A055870 -id:A055870 - cp's OEIS frontend

A056567 Fibonomial coefficients.

1, 55, 4895, 352440, 27372840, 2063912136, 157373300370, 11948265189630, 908637119420910, 69056421075989160, 5249543573067466872, 399024295188779925720, 30331388438447118520355, 2305576054220330112077285, 175254358052498673797874685, 13321629629800423781409595728

Offset: 0

Wolfdieter Lang, Jul 10 2000

Cf. A010048, A000045, A001654-8, A056565-6, A001076 (signed), A049666, A049667 (signed), A049669.

a[n_] := (1/2227680) Times @@ Fibonacci[n + Range[9]]; Array[a, 20, 0] (* Giovanni Resta, May 08 2016 *)

b(n, k)=prod(j=1, k, fibonacci(n+j)/fibonacci(j));
vector(20, n, b(n-1, 9))  \\ Joerg Arndt, May 08 2016

a(n) = A010048(n+9, 9) = Fibonomial(n+9, 9).
G.f.: 1/p(10, n) with p(10, n)= 1 - 55*x - 1870*x^2 + 19635*x^3 + 85085*x^4 - 136136*x^5 - 85085*x^6 + 19635*x^7 + 1870*x^8 - 55*x^9 - x^10 = (1 - x - x^2)*(1 + 4*x - x^2)*(1 - 11*x - x^2)*(1 + 29*x - x^2)*(1 - 76*x - x^2) (n=10 row polynomial of signed Fibonomial triangle A055870; see this entry for Knuth and Riordan references).
Recursion: a(n) = 76*a(n-1) + a(n-2)+((-1)^n)*A056565(n), n >= 2, a(0)=1, a(1)=55.
G.f.: exp( Sum_{k>=1} F(10*k)/F(k) * x^k/k ), where F(n) = A000045(n). - Seiichi Manyama, May 07 2025

A056586 Ninth power of Fibonacci numbers A000045.

Original entry on oeis.org

0, 1, 1, 512, 19683, 1953125, 134217728, 10604499373, 794280046581, 60716992766464, 4605366583984375, 350356403707485209, 26623333280885243904, 2023966356928852115753, 153841020405122283630137

Offset: 0

Wolfdieter Lang, Jul 10 2000

Divisibility sequence; that is, if n divides m, then a(n) divides a(m).

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 1, p. 85, (exercise 1.2.8. Nr. 30) and p. 492 (solution).

Vincenzo Librandi, Table of n, a(n) for n = 0..101
A. Brousseau, A sequence of power formulas, Fib. Quart., 6 (1968), 81-83.
J. Riordan, Generating functions for powers of Fibonacci numbers, Duke. Math. J. 29 (1962) 5-12.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (55,1870,-19635,-85085,136136,85085,-19635,-1870,55,1).

Cf. A000045, A007598, A056570, A056571, A056572, A056573, A056574, A056585, A056588, A055870.

[Fibonacci(n)^9: n in [0..20]]; // Vincenzo Librandi, Jun 04 2011

a(n) = fibonacci(n)^9; \\ Michel Marcus, Sep 06 2017

a(n) = F(n)^9, F(n)=A000045(n).
G.f.: x*p(9, x)/q(9, x) with p(9, x) := sum_{m=0..8} A056588(8, m)*x^m = 1 - 54*x - 1413*x^2 + 9288*x^3 + 17840*x^4 - 9288*x^5 - 1413*x^6 + 54*x^7 + x^8 and q(9, x) := sum_{m=0..10} A055870(10, m)*x^m = (1 - x - x^2)*(1 + 4*x - x^2)*(1 - 11*x - x^2)*(1 + 29*x - x^2)*(1 - 76*x - x^2) (factorization deduced from Riordan result).
Recursion (cf. Knuth's exercise): sum_{m=0..10} A055870(10, m)*a(n-m) = 0, n >= 10; inputs: a(n), n=0..9. a(n) = 55*a(n-1) + 1870*a(n-2) - 19635*a(n-3) - 85085*a(n-4) + 136136*a(n-5) + 85085*a(n-6) - 19635*a(n-7) - 1870*a(n-8) + 55*a(n-9) + a(n-10).

