A030191 -id:A030191 - cp's OEIS frontend

A328647 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)/(x^2-3x+1)).

1, 1, 4, -2, -1, 11, -12, 3, 1, 29, -44, 24, -4, -1, 76, -145, 110, -40, 5, 1, 199, -456, 435, -220, 60, -6, -1, 521, -1393, 1596, -1015, 385, -84, 7, 1, 1364, -4168, 5572, -4256, 2030, -616, 112, -8, -1, 3571, -12276, 18756, -16716, 9576, -3654, 924, -144

Offset: 0

Clark Kimberling, Nov 01 2019

The first 201 polynomials are irreducible. Column 1 of the array: A002879 (odd-indexed Lucas numbers). Row sums: A000032 (Lucas numbers). Alternating row sums: essentially 5*A030191.

			First eight rows:
     1,     1;
     4,    -2,   -1;
    11,   -12,    3,     1;
    29,   -44,   24,    -4,   -1;
    76,  -145,  110,   -40,    5,    1;
   199,  -456,  435,  -220,   60,   -6,  -1;
   521, -1393, 1596, -1015,  385,  -84,   7,  1;
  1364, -4168, 5572, -4256, 2030, -616, 112, -8, -1;
First eight polynomials:
1 + x
4 - 2 x - x^2
11 - 12 x + 3 x^2 + x^3
29 - 44 x + 24 x^2 - 4 x^3 - x^4
76 - 145 x + 110 x^2 - 40 x^3 + 5 x^4 + x^5
199 - 456 x + 435 x^2 - 220 x^3 + 60 x^4 - 6 x^5 - x^6
521 - 1393 x + 1596 x^2 - 1015 x^3 + 385 x^4 - 84 x^5 + 7 x^6 + x^7
1364 - 4168 x + 5572 x^2 - 4256 x^3 + 2030 x^4 - 616 x^5 + 112 x^6 - 8 x^7 - x^8

Cf. A328646, A002879, A000032.

g[x_, n_] := Numerator[ Factor[D[(1 + x)/(x^2 - 3 x + 1), {x, n}]]]
Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]] (* polynomials *)
h[n_] := CoefficientList[g[x, n]/n!, x]
Table[h[n], {n, 0, 10}]
Column[%]  (* A328647 array *)

A057281 Coefficient triangle of polynomials (falling powers) related to Fibonacci convolutions. Companion triangle to A057282.

Original entry on oeis.org

1, 5, 16, 20, 160, 300, 75, 1075, 4850, 6840, 275, 6100, 48175, 159650, 186120, 1000, 31550, 379700, 2168650, 5846700, 5916240, 3625, 153875, 2605175, 22426825, 103057800, 238437900, 215717040, 13125, 720375, 16273875, 195469125

Offset: 0

Wolfdieter Lang, Sep 13 2000

The row polynomials are p(k,x) := sum(a(k,m)*x^(k-m),m=0..k), k=0,1,2,..
The k-th convolution of F0(n) := A000045(n+1), n >= 0, (Fibonacci numbers starting with F0(0)=1) with itself is Fk(n) := A037027(n+k,k) = (p(k-1,n)*(n+1)*F0(n+1) + q(k-1,n)*(n+2)*F0(n))/(k!*5^k), k=1,2,..., where the companion polynomials q(k,n) := sum(b(k,m)*n^(k-m),m=0..k), k >= 0, are the row polynomials of triangle b(k,m)= A057282(k,m).
a(k,0)= A030191(k), k >= 0.

			k=2: F2(n)=((5*n^2+21*n+16)*F(n+2)+(5*n^2+27*n+34)*F(n+1))/50, F(n)=A000045(n); see A001628.

Cf. A000045, A037027, A057995, A057280, A057282.

A084329 a(0)=0, a(1)=1, a(n)=20a(n-1)-20a(n-2).

