, 25, 0, 41, 0, 13, 0, 1, 0, 9, 0, 63, 0, 61, 0, 15, 0, 1, 1, 0, 41, 0, 129, 0, 85, 0, 17, 0, 1, 0, 11, 0, 129, 0, 231, 0, 113, 0, 19, 0, 1, 1, 0, 61, 0, 321, 0, 377, 0, 145, 0, 21, 0, 1

Offset: 0

Paul Barry, Sep 17 2005

Diagonal sums are aerated Pell numbers.

			Rows begin
  1;
  0,  1;
  1,  0,  1;
  0,  3,  0,  1;
  1,  0,  5,  0,  1;
  0,  5,  0,  7,  0,  1;
  1,  0, 13,  0,  9,  0,  1;
  0,  7,  0, 25,  0, 11,  0,  1;
  1,  0, 25,  0, 41,  0, 13,  0,  1;

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

Cf. A000073, A008288, A114123, A216182.

function T(n,k)
  if k lt 0 or k gt n then return 0;
  elif k eq n then return 1;
  elif k eq 0 then return (1+(-1)^n)/2;
  else return T(n-1,k-1) + T(n-2,k) + T(n-3,k-1);
  end if;
  return T;
end function;
[T(n,k): k in [0..n], n in [0..14]]; // G. C. Greubel, Nov 20 2021

A008288[n_, k_]:= Hypergeometric2F1[-n, -k, 1, 2];
T[n_, k_]:= T[n, k]= (1+(-1)^(n-k))*A008288[(n-k)/2, k]/2;
Table[T[n, k], {n,0,14}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 20 2021 *)

def A008288(n, k): return simplify( hypergeometric([-n, -k], [1], 2) )
def A112743(n, k): return (1 + (-1)^(n-k))*A008288((n-k)/2, k)/2
flatten([[A112743(n,k) for k in (0..n)] for n in (0..14)]) # G. C. Greubel, Nov 20 2021

Riordan array (1/(1-x^2), x*(1+x^2)/(1-x^2)).
T(n,k) = Sum_{j=0..k} (1+(-1)^(n-k))*binomial(k,j)*binomial((n-k)/2,j)*2^(j-1).
Sum_{k=0..n} T(n, k) = A000073(n).
T(n,k) = T(n-1,k-1) + T(n-2,k) + T(n-3,k-1). - Philippe Deléham, Mar 11 2013

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A099463 Bisection of tribonacci numbers.

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A116404 Expansion of (1-x)/((1-x)^2 - x^2*(1+x)^2).

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A184883 Number triangle T(n,k) = [k<=n]*Hypergeometric2F1([-k, 2k-2n], [1], 2).

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A216182 Riordan array ((1+x)/(1-x)^2, x(1+x)^2/(1-x)^2).

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A112743 An aerated Delannoy triangle.

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