A383715 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) = (-1)^floor((k+1)/2) * A099927(n,k).

Original entry on oeis.org

1, 1, -1, 1, -2, -1, 1, -5, -5, 1, 1, -12, -30, 12, 1, 1, -29, -174, 174, 29, -1, 1, -70, -1015, 2436, 1015, -70, -1, 1, -169, -5915, 34307, 34307, -5915, -169, 1, 1, -408, -34476, 482664, 1166438, -482664, -34476, 408, 1, 1, -985, -200940, 6791772, 39618670, -39618670, -6791772, 200940, 985, -1

Offset: 0

Seiichi Manyama, May 07 2025

			Triangle starts:
  1;
  1,   -1;
  1,   -2,    -1;
  1,   -5,    -5,     1;
  1,  -12,   -30,    12,     1;
  1,  -29,  -174,   174,    29,    -1;
  1,  -70, -1015,  2436,  1015,   -70,   -1;
  1, -169, -5915, 34307, 34307, -5915, -169, 1;
  ...

Cf. A055870, A099927.

pell(n) = ([2, 1; 1, 0]^n)[2, 1];
p(n, k) = prod(j=0, k-1, pell(n-j));
a099927(n, k) = p(n, k)/p(k, k);
T(n, k) = (-1)^((k+1)\2)*a099927(n, k);

Let f(n, x) be defined as f(n, x) = Sum_{k=0..n} T(n,k) * x^k.
f(n, x) = exp( -Sum_{k>=1} Pell(n*k)/Pell(k) * x^k/k ).
Sum_{k>=0} A099927(n+k,n) * x^k = 1/f(n+1, x).

A056568 Fibonomial coefficients.

Original entry on oeis.org

1, 89, 12816, 1493064, 187628376, 22890661872, 2824135408458, 346934172869802, 42689423937884208, 5249543573067466872, 645693859487298425256, 79413089729752455762384, 9767258556969762111163771, 1201288963378036364032704659, 147748983166877427393815516256

Offset: 0

Wolfdieter Lang, Jul 10 2000

Alois P. Heinz, Table of n, a(n) for n = 0..150

Cf. A010048, A000045, A001654-8, A056565-7, A001906, A004187 (signed), A049660, A049668 (signed), A049670.

[&*[Fibonacci(n+i): i in [0..9]]/122522400: n in [1..15]]; // Vincenzo Librandi, Oct 31 2014

F:= combinat[fibonacci]: a:= n-> mul(F(n+i), i=0..9)/122522400: seq(a(n), n=1..18); # Zerinvary Lajos, Oct 07 2007
a:= n-> (Matrix(11, (i,j)-> if (i=j-1) then 1 elif j=1 then [1514513, -582505, -83215, 4895, 89, -1][abs(i-11/2)+1/2] else 0 fi)^n)[1, 1]; seq(a(n), n=0..18);  # Alois P. Heinz, Aug 15 2008

Times@@@Partition[Fibonacci[Range[30]],10,1]/122522400 (* Harvey P. Dale, Jul 27 2019 *)

a(n)=prod(k=0,9,fibonacci(n+k))/122522400; \\ Joerg Arndt, Oct 31 2014

a(n) = A010048(n+10,10) =: Fibonomial(n+10,10).
G.f.: 1/p(11,n) with p(11,n) = 1-89*x -4895*x^2 +83215*x^3 +582505*x^4 -1514513*x^5 -1514513*x^6 +582505*x^7 +83215*x^8 -4895*x^9 -89*x^10 +x^11 = (1+x) *(1-3*x+x^2) *(1+7*x+x^2) *(1-18*x+x^2) *(1+47*x+x^2) *(1-123*x+x^2) (n=8 row polynomial of signed Fibonomial triangle A055870; see this entry for Knuth and Riordan references).
Recursion: a(n)=123*a(n-1)-a(n-2)+((-1)^n)*A056566(n), n >= 2, a(0)=1, a(1)=89.
G.f.: exp( Sum_{k>=1} F(11*k)/F(k) * x^k/k ), where F(n) = A000045(n). - Seiichi Manyama, May 07 2025

A056587 Tenth power of Fibonacci numbers A000045.

