A094441 -id:A094441 - cp's OEIS frontend

A094444 Triangular array T(n,k) = Fibonacci(n+4-k)*C(n,k), k=0..n, n>=0.

3, 5, 3, 8, 10, 3, 13, 24, 15, 3, 21, 52, 48, 20, 3, 34, 105, 130, 80, 25, 3, 55, 204, 315, 260, 120, 30, 3, 89, 385, 714, 735, 455, 168, 35, 3, 144, 712, 1540, 1904, 1470, 728, 224, 40, 3, 233, 1296, 3204, 4620, 4284, 2646, 1092, 288, 45, 3, 377, 2330, 6480, 10680, 11550, 8568, 4410, 1560, 360, 50, 3

Offset: 0

Clark Kimberling, May 03 2004

Row sums are Fibonacci numbers.

Row sums with alternating signs are Fibonacci numbers or their negatives.

			First few rows:
   3;
   5,   3;
   8,  10,   3;
  13,  24,  15,  3;
  21,  52,  48, 20,  3;
  34, 105, 130, 80, 25, 3;

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

Cf. A000045, A094435, A094436, A094437, A094438, A094439, A094440, A094441, A094442, A094443.

Flat(List([0..12], n-> List([0..n], k-> Binomial(n,k)*Fibonacci(n-k+4) ))); # G. C. Greubel, Oct 30 2019

[Binomial(n,k)*Fibonacci(n-k+4): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 30 2019

with(combinat); seq(seq(fibonacci(n-k+4)*binomial(n,k), k=0..n), n=0..12); # G. C. Greubel, Oct 30 2019

Table[Fibonacci[n-k+4]*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+4);
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Oct 30 2019

[[binomial(n,k)*fibonacci(n-k+4) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Oct 30 2019

From G. C. Greubel, Oct 30 2019: (Start)
T(n,k) = binomial(n,k)*Fibonacci(n-k+4).
Sum_{k=0..n} T(n,k) = Fibonacci(2*n+4).
Sum_{k=0..n} (-1)^(k+1) * T(n,k) = (-1)^n * Fibonacci(n-4). (End)

A081569 Fourth binomial transform of F(n+1).

Original entry on oeis.org

1, 5, 26, 139, 757, 4172, 23165, 129217, 722818, 4050239, 22718609, 127512940, 715962889, 4020920141, 22584986378, 126867394723, 712691811325, 4003745802188, 22492567804517, 126361939999081, 709898671705906, 3988211185370615, 22405825905923321, 125876420631268204

Offset: 0

Paul Barry, Mar 22 2003

Binomial transform of A081568.
Case k = 4 of family of recurrences a(n) = (2*k+1)*a(n-1) - A028387(k-1)*a(n-2) for n >= 2, with a(0) = 1 and a(1) = k + 1.
a(n) = 5^n * a(n;1/5) = Sum_{k=0..n} binomial(n,k) * (-1)^k * F(k-1) * 5^(n-k), which implies also Deléham's formula given below and where a(n;d), n=0,1,...,d, denote the delta-Fibonacci numbers defined in comments to A000045 (see also Witula's et al. papers). - Roman Witula, Jul 12 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Edyta Hetmaniok, Bożena Piątek, and Roman Wituła, Binomials Transformation Formulae of Scaled Fibonacci Numbers, Open Mathematics, 15(1) (2017), 477-485.
Roman Witula and Damian Slota, delta-Fibonacci numbers, Appl. Anal. Discr. Math 3 (2009), 310-329, MR2555042.
Index entries for linear recurrences with constant coefficients, signature (9,-19).

Cf. A000045, A028387, A081568, A081574, A094441.

a:=[1,5];; for n in [3..30] do a[n]:=9*a[n-1]-19*a[n-2]; od; a; # G. C. Greubel, Aug 12 2019

I:=[1, 5]; [n le 2 select I[n] else 9*Self(n-1)-19*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 09 2013

seq(coeff(series((1-4*x)/(1-9*x+19*x^2), x, n+1), x, n), n = 0 .. 30); # G. C. Greubel, Aug 12 2019

CoefficientList[Series[(1-4x)/(1 -9x +19x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 09 2013 *)

Vec((1-4*x)/(1-9*x+19*x^2) + O(x^30)) \\ Altug Alkan, Dec 10 2015

def A081569_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-4*x)/(1-9*x+19*x^2)).list()
A081569_list(30) # G. C. Greubel, Aug 12 2019

a(n) = 9*a(n-1) - 19*a(n-2) for n >= 2, with a(0) = 1 and a(1) = 5.
a(n) = (1/2 - sqrt(5)/10)*(9/2 - sqrt(5)/2)^n + (sqrt(5)/10 + 1/2)*(sqrt(5)/2 + 9/2)^n.
G.f.: (1 - 4*x)/(1 - 9*x + 19*x^2).
a(n) = Sum_{k=0..n} A094441(n,k)*4^k. - Philippe Deléham, Dec 14 2009
a(n) = A081574(n) - 4*A081574(n-1). - R. J. Mathar, Jul 19 2012
E.g.f.: exp(9*x/2)*(5*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Jun 03 2024

A081568 Third binomial transform of Fibonacci(n+1).