Original entry on oeis.org

0, 1, 20, 380, 7200, 136400, 2584000, 48952000, 927360000, 17568160000, 332816000000, 6304956800000, 119442816000000, 2262757184000000, 42866287360000000, 812070603520000000, 15384086323200000000

Offset: 0

Benoit Cloitre, Jun 21 2003

nonn

Index entries for linear recurrences with constant coefficients, signature (20,-20).

Cf. A030191.

Union[Flatten[NestList[{#[[2]],20(#[[2]]-#[[1]])}&,{0,1},20]]]  (* Harvey P. Dale, Feb 24 2011 *)
LinearRecurrence[{20,-20},{0,1},20] (* Harvey P. Dale, Nov 29 2019 *)

a(n)=(1/8)*sum(k=0,n,binomial(n,k)*fibonacci(6*k))

PARI
```
a(n)=imag((6+8*quadgen(5))^n)/8
```

a(n)=(1/8)*sum(k=0, n, binomial(n, k)*F(6*k)) where F(k) denotes the k-th Fibonacci number.

G.f.: x/(1-20x+20x^2).

A098111 Inverse binomial transform of A098149.

Original entry on oeis.org

1, 0, -5, -25, -100, -375, -1375, -5000, -18125, -65625, -237500, -859375, -3109375, -11250000, -40703125, -147265625, -532812500, -1927734375, -6974609375, -25234375000, -91298828125, -330322265625, -1195117187500, -4323974609375, -15644287109375, -56601562500000

Offset: 0

Creighton Dement, Sep 23 2004

A030191(n) + 2*a(n) + A093129(n+2) = 4*A093129(n+1). - Creighton Dement, Oct 18 2004
From Wolfdieter Lang, Oct 02 2013: (Start)
These numbers a(n) and those of A030191(n) =: b(n), both interspersed with zeros, appear in the formula for nonnegative powers of the algebraic number rho(10) := 2*cos(pi/10) = phi*sqrt(3-phi), with the golden section phi, in terms of the power basis of the number field Q(rho(10)) of degree 4 (see A187360, n=10). In a (regular) decagon rho(10) is the length ratio of a smallest diagonal to the side. rho(10)^n = sum(A(n,k)*rho(10)^k, k=0..3), with A(2*k+1,0) = 0, A(2*k,0) = a(k), k >= 0; A(2*k,1) = 0, A(2*k+1,1) = a(k), k >= 0; A(2*k+1,2) = 0, k >= 0, A(0,2) = 0, A(2*k,2) = b(k-1), k >= 1; and A(2*k,3) = 0, k >= 0, A(1,3) = 0, A(2*k+1,3) = b(k-1), k >= 1. (End)

			Powers of rho(10) in the Q(rho(10)) power basis for n = 5: rho(10)^5 = 0*1 + a(2)*rho(10) + 0*rho(10)^2 + b(1)*rho(10)^3 = -5*rho(10) + 5*rho(10)^3.  - _Wolfdieter Lang_, Oct 02 2013

Index entries for linear recurrences with constant coefficients, signature (5,-5).
Index entries for sequences related to Chebyshev polynomials.

Cf. A030191, A049310, A098149, A187360.

LinearRecurrence[{5,-5},{1,0},40] (* Harvey P. Dale, Dec 08 2015 *)

G.f.: (1-5x)/(1-5x+5x^2).
From Wolfdieter Lang, Oct 02 2013: (Start)
a(n) = b(n) - 5*b(n-1), n >= 0,  with b(n) = A030191(n) = (sqrt(5))^n*S(n, sqrt(5)), with Chebyshev S-polynomials (see A049310).
a(n) = 5*(a(n-1) - a(n-2)), n >= 1, a(-1) = 1 = a(0). (End)

