A081570 -id:A081570 - cp's OEIS frontend

A094441 Triangular array T(n,k) = Fibonacci(n+1-k)*C(n,k), 0 <= k <= n.

1, 1, 1, 2, 2, 1, 3, 6, 3, 1, 5, 12, 12, 4, 1, 8, 25, 30, 20, 5, 1, 13, 48, 75, 60, 30, 6, 1, 21, 91, 168, 175, 105, 42, 7, 1, 34, 168, 364, 448, 350, 168, 56, 8, 1, 55, 306, 756, 1092, 1008, 630, 252, 72, 9, 1, 89, 550, 1530, 2520, 2730, 2016, 1050, 360, 90, 10, 1

Offset: 0

Clark Kimberling, May 03 2004

Triangle of coefficients of polynomials u(n,x) jointly generated with A209415; see the Formula section.
Column 1:  Fibonacci numbers:  F(n)=A000045(n)
Column 2:  n*F(n)
Row sums:  odd-indexed Fibonacci numbers
Alternating row sums: signed Fibonacci numbers
Coefficient of x^n in u(n,x):  1
Coefficient of x^(n-1) in u(n,x):  n
Coefficient of x^(n-2) in u(n,x):  n(n+1)
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 27 2012
Row n shows the coefficients of the numerator of the n-th derivative of (1/n!)*(x+1)/(1-x-x^2); see the Mathematica program. - Clark Kimberling, Oct 22 2019

			First five rows:
  1;
  1,  1;
  2,  2,  1;
  3,  6,  3,  1;
  5, 12, 12,  4,  1;
First three polynomials v(n,x): 1, 1 + x, 2 + 2x + x^2.
From _Philippe Deléham_, Mar 27 2012: (Start)
(0, 1, 1, -1, 0, 0, 0, ...) DELTA (1, 0, 0, 1, 0, 0, 0, ...) begins:
  1;
  0,  1;
  0,  1,  1;
  0,  2,  2,  1;
  0,  3,  6,  3,  1;
  0,  5, 12, 12,  4,  1. (End)

G. C. Greubel, Rows n = 0..100 of triangle, flattened
E. Kiliç, H. Belbachir, Generalized double binomial sums families by generating functions, 2014.

Cf. A000045, A094435, A094436, A094437, A094438, A094439, A094440, A094442, A094443, A094444.

Flat(List([0..12], n-> List([0..n], k-> Binomial(n,k)*Fibonacci(n-k+1) ))); # G. C. Greubel, Oct 30 2019

[Binomial(n,k)*Fibonacci(n-k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 30 2019

with(combinat); seq(seq(fibonacci(n-k+1)*binomial(n,k), k=0..n), n=0..12); # G. C. Greubel, Oct 30 2019

(* First program *)
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
v[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A094441 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A094442 *)
(* Next program outputs polynomials having coefficients T(n,k) *)
g[x_, n_] := Numerator[(-1)^(n + 1) Factor[D[(x + 1)/(1 - x - x^2), {x, n}]]]
Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]] (* Clark Kimberling, Oct 22 2019 *)
(* Second program *)
Table[Fibonacci[n-k+1]*Binomial[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 30 2019 *)

T(n,k) = binomial(n,k)*fibonacci(n-k+1);
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Oct 30 2019

[[binomial(n,k)*fibonacci(n-k+1) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Oct 30 2019

Sum_{k=0..n} T(n,k)*x^k = A039834(n-1), A000045(n+1), A001519(n+1), A081567(n), A081568(n), A081569(n), A081570(n), A081571(n) for x = -1, 0, 1, 2, 3, 4, 5, 6 respectively. - Philippe Deléham, Dec 14 2009
From Clark Kimberling, Mar 09 2012: (Start)
A094441 shows the coefficient of the polynomials u(n,x) which are jointly generated with polynomials v(n,x) by these rules:
u(n,x) = x*u(n-1,x) + v(n-1,x),
v(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
(End)
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-1) - T(n-2,k-2), T(1,0) = T(2,0) = T(2,1) = 1 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 27 2012
G.f. (1-x*y)/(1 - 2*x*y - x - x^2 + x^2*y + x^2*y^2). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Oct 30 2019: (Start)
T(n,k) = binomial(n,k)*Fibonacci(n-k+1).
Sum_{k=0..n} T(n,k) = Fibonacci(2*n+1).
Sum_{k=0..n} (-1)^k * T(n,k) = (-1)^n * Fibonacci(n-1). (End)

A081575 Fifth binomial transform of Fibonacci numbers F(n).

