A114197 -id:A114197 - cp's OEIS frontend

A109906 A triangle based on A000045 and Pascal's triangle: T(n,m) = Fibonacci(n-m+1) * Fibonacci(m+1) * binomial(n,m).

1, 1, 1, 2, 2, 2, 3, 6, 6, 3, 5, 12, 24, 12, 5, 8, 25, 60, 60, 25, 8, 13, 48, 150, 180, 150, 48, 13, 21, 91, 336, 525, 525, 336, 91, 21, 34, 168, 728, 1344, 1750, 1344, 728, 168, 34, 55, 306, 1512, 3276, 5040, 5040, 3276, 1512, 306, 55, 89, 550, 3060, 7560, 13650, 16128, 13650, 7560, 3060, 550, 89

Offset: 0

Roger L. Bagula and Gary W. Adamson, Aug 24 2008

Row sums give A081057.

			Triangle T(n,k) begins:
   1;
   1,   1;
   2,   2,    2;
   3,   6,    6,    3;
   5,  12,   24,   12,     5;
   8,  25,   60,   60,    25,     8;
  13,  48,  150,  180,   150,    48,    13;
  21,  91,  336,  525,   525,   336,    91,   21;
  34, 168,  728, 1344,  1750,  1344,   728,  168,   34;
  55, 306, 1512, 3276,  5040,  5040,  3276, 1512,  306,  55;
  89, 550, 3060, 7560, 13650, 16128, 13650, 7560, 3060, 550, 89;
  ...

Reinhard Zumkeller, Rows n = 0..120 of table, flattened
Peter McCalla, Asamoah Nkwanta, Catalan and Motzkin Integral Representations, arXiv:1901.07092 [math.NT], 2019.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A141611, A141617, A000045, A081057.

Some other Fibonacci-Pascal triangles: A027926, A036355, A037027, A074829, A105809, A111006, A114197, A162741, A228074.

a109906 n k = a109906_tabl !! n !! k
a109906_row n = a109906_tabl !! n
a109906_tabl = zipWith (zipWith (*)) a058071_tabl a007318_tabl
-- Reinhard Zumkeller, Aug 15 2013

f:= n-> combinat[fibonacci](n+1):
T:= (n, k)-> binomial(n, k)*f(k)*f(n-k):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Apr 26 2023

Clear[t, n, m] t[n_, m_] := Fibonacci[(n - m + 1)]*Fibonacci[(m + 1)]*Binomial[n, m]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]

T(n,m) = Fibonacci(n-m+1)*Fibonacci(m+1)*binomial(n,m).

T(n,k) = A058071(n,k) * A007318(n,k). - Reinhard Zumkeller, Aug 15 2013

Offset changed by Reinhard Zumkeller, Aug 15 2013

A114198 a(n) = Sum_{k=0..n} binomial(n,k)^2*F(k+1).

Original entry on oeis.org

1, 2, 7, 31, 142, 659, 3113, 14918, 72199, 351983, 1726022, 8504509, 42070429, 208812722, 1039387519, 5186451311, 25935769702, 129942777227, 652133298421, 3277734587302, 16496741964221, 83129076840317, 419362231888882

Offset: 0

Paul Barry, Nov 16 2005

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Eric Weisstein's World of Mathematics, Legendre Polynomial.

Cf. A000045, A219672, A219673.

a:= proc(n) option remember; `if`(n<4, [1, 2, 7, 31][n+1],
     ((3*(n-1))*(2*n-5)*(13*n^2-26*n+10) *a(n-1)
      -(7*n^2-14*n+6)*(13*n^2-52*n+49) *a(n-2)
      +(n-2)*(182*n^3-819*n^2+1050*n-351) *a(n-3)
      -(n-2)*(n-3)*(13*n^2-26*n+10) *a(n-4))/
      (n*(n-1)*(13*n^2-52*n+49)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 28 2016

FullSimplify@Table[(GoldenRatio^(n - 1) LegendreP[n, Sqrt[5] - 2] - (1 - GoldenRatio)^(n - 1) LegendreP[n, -Sqrt[5] - 2])/Sqrt[5], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 28 2016 *)

a(n) = Sum_{k=0..n} C(n, k)^2 * F(k+1); a(n) = A114197(2n, n).
a(n) = (phi^(n-1) * P_n(sqrt(5)-2) - (1-phi)^(n-1) * P_n(-sqrt(5)-2))/sqrt(5), where phi = (1+sqrt(5))/2, P_n(x) is the Legendre polynomial.
a(n) ~ sqrt((6 + 2*sqrt(5) + sqrt(2*(29 + 13*sqrt(5))))/10)/2 * ((3 + sqrt(5))/2 + sqrt(2*(1+sqrt(5))))^n / sqrt(Pi*n). - Vaclav Kotesovec, May 06 2017
a(n) ~ sqrt(2*phi^2 + phi^(7/2)) * (2*phi^(1/2) + phi^2)^n / (2*sqrt(5*Pi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 22 2017
D-finite with recurrence +n*(n-1)*a(n) -5*n*(n-1)*a(n-1) +2*(-n^2+17*n-27)*a(n-2) +(11*n^2-135*n+270)*a(n-3) +2*(-17*n^2+121*n-215)*a(n-4) +(n-4)*(43*n-191)*a(n-5) -3*(n-4)*(n-5)*a(n-6)=0. - R. J. Mathar, May 11 2022

A114199 Row sums of a Pascal-Fibonacci triangle.

Original entry on oeis.org

1, 2, 4, 8, 17, 38, 87, 200, 458, 1044, 2373, 5388, 12233, 27782, 63112, 143392, 325805, 740266, 1681935, 3821412, 8682310, 19726316, 44818473, 101828344, 231355953, 525645354, 1194276812, 2713420728, 6164945513, 14006877390

Offset: 0

Paul Barry, Nov 16 2005

Binomial transform of double Fibonacci sequence A103609(n+2). Row sums of A114197.

Michael De Vlieger, Table of n, a(n) for n = 0..2806
Sergio Falcón, Binomial Transform of the Generalized k-Fibonacci Numbers, Communications in Mathematics and Applications (2019) Vol. 10, No. 3, 643-651.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2,1).

Cf. A000045, A103609, A114197.

[n le 4 select 2^(n-1) else 4*Self(n-1) -5*Self(n-2) +2*Self(n-3) +Self(n-4): n in [1..30]]; // G. C. Greubel, Oct 23 2024

LinearRecurrence[{4,-5,2,1},{1,2,4,8},30] (* Harvey P. Dale, Dec 07 2015 *)

@CachedFunction # a = A114199
def a(n): return 2^n if n<4 else 4*a(n-1) -5*a(n-2) +2*a(n-3) +a(n-4)
[a(n) for n in range(71)] # G. C. Greubel, Oct 23 2024

G.f.: (1-x)^2/(1-4*x+5*x^2-2*x^3-x^4).
a(n) = Sum_{k=0..n} Sum_{j=0..n-k} C(n-k, j)*C(k, j)*Fibonacci(j).
a(n) = Sum_{k=0..n} C(n, k)*Fibonacci(floor((k+2)/2)).

cp's OEIS Frontend

A109906 A triangle based on A000045 and Pascal's triangle: T(n,m) = Fibonacci(n-m+1) * Fibonacci(m+1) * binomial(n,m).

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A114198 a(n) = Sum_{k=0..n} binomial(n,k)^2*F(k+1).

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A114199 Row sums of a Pascal-Fibonacci triangle.

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