More terms from David Wasserman, Jan 16 2008

A202551 Triangle T(n,k), read by rows, given by (1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, -1, 0, -1, 1, -1, 1, 1, -1, -1, 3, -2, -1, 1, 0, 2, -5, 3, 1, -1, 1, -2, -2, 7, -4, -1, 1, 1, -5, 7, 1, -9, 5, 1, -1, 0, -3, 12, -15, 1, 11, -6, -1, 1, -1, 3, 3, -21, 26, -4, -13, 7, 1, -1

Offset: 0

Philippe Deléham, Dec 21 2011

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

			Triangle begins :
1
1, -1
0, -1, 1
-1, 1, 1, -1
-1, 3, -2, -1, 1
0, 2, -5, 3, 1, -1

Cf. A129267, A199324

T(n,k) = T(n-1,k) + T(n-2,k-1) - T(n-1,k-1) - T(n-2,k).
G.f.: 1/(1+(y-1)*x+(1-y)*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A190873(n+1), A190871(n+1), A057086(n), A057085(n+1), A057084(n), A030240(n), A030192(n), A030191(n), A001787(n+1), A057083(n), A099087(n), A010892(n), A000007(n), (-1)^n*A000045(n+1) for x = -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2 respectively.

A084328 a(0)=0, a(1)=1; a(n) = 13a(n-1) - 11a(n-2).

Original entry on oeis.org

0, 1, 13, 158, 1911, 23105, 279344, 3377317, 40832337, 493669894, 5968552915, 72160819061, 872436565728, 10547906344793, 127525980259301, 1541810773578190, 18640754273664159, 225369887048273977

Offset: 0

Benoit Cloitre, Jun 21 2003

Index entries for linear recurrences with constant coefficients, signature (13,-11).

Cf. A030191.

Join[{a=0,b=1},Table[c=13*b-11*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2011 *)

a(n)=(1/5)*sum(k=0,n,binomial(n,k)*fibonacci(5*k));

[lucas_number1(n,13,11) for n in range(0, 18)] # Zerinvary Lajos, Apr 29 2009

a(n) = (1/5)*Sum_{k=0..n} binomial(n, k)*F(5*k) where F(k) denotes the k-th Fibonacci number.

G.f.: x / (11*x^2-13*x+1). - Colin Barker, Jun 26 2013

A087426 a(n) = S(n,4) where S(n,m) = sum(k=0,n,binomial(n,k)Fibonacci(mk)).

Original entry on oeis.org

0, 3, 27, 216, 1701, 13365, 104976, 824499, 6475707, 50860872, 399466485, 3137450517, 24641856288, 193539651939, 1520080160859, 11938864580280, 93769059774789, 736471756750581, 5784324272782128, 45430672644283923

Offset: 0

Benoit Cloitre, Oct 23 2003

M. Griffiths, Families of Sequences From a Class of Multinomial Sums, J. Int. Seq. 15 (2012) # 12.1.8
Index entries for linear recurrences with constant coefficients, signature (9,-9).

Cf. A001906 (S(n,1)), A030191 (S(n,2)).

a(n) = 9*a(n-1)-9*a(n-2).
a(n) = (1/sqrt(5))*(((9+3*sqrt(5))/2)^n-((9-3*sqrt(5))/2)^n).
a(n) = 3^n*F(2n). - Benoit Cloitre, Sep 13 2005
G.f.: 3*x / (9*x^2-9*x+1). - Colin Barker, Jun 26 2013
E.g.f.: 2*exp(9*x/2)*sinh(3*sqrt(5)*x/2)/sqrt(5). - Stefano Spezia, Feb 23 2025

A260304 a(n) = 5a(n-1) - 5a(n-2) for n>1, a(0)=2, a(1)=3.

Original entry on oeis.org

2, 3, 5, 10, 25, 75, 250, 875, 3125, 11250, 40625, 146875, 531250, 1921875, 6953125, 25156250, 91015625, 329296875, 1191406250, 4310546875, 15595703125, 56425781250, 204150390625, 738623046875, 2672363281250, 9668701171875, 34981689453125, 126564941406250

Offset: 0

Ilya Gutkovskiy, Nov 21 2015

Lim_{n -> infinity} a(n + 1)/a(n) = 2 + phi = 3.6180339887..., where phi is the golden ratio (A001622).