Original entry on oeis.org

0, 1, 1, 1024, 59049, 9765625, 1073741824, 137858491849, 16679880978201, 2064377754059776, 253295162119140625, 31181719929966183601, 3833759992447475122176, 471584161164422542970449

Offset: 0

Wolfdieter Lang, Jul 10 2000

Divisibility sequence; that is, if n divides m, then a(n) divides a(m).

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 1, p. 85, (exercise 1.2.8. Nr. 30) and p. 492 (solution).

Vincenzo Librandi, Table of n, a(n) for n = 0..96
A. Brousseau, A sequence of power formulas, Fib. Quart., 6 (1968), 81-83.
J. Riordan, Generating functions for powers of Fibonacci numbers, Duke. Math. J. 29 (1962) 5-12.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (89,4895,-83215,-582505,1514513,1514513,-582505,-83215,4895,89,-1).

Cf. A000045, A007598, A056570, A056571, A056572, A056573, A056574, A056585, A056586, A056588, A055870.

[Fibonacci(n)^10: n in [0..20]]; // Vincenzo Librandi, Jun 04 2011

Fibonacci[Range[0,15]]^10 (* Harvey P. Dale, Jul 29 2018 *)

a(n) = fibonacci(n)^10; \\ Michel Marcus, Sep 06 2017

a(n) = F(n)^10, F(n)=A000045(n).
G.f.: x*p(10, x)/q(10, x) with p(10, x) := sum_{m=0..9} A056588(9, m)*x^m = (1-x)*(1 - 87*x - 4047*x^2 + 42186*x^3 + 205690*x^4 + 42186*x^5 - 4047*x^6 - 87*x^7 + x^8) and q(10, x) := sum_{m=0..11} A055870(11, m)*x^m = (1+x)*(1 - 3*x + x^2)*(1 + 7*x + x^2)*(1 - 18*x + x^2)*(1 + 47*x + x^2)*(1 - 123*x + x^2) (denominator factorization deduced from Riordan result).
Recursion (cf. Knuth's exercise): sum_{m=0..11} A055870(11, m)*a(n-m) = 0, n >= 11; inputs: a(n), n=0..10.  a(n) = 89*a(n-1) + 4895*a(n-2) - 83215*a(n-3) - 582505*a(n-4) + 1514513*a(n-5) + 1514513*a(n-6) - 582505*a(n-7) -83215*a(n-8) + 4895*a(n-9) + 89*a(n-10) - a(n-11).

More terms from Larry Reeves (larryr(AT)acm.org), Jul 17 2001

A156133 Denominator coefficients of infinite over the Fibonacci sequence: p(x,n)=(1 - x)Sum[Fibonacci[k]^nx^k, {k, 0, Infinity}]; t(n,m)=Coefficients(Denominator(p(x,n)).

Original entry on oeis.org

1, -1, 1, 1, 1, -2, -2, 1, 1, -3, -6, 3, 1, 1, -4, -19, -4, 1, -1, 8, 40, -60, -40, 8, 1, 1, -13, -104, 260, 260, -104, -13, 1, 1, -21, -273, 1092, 1820, -1092, -273, 21, 1, 1, -33, -747, 3894, 16270, 3894, -747, -33, 1, -1, 55, 1870, -19635, -85085, 136136, 85085

Offset: 0

Roger L. Bagula, Feb 04 2009

Row sums are:
{1, 1, -2, -4, -25, -44, 288, 1276, 22500, 96976, -1707552,...}.
The denominator and numerator polynomials appear to be new.