Original entry on oeis.org

0, 1, 11, 92, 693, 4955, 34408, 234793, 1584891, 10624804, 70911005, 471901739, 3134499984, 20794349393, 137837343787, 913174649260, 6047638172037, 40041955063867, 265079998713464, 1754663288995961, 11613976216265115

Offset: 0

Paul Barry, Mar 24 2003

Binomial transform of A081574.

Case k=5 of family of recurrences a(n) = (2k+1)*a(n-1) - A028387(k-1)*a(n-2), a(0)=0, a(1)=1.

S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
Index entries for linear recurrences with constant coefficients, signature (11,-29).

Cf. A000045, A081570.

a:=[0,1];; for n in [3..30] do a[n]:=11*a[n-1]-29*a[n-2]; od; a; # G. C. Greubel, Aug 13 2019

[n le 2 select (n-1) else 11*Self(n-1)-29*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Aug 09 2013

seq(coeff(series(x/(1-11*x+29*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Aug 13 2019

LinearRecurrence[{11,-29}, {0,1}, 30] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2011; modified by G. C. Greubel, Aug 13 2019 *)
CoefficientList[Series[x/(1 -11x +29x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 09 2013 *)

my(x='x+O('x^30)); Vec(x/(1-11*x+29*x^2)) \\ G. C. Greubel, Aug 13 2019

[lucas_number1(n,11,29) for n in range(0, 21)] # Zerinvary Lajos, Apr 27 2009

a(n) = 11*a(n-1) - 29*a(n-2), a(0)=0, a(1)=1.
a(n) = ((sqrt(5)/2 + 11/2)^n - (11/2 - sqrt(5)/2)^n)/sqrt(5).
G.f.: x/(1 - 11*x + 29*x^2). - adapted by Vincenzo Librandi, Aug 09 2013
E.g.f.: 2*exp(11*x/2)*sinh(sqrt(5)*x/2)/sqrt(5). - Ilya Gutkovskiy, Aug 12 2017

A270863 Self-composition of the Fibonacci sequence.

Original entry on oeis.org

0, 1, 2, 6, 17, 50, 147, 434, 1282, 3789, 11200, 33109, 97878, 289354, 855413, 2528850, 7476023, 22101326, 65338038, 193158521, 571033600, 1688143881, 4990651642, 14753839486, 43616704857, 128943855250, 381196100507, 1126928202714, 3331532438042, 9848993360069

Offset: 0

Oboifeng Dira, Mar 24 2016

This sequence has the same relation to the Fibonacci numbers A000045 as A030267 has to the natural numbers A000027.
From Oboifeng Dira, Jun 28 2020: (Start)
This sequence can be generated from a family of composition pairs of generating functions g(f(x)), where k is an integer and where
f(x) = x/(1-k*x-x^2) and g(x) = (x+(k-1)*x^2)/(1-(3-2*k)*x-(3*k-k^2-1)*x^2).
Some cases of k values are:
k=-5, f(x) g.f. 0,A052918(-1)^n and g(x) g.f. 0,A081571
k=-4, f(x) g.f. A001076(-1)^(n+1) and g(x) g.f. 0,A081570
k=-3, f(x) g.f. A006190(-1)^(n+1) and g(x) g.f. 0,A081569
k=-2, f(x) g.f. A215936(n+2) and g(x) g.f. 0,A081568
k=-1, f(x) g.f. A039834(n+2) and g(x) g.f. 0,A081567
k=0,  f(x) g.f. A000035 and g(x) g.f. 0,A001519(n+1)
k=1,  f(x) g.f. A000045 and g(x) g.f. A000045
k=2,  f(x) g.f. A000129 and g(x) g.f. 0,A039834(n+1)
k=3,  f(x) g.f. A006190 and g(x) g.f. 0,A001519(-1)^n
k=4,  f(x) g.f. A001076 and g(x) g.f. 0,A093129(-1)^n
k=5,  f(x) g.f. 0,A052918 and g(x) g.f. 0,A192240(-1)^n
k=6,  f(x) g.f. A005668 and g(x)=(x+5*x^2)/(1+9*x+19*x^2)
k=7,  f(x) g.f. 0,A054413 and g(x)=(x+6*x^2)/(1+11*x+29*x^2).
(End)