Original entry on oeis.org

1, 4, 17, 75, 338, 1541, 7069, 32532, 149965, 691903, 3193706, 14745009, 68084297, 314394980, 1451837593, 6704518371, 30961415074, 142980203437, 660285858245, 3049218769908, 14081386948661, 65028302171639, 300302858766202, 1386808687475385, 6404329365899473

Offset: 0

Paul Barry, Mar 22 2003

Binomial transform of A081567.
Case k=3 of family of recurrences a(n) = (2k+1)*a(n-1) - A028387(k-1)*a(n-2) for n >= 2, with a(0) = 1 and a(1) = k + 1.
a(n) = 4^n*a(n;1/4) = Sum_{k=0..n} binomial(n,k) * (-1)^k * F(k-1) * 4^(n-k), which also implies Deléham's formula given below and where a(n;d), n = 0, 1, ..., d, denote the delta-Fibonacci numbers defined in comments to A000045 (see also Witula's et al. papers). - Roman Witula, Jul 12 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Edyta Hetmaniok, Bożena Piątek, and Roman Wituła, Binomials Transformation Formulae of Scaled Fibonacci Numbers, Open Mathematics, 15(1) (2017), 477-485.
Roman Witula and Damian Slota, delta-Fibonacci numbers, Appl. Anal. Discr. Math 3 (2009), 310-329, MR2555042.
Index entries for linear recurrences with constant coefficients, signature (7,-11).

Cf. A000045, A161731 (INVERT transform), A007582 (INVERTi transform), A028387, A081567, A081569 (binomial transform), A094441, A099453.

a:=[1,4];; for n in [3..30] do a[n]:=7*a[n-1]-11*a[n-2]; od; a; # G. C. Greubel, Aug 12 2019

I:=[1, 4]; [n le 2 select I[n] else 7*Self(n-1)-11*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 09 2013

seq(coeff(series((1-3*x)/(1-7*x+11*x^2), x, n+1), x, n), n = 0 .. 30); # G. C. Greubel, Aug 12 2019

CoefficientList[Series[(1-3x)/(1 -7x +11x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 09 2013 *)
LinearRecurrence[{7,-11},{1,4},30] (* Harvey P. Dale, Feb 01 2015 *)

Vec((1-3*x)/(1-7*x+11*x^2) + O(x^30)) \\ Altug Alkan, Dec 10 2015

def A081568_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-3*x)/(1-7*x+11*x^2)).list()
A081568_list(30) # G. C. Greubel, Aug 12 2019

a(n) = 7*a(n-1) - 11*a(n-2) for n >= 2, with a(0) = 1 and a(1) = 4.
a(n) = (1/2 - sqrt(5)/10)*(7/2 - sqrt(5)/2)^n + (sqrt(5)/10 + 1/2)*(sqrt(5)/2 + 7/2)^n = A099453(n) - 3*A099453(n-1).
G.f.: (1 - 3*x)/(1 - 7*x + 11*x^2).
a(n) = Sum_{k=0..n} A094441(n,k)*3^k. - Philippe Deléham, Dec 14 2009
G.f.: Q(0,u)/x - 1/x, where u = x/(1 - 3*x), Q(k,u) = 1 + u^2 + (k+2)*u - u*(k + 1 + u)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 07 2013
E.g.f.: exp(7*x/2)*(5*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Jun 03 2024

A081570 Fifth binomial transform of F(n+1).

Original entry on oeis.org

1, 6, 37, 233, 1490, 9633, 62753, 410926, 2700349, 17786985, 117346714, 774991289, 5121849473, 33865596822, 223987930325, 1481764925737, 9803764203682, 64870223394129, 429263295428641, 2840659771285310, 18798621916707821

Offset: 0

Paul Barry, Mar 22 2003

Binomial transform of A081569.

Case k=5 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 (11,-29).