Index entries for linear recurrences with constant coefficients, signature (5,-5).

Cf. A020876, A030191, A098111.

Cf. A093129: initial values 1,2; A081567: initial values 1,3.

[n le 2 select n+1 else 5*Self(n-1)-5*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 23 2015

Table[((5 + 2 Sqrt[5]) ((5 - Sqrt[5])/2)^n + (5 - 2 Sqrt[5]) ((5 + Sqrt[5])/2)^n)/5, {n, 0, 30}]
RecurrenceTable[{a[0] == 2, a[1] == 3, a[n] == 5 a[n - 1] - 5 a[n - 2]}, a, {n, 0, 30}] (* Bruno Berselli, Nov 23 2015 *)

a(n)=([0,1; -5,5]^n*[2;3])[1,1] \\ Charles R Greathouse IV, Jul 26 2016

G.f.: (2 - 7*x)/(1 - 5*x + 5*x^2).
a(n) = ((5 + 2*sqrt(5))*((5 - sqrt(5))/2)^n + (5 - 2*sqrt(5))*((5 + sqrt(5))/2)^n)/5.
a(n) = 2*A030191(n) - 7*A030191(n-1). - Bruno Berselli, Nov 23 2015

Edited by Bruno Berselli, Nov 23 2015

A336602 a(n) = 8a(n-1) - 21a(n-2) + 20a(n-3) - 5a(n-4), with initial terms a(0)=1, a(1)=7, a(2)=35, a(3)=154.

Original entry on oeis.org

1, 7, 35, 154, 632, 2487, 9529, 35875, 133471, 492538, 1807268, 6604891, 24069905, 87539199, 317907067, 1153307002, 4180842064, 15147734815, 54860799881, 198634274203, 719047882103, 2602540622106, 9418700937340, 34084040705539, 123335178991777, 446277892754167, 1614771692630099

Offset: 0

Peter Morris, Dec 20 2020

Index entries for linear recurrences with constant coefficients, signature (8,-21,20,-5).

Extension of patterns illustrated in A001519, A033191, A033190, A094667, A030191, A094788.

G.f.: ( 1-x+x^3 ) / ( (5*x^2-5*x+1)*(x^2-3*x+1) ). - R. J. Mathar, May 05 2023

Offset corrected by Jon E. Schoenfield, Feb 05 2021

cp's OEIS Frontend

A328647 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)/(x^2-3x+1)).

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A057281 Coefficient triangle of polynomials (falling powers) related to Fibonacci convolutions. Companion triangle to A057282.

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A084329 a(0)=0, a(1)=1, a(n)=20a(n-1)-20a(n-2).

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A098111 Inverse binomial transform of A098149.

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A202551 Triangle T(n,k), read by rows, given by (1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A084328 a(0)=0, a(1)=1; a(n) = 13*a(n-1) - 11*a(n-2).

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A087426 a(n) = S(n,4) where S(n,m) = sum(k=0,n,binomial(n,k)*Fibonacci(m*k)).

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A260304 a(n) = 5*a(n-1) - 5*a(n-2) for n>1, a(0)=2, a(1)=3.

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A336602 a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4), with initial terms a(0)=1, a(1)=7, a(2)=35, a(3)=154.

A084328 a(0)=0, a(1)=1; a(n) = 13a(n-1) - 11a(n-2).

A087426 a(n) = S(n,4) where S(n,m) = sum(k=0,n,binomial(n,k)Fibonacci(mk)).

A260304 a(n) = 5a(n-1) - 5a(n-2) for n>1, a(0)=2, a(1)=3.

A336602 a(n) = 8a(n-1) - 21a(n-2) + 20a(n-3) - 5a(n-4), with initial terms a(0)=1, a(1)=7, a(2)=35, a(3)=154.