			a(5) = 3*a(4)+a(3)-3*a(2)-a(1) = 51+6-6-1 = 50.

Colin Barker, Table of n, a(n) for n = 0..1000
Oboifeng Dira, A Note on Composition and Recursion, Southeast Asian Bulletin of Mathematics (2017), Vol. 41, Issue 6, 849-853.
Oboifeng Dira, Family of composition pairs g(f(x)) generating A270683
Index entries for linear recurrences with constant coefficients, signature (3,1,-3,-1).

Cf. A000027, A000045, A001622, A030267, A000035, A001519, A000129, A039834.

I:=[0, 1, 2, 6]; [m le 4 select I[m] else 3*Self(m-1)+Self(m-2)-3*Self(m-3)-Self(m-4): m in [1..30]]; // Marius A. Burtea, Aug 03 2019

f:= x-> x/(1-x-x^2):
a:= n-> coeff(series(f(f(x)), x, n+1), x, n):
seq(a(n), n=0..30);

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,-3,1,3]^(n-1)*[1;2;6;17])[1,1] \\ Charles R Greathouse IV, Mar 24 2016

concat(0, Vec(x*(1-x-x^2)/(1-3*x-x^2+3*x^3+x^4) + O(x^40))) \\ Colin Barker, Mar 24 2016

a(n) = 3*a(n-1)+a(n-2)-3*a(n-3)-a(n-4) for n > 3, a(0)=0, a(1)=1, a(2)=2, a(3)=6.
G.f.: x*(1-x-x^2) / (1-3*x-x^2+3*x^3+x^4). - Colin Barker, Mar 24 2016
G.f.: B(B(x)) where B(x) is the g.f. of A000045. - Joerg Arndt, Mar 25 2016
a(n) = (phi*((phi^2 + 5^(1/4)*sqrt(3*phi))^n - (phi^2 - 5^(1/4)*sqrt(3*phi))^n) + (psi^2 + 5^(1/4)*sqrt(3*psi))^n - (psi^2 - 5^(1/4)*sqrt(3*psi))^n)/(2^n * 5^(3/4) * sqrt(3*phi)), where phi = (sqrt(5) + 1)/2 is the golden ratio, and psi = 1/phi = (sqrt(5) - 1)/2. - Vladimir Reshetnikov, Aug 01 2019
0 = a(n)*(a(n) +6*a(n+1) -a(n+2)) +a(n+1)*(8*a(n+1) -9*a(n+2) +a(n+3)) +a(n+2)*(-8*a(n+2) +6*a(n+3)) +a(n+3)*(-a(n+3)) if n>=0. - Michael Somos, Feb 05 2022

A081571 Sixth binomial transform of F(n+1).

Original entry on oeis.org

1, 7, 50, 363, 2669, 19814, 148153, 1113615, 8402722, 63577171, 481991621, 3659227062, 27808295345, 211479529943, 1609093780114, 12247558413819, 93245414394973, 710040492168070, 5407464407991017, 41185377124992351, 313703861897268866, 2389549742539808867

Offset: 0

Paul Barry, Mar 22 2003

Binomial transform of A081570.