Cf. A000045.

a:=[1,6];; for n in [3..30] do a[n]:=11*a[n-1]-29*a[n-2]; od; a; # G. C. Greubel, Aug 12 2019

I:=[1, 6]; [n le 2 select I[n] else 11*Self(n-1)-29*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 09 2013

seq(coeff(series((1-5*x)/(1-11*x+29*x^2), x, n+1), x, n), n = 0 .. 30); # G. C. Greubel, Aug 12 2019

CoefficientList[Series[(1-5x)/(1 -11x +29x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 09 2013 *)
LinearRecurrence[{11,-29},{1,6},30] (* Harvey P. Dale, Aug 04 2022 *)

my(x='x+O('x^30)); Vec((1-5*x)/(1-11*x+29*x^2)) \\ G. C. Greubel, Aug 12 2019

def A081570_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-5*x)/(1-11*x+29*x^2)).list()
A081570_list(30) # G. C. Greubel, Aug 12 2019

a(n) = 11*a(n-1) - 29*a(n-2), a(0)=1, a(1)=6.
a(n) = (1/2 - sqrt(5)/10)*(11/2 - sqrt(5)/2)^n + (sqrt(5)/10 + 1/2)*(sqrt(5)/2 + 11/2)^n .
G.f.: (1-5*x)/(1-11*x+29*x^2).
a(n) = Sum_{k=0..n} A094441(n,k)*5^k. - Philippe Deléham, Dec 14 2009

A326925 Irregular triangular array read by rows: row n shows the coefficients of this polynomial of degree n: (1/n!)*(numerator of n-th derivative of (1-x)/(1-x-x^2)).

Original entry on oeis.org

1, -1, 0, 2, -1, 1, 0, 3, -1, 1, 4, 0, 4, -1, 2, 5, 10, 0, 5, -1, 3, 12, 15, 20, 0, 6, -1, 5, 21, 42, 35, 35, 0, 7, -1, 8, 40, 84, 112, 70, 56, 0, 8, -1, 13, 72, 180, 252, 252, 126, 84, 0, 9, -1, 21, 130, 360, 600, 630, 504, 210, 120, 0, 10, -1, 34, 231, 715

Offset: 1

Clark Kimberling, Oct 22 2019

Column 1: Fibonacci numbers, F(m), for m >= -1, as in A000045. For n >= 0, the n-th row sum = F(2n), as in A001906.

Conjecture: The odd degree polynomials are irreducible; the even degree (= 2k) polynomials have exactly two irreducible factors, each of degree k.

			First 7 rows:
1    -1
0     2   -1
1     0    3   -1
1     4    0    4   -1
2     5    0   10    5   -1
3    12   15   20    0    6   -1
5    21   42   35   35    0    7   -1
First 7 polynomials:
1 - x
2 x - x^2
1 + 3 x^2 - x^3
1 + 4 x + 4 x^3 - x^4
2 + 5 x + 10 x^2 + 5 x^4 - x^5
3 + 12 x + 15 x^2 + 20 x^3 + 6 x^5 - x^6
5 + 21 x + 42 x^2 + 35 x^3 + 35 x^4 + 7 x^6 - x^7
Factorizations of even-degree polynomials:
degree 2:  (2 - x)*x
degree 4:  (1 + x^2)*(1 + 4x - x^2)
degree 6:  (1 + 3x + x^3)*(3 + 3x + 6x^2 - x^3)
degree 8:  (2 + 4x + 6x^2 + x^4)*(4 + 12 x + 6x^2 + 8x^3 - x^4)
degree 10: (3 + 10 x + 10 x^2 + 10 x^3 + x^5)*(7 + 20 x + 30 x^2 + 10 x^3 + 10 x^4 - x^5)
It appears that the constant terms of the factors are Fibonacci numbers (A000045) and Lucas numbers (A000032).

Clark Kimberling, Table of n, a(n) for n = 1..10010

Cf. A000045, A001906, A094440, A094441, A328610, A328611.

g[x_, n_] := Numerator[(-1)^(n + 1) Factor[D[(1 - x)/(1 - x - x^2), {x, n}]]]
Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]] (* polynomials *)
h[n_] := CoefficientList[g[x, n]/n!, x] (* A326925 *)
Table[h[n], {n, 0, 10}]
Column[%]

G.f. as array: ((y^2 + y - 1)*x - y + 1)/(1 + (y^2 + y - 1)*x^2 + (-2*y - 1)*x). - Robert Israel, Oct 31 2019

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

cp's OEIS Frontend

A094444 Triangular array T(n,k) = Fibonacci(n+4-k)*C(n,k), k=0..n, n>=0.

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A081569 Fourth binomial transform of F(n+1).

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A081568 Third binomial transform of Fibonacci(n+1).

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A081570 Fifth binomial transform of F(n+1).

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A326925 Irregular triangular array read by rows: row n shows the coefficients of this polynomial of degree n: (1/n!)*(numerator of n-th derivative of (1-x)/(1-x-x^2)).

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

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