			{1},
{-1, 1, 1},
{1, -2, -2, 1},
{1, -3, -6, 3, 1},
{1, -4, -19, -4, 1},
{-1, 8, 40, -60, -40, 8, 1},
{1, -13, -104, 260, 260, -104, -13, 1},
{1, -21, -273, 1092, 1820, -1092, -273, 21, 1},
{1, -33, -747, 3894, 16270, 3894, -747, -33, 1},
{-1, 55, 1870, -19635, -85085, 136136, 85085, -19635, -1870, 55, 1},
{1, -89, -4895, 83215, 582505, -1514513, -1514513, 582505, 83215, -4895, -89, 1}

A000045, A055870

Clear[t0, p, x, n, m];
p[x_, n_] = (1 - x)*Sum[Fibonacci[k]^n*x^k, {k, 0, Infinity}]
Table[Denominator[FullSimplify[ExpandAll[p[x, n]]]], {n, 0, 10}];
Flatten[%]

p(x,n)=(1 - x)*Sum[Fibonacci[k]^n*x^k, {k, 0, Infinity}]; t(n,m)=Coefficients(Denominator(p(x,n)).

A317360 Triangle a(n, k) read by rows: coefficient triangle that gives Lucas powers and sums of Lucas powers.

Original entry on oeis.org

1, 1, 2, 1, 7, -4, 1, 24, -23, -8, 1, 76, -164, -79, 16, 1, 235, -960, -1045, 255, 32, 1, 716, -5485, -11155, 5940, 831, -64, 1, 2166, -29816, -116480, 109960, 32778, -2687, -128, 1, 6527, -158252, -1143336, 2024920, 1029844, -176257, -8703, 256, 1, 19628, -822291, -10851888, 34850816, 32711632, -9230829, -937812, 28159, 512

Offset: 0

Tony Foster III, Jul 26 2018

			n\k|  0  1      2       3        4       5         6        7      8     9
---+-------------------------------------------------------------------------
0  |  1
1  |  1  2
2  |  1  7     -4
3  |  1  24    -23     -8
4  |  1  76    -164    -79       16
5  |  1  235   -960    -1045     255     32
6  |  1  716   -5485   -11155    5940    831      -64
7  |  1  2166  -29816  -116480   109960  32778    -2687    -128
8  |  1  6527  -158252 -1143336  2024920 1029844  -176257  -8703   256
9  |  1  19628 -822291 -10851888 4850816 32711632 -9230829 -937812 28159 512

Cf. A000032, A000045, A055870, A010048, A027961, A005970, A305695.

lucas(p)=2*fibonacci(p+1)-fibonacci(p);
S(n, k) = (-1)^floor((k+1)/2)*(prod(j=0, k-1, fibonacci(n-j))/prod(j=1, k, fibonacci(j)));
T(n, k) = sum(j=0, k, lucas(k+1-j)^n * S(n+1, j));
tabl(m) = for (n=0, m, for (k=0, n, print1(T(n, k), ", ")); print);
tabl(9);

a(n, k) = Sum_{j=0..k} Lucas(k+1-j)^n * A055870(n+1, j).
Sum_{j=0..n} a(n, n-j) * A010048(k-1+j, n) = Lucas(k)^n.
Sum_{j=0..n} a(n, n-j) * A305695(k-2+j, n-1) = Sum_{t=1..k} Lucas(t)^n.

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A056567 Fibonomial coefficients.

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A056586 Ninth power of Fibonacci numbers A000045.

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A383715 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) = (-1)^floor((k+1)/2) * A099927(n,k).

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A056568 Fibonomial coefficients.

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A056587 Tenth power of Fibonacci numbers A000045.

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A156133 Denominator coefficients of infinite over the Fibonacci sequence: p(x,n)=(1 - x)*Sum[Fibonacci[k]^n*x^k, {k, 0, Infinity}]; t(n,m)=Coefficients(Denominator(p(x,n)).

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A317360 Triangle a(n, k) read by rows: coefficient triangle that gives Lucas powers and sums of Lucas powers.

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A156133 Denominator coefficients of infinite over the Fibonacci sequence: p(x,n)=(1 - x)Sum[Fibonacci[k]^nx^k, {k, 0, Infinity}]; t(n,m)=Coefficients(Denominator(p(x,n)).