Case k=6 of family of recurrences a(n) = (2k+1)*a(n-1) - A028387(k-1)*a(n-2), a(0)=1, a(1)=k+1.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (13,-41).

Cf. A000045, A028387, A081570.

a:=[1,7];; for n in [3..30] do a[n]:=13*a[n-1]-41*a[n-2]; od; a; # G. C. Greubel, Aug 12 2019

I:=[1, 7]; [n le 2 select I[n] else 13*Self(n-1)-41*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 09 2013

seq(coeff(series((1-6*x)/(1-13*x+41*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Aug 12 2019

CoefficientList[Series[(1-6x)/(1 -13x +41x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 09 2013 *)

my(x='x+O('x^30)); Vec((1-6*x)/(1-13*x+41*x^2)) \\ G. C. Greubel, Aug 12 2019

def A081571_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-6*x)/(1-13*x+41*x^2)).list()
A081571_list(30) # G. C. Greubel, Aug 12 2019

a(n) = 13*a(n-1) - 41*a(n-2), a(0)=1, a(1)=7.
a(n) = (1/2 - sqrt(5)/10)*(13/2 - sqrt(5)/2)^n + (sqrt(5)/10 + 1/2)*(sqrt(5)/2 + 13/2)^n.
G.f.: (1 - 6*x)/(1 - 13*x + 41*x^2).
a(n) = Sum_{k=0..n} A094441(n,k)*6^k. - Philippe Deléham, Dec 14 2009
E.g.f.: exp(13*x/2)*(5*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Mar 30 2023

A081572 Square array of binomial transforms of Fibonacci numbers, read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 4, 10, 13, 5, 1, 5, 17, 35, 34, 8, 1, 6, 26, 75, 125, 89, 13, 1, 7, 37, 139, 338, 450, 233, 21, 1, 8, 50, 233, 757, 1541, 1625, 610, 34, 1, 9, 65, 363, 1490, 4172, 7069, 5875, 1597, 55, 1, 10, 82, 535, 2669, 9633, 23165, 32532, 21250, 4181, 89

Offset: 0

Paul Barry, Mar 22 2003

Array rows are solutions of the recurrence a(n) = (2*k+1)*a(n-1) - A028387(k-1)*a(n-2), where a(0) = 1 and a(1) = k+1.

			The array rows begins as:
  1, 1,  2,   3,    5,     8,     13, ... A000045;
  1, 2,  5,  13,   34,    89,    233, ... A001519;
  1, 3, 10,  35,  125,   450,   1625, ... A081567;
  1, 4, 17,  75,  338,  1541,   7069, ... A081568;
  1, 5, 26, 139,  757,  4172,  23165, ... A081569;
  1, 6, 37, 233, 1490,  9633,  62753, ... A081570;
  1, 7, 50, 363, 2669, 19814, 148153, ... A081571;
Antidiagonal triangle begins as:
  1;
  1, 1;
  1, 2,  2;
  1, 3,  5,   3;
  1, 4, 10,  13,   5;
  1, 5, 17,  35,  34,    8;
  1, 6, 26,  75, 125,   89,   13;
  1, 7, 37, 139, 338,  450,  233,  21;
  1, 8, 50, 233, 757, 1541, 1625, 610, 34;

G. C. Greubel, Antidiagonal rows n = 0..50, flattened

Array row n: A000045 (n=0), A001519 (n=1), A081567 (n=2), A081568 (n=3), A081569 (n=4), A081570 (n=5), A081571 (n=6).
Array column k: A000027 (k=1), A002522 (k=2).
Different from A073133.
Cf. A028387.

A081572:= func< n,k | (&+[Binomial(k,j)*Fibonacci(j+1)*(n-k)^(k-j): j in [0..k]]) >;
[A081572(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 27 2021

T[n_, k_]:= If[n==0, Fibonacci[k+1], Sum[Binomial[k, j]*Fibonacci[j+1]*n^(k-j), {j, 0, k}]]; Table[T[n-k, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 26 2021 *)

def A081572(n,k): return sum( binomial(k,j)*fibonacci(j+1)*(n-k)^(k-j) for j in (0..k) )
flatten([[A081572(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 27 2021

Rows are successive binomial transforms of F(n+1).
T(n, k) = ((5+sqrt(5))/10)*( (2*n + 1 + sqrt(5))/2)^k + ((5-sqrt(5)/10)*( 2*n + 1 - sqrt(5))/2 )^k.
From G. C. Greubel, May 27 2021: (Start)
T(n, k) = Sum_{j=0..k} binomial(k,j)*n^(k-j)*Fibonacci(j+1) (square array).
T(n, k) = Sum_{j=0..k} binomial(k,j)*(n-k)^(k-j)*Fibonacci(j+1) (antidiagonal triangle). (End)

A106198 Triangle, columns = successive binomial transforms of Fibonacci numbers.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 5, 13, 10, 4, 1, 8, 34, 35, 17, 5, 1, 13, 89, 125, 75, 26, 6, 1, 21, 233, 450, 338, 139, 37, 7, 1, 34, 610, 1625, 1541, 757, 233, 50, 8, 1

Offset: 0

Gary W. Adamson, Apr 24 2005

Column 0 = Fibonacci numbers, column 1 = odd-indexed Fibonacci numbers (first binomial transform of 1, 1, 2, 3, 5, ...); column 2 = second binomial transform of Fibonacci numbers, etc.

			First few rows of the triangle are:
   1;
   1,   1;
   2,   2,   1;
   3,   5,   3,   1;
   5,  13,  10,   4,   1;
   8,  34,  35,  17,   5,   1;
  13,  89, 125,  75,  26,   6,   1;
  21, 233, 450, 338, 139,  37,   7,   1;
  ...
Column 2 = A081567, second binomial transform of Fibonacci numbers: 1, 3, 10, 35, 125, ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A000045, A081567, A081568, A081569, A081570.

T:= function(n,k)
    if k=0 then return Fibonacci(n+1);
    else return Sum([0..n-k], j-> Binomial(n-k,j)*Fibonacci(j+1)*k^(n-k-j));
    fi; end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Dec 11 2019

function T(n,k)
  if k eq 0 then return Fibonacci(n+1);
  else return (&+[Binomial(n-k,j)*Fibonacci(j+1)*k^(n-k-j): j in [0..n-k]]);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 11 2019

with(combinat);
T:= proc(n, k) option remember;
      if k=0 then fibonacci(n+1)
    else add( binomial(n-k,j)*fibonacci(j+1)*k^(n-k-j), j=0..n-k)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Dec 11 2019

Table[If[k==0, Fibonacci[n+1], Sum[Binomial[n-k, j]*Fibonacci[j+1]*k^(n-k-j), {j,0,n-k}]], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 11 2019 *)

T(n,k) = if(k==0, fibonacci(n+1), sum(j=0,n-k, binomial(n-k,j)*fibonacci( j+1)*k^(n-k-j)) ); \\ G. C. Greubel, Dec 11 2019

@CachedFunction
def T(n, k):
    if (k==0): return fibonacci(n+1)
    else: return sum(binomial(n-k,j)*fibonacci(j+1)*k^(n-k-j) for j in (0..n-k))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 11 2019

Offset column k = k-th binomial transform of the Fibonacci numbers, given leftmost column = Fibonacci numbers.

cp's OEIS Frontend

A094441 Triangular array T(n,k) = Fibonacci(n+1-k)*C(n,k), 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A081575 Fifth binomial transform of Fibonacci numbers F(n).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A270863 Self-composition of the Fibonacci sequence.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

PARI

PARI

Formula

A081571 Sixth binomial transform of F(n+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A081572 Square array of binomial transforms of Fibonacci numbers, read by ascending antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A106198 Triangle, columns = successive binomial transforms of Fibonacci numbers.

Views

Author

Keywords

Comments