A117716 -id:A117716 - cp's OEIS frontend

A117715 Triangle, read by rows, T(n, k) = Fibonacci(n, k), where Fibonacci(n, x) is the Fibonacci polynomial.

0, 1, 1, 0, 1, 2, 1, 2, 5, 10, 0, 3, 12, 33, 72, 1, 5, 29, 109, 305, 701, 0, 8, 70, 360, 1292, 3640, 8658, 1, 13, 169, 1189, 5473, 18901, 53353, 129949, 0, 21, 408, 3927, 23184, 98145, 328776, 927843, 2298912, 1, 34, 985, 12970, 98209, 509626, 2026009, 6624850, 18674305, 46866034

Offset: 0

Roger L. Bagula, Apr 13 2006

			Triangle begins as:
  0;
  1,  1;
  0,  1,   2;
  1,  2,   5,   10;
  0,  3,  12,   33,   72;
  1,  5,  29,  109,  305,   701;
  0,  8,  70,  360, 1292,  3640,  8658;
  1, 13, 169, 1189, 5473, 18901, 53353, 129949;

Steven Wolfram, The Mathematica Book, Cambridge University Press, 3rd ed. 1996, page 728

Alois P. Heinz, Rows n = 0..140, flattened
Eric Weisstein's World of Mathematics, Fibonacci Polynomial.
Wikipedia, Fibonacci Polynomial

Cf. A000045, A117716, A049310, A073133, A157103 (antidiagonals).
Main diagonal and first lower diagonal give: A084844, A084845.
Cf. A352361.

A117715:= func< n, k | k eq 0 select (n mod 2) else Evaluate(DicksonSecond(n-1, -1), k) >;
[A117715(n, k): k in [0..n], n in [0..13]]; // G. C. Greubel, Oct 01 2024

with(combinat):for n from 0 to 9 do seq(fibonacci(n,m), m = 0 .. n) od; # Zerinvary Lajos, Apr 09 2008

Table[Fibonacci[n, k], {n,0,12}, {k,0,n}]//Flatten

from sympy import fibonacci
def T(n, m): return 0 if n==0 else fibonacci(n, m)
for n in range(21): print([T(n, m) for m in range(n + 1)]) # Indranil Ghosh, Aug 12 2017

def A117715(n,k): return lucas_number1(n, k, -1)
flatten([[A117715(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 01 2024

T(n, 1) = A000045(n).
T(n, 3) = A006190(n).
T(n, 4) = A001076(n).
T(n, 5) = A052918(n-1).
T(5, k) = A057721(k).
T(6, k) = A124152(k).
T(n, k) = (-1)^(n-1)*A352361(n-k, n). - G. C. Greubel, Oct 01 2024

Definition simplified by the Assoc. Editors of the OEIS, Nov 17 2009

A117724 Triangle T(n,k) = coefficient [x^n] of x^2/(1-(k+1)*x^2-x^3) for row n, and columns k = 0..n, read by rows.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 2, 3, 4, 5, 1, 1, 1, 1, 1, 1, 1, 4, 9, 16, 25, 36, 49, 2, 4, 6, 8, 10, 12, 14, 16, 2, 9, 28, 65, 126, 217, 344, 513, 730, 3, 12, 27, 48, 75, 108, 147, 192, 243, 300, 4, 22, 90, 268, 640, 1314, 2422, 4120, 6588, 10030, 14674

Offset: 0

Roger L. Bagula, Apr 13 2006

			The table starts:
  0;
  0,  0;
  1,  1,  1;
  0,  0,  0,  0;
  1,  2,  3,  4,   5;
  1,  1,  1,  1,   1,   1;
  1,  4,  9, 16,  25,  36,  49;
  2,  4,  6,  8,  10,  12,  14,  16;
  2,  9, 28, 65, 126, 217, 344, 513, 730;
  3, 12, 27, 48,  75, 108, 147, 192, 243, 300;

Nathaniel Johnston, Rows n = 0..50, flattened

Cf. A000931, A008346, A052931, A117716, A119282.

m:=12;
R:=PowerSeriesRing(Integers(), m+2);
A117724:= func< n, k | Coefficient(R!( x^2/(1-(k+1)*x^2-x^3) ), n) >;
[A117724(n, k): k in [0..n], n in [0..m]]; // G. C. Greubel, Jul 23 2023

t:=taylor(x^2/(1-(k+1)*x^2-x^3), x, 15):
seq(seq(coeff(t,x,n), k=0..n),n=0..12); # Nathaniel Johnston, Apr 27 2011

T[n_, k_]:= T[n, k]= Coefficient[Series[x^2/(1-(k+1)*x^2-x^3), {x,0,n+ 2}], x, n];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten

def A117724(n, k):
    P. = PowerSeriesRing(QQ)
    return P( x^2/(1-(k+1)*x^2-x^3) ).list()[n]
flatten([[A117724(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 23 2023

T(n,k) = coefficient [x^n] ( x^2/(1-(k+1)*x^2-x^3) ).
T(n, 0) = A000931(n+1).
T(n, 1) = A008346(n-2) = (-1)^(n-1)*A119282(n-1).
T(n, 2) = A052931(n-2).

Edited by G. C. Greubel, Jul 23 2023

A368892 a(n) = Sum_{k=0..floor(n/3)} n^(n-3k) binomial(n-2*k,k).

Original entry on oeis.org

1, 1, 4, 28, 264, 3200, 47521, 835569, 16974208, 391147867, 10080150040, 287244283821, 8967781893889, 304393809948904, 11160668048222588, 439582708115133751, 18509867068477014112, 829768603643818659302, 39454459640462073466945

Offset: 0

Seiichi Manyama, Jan 09 2024

Cf. A368891, A368893.

Cf. A117716, A084845.

Join[{1}, Table[n^n * HypergeometricPFQ[{1/3 - n/3, 2/3 - n/3, -n/3}, {1/2 - n/2, -n/2}, -27/(4*n^3)], {n, 1, 20}]] (* Vaclav Kotesovec, Jan 09 2024 *)

a(n) = sum(k=0, n\3, n^(n-3*k)*binomial(n-2*k, k));

a(n) = [x^n] 1/(1 - n*x - x^3).

a(n) ~ n^n. - Vaclav Kotesovec, Jan 09 2024

cp's OEIS Frontend

A117715 Triangle, read by rows, T(n, k) = Fibonacci(n, k), where Fibonacci(n, x) is the Fibonacci polynomial.

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A117724 Triangle T(n,k) = coefficient [x^n] of x^2/(1-(k+1)*x^2-x^3) for row n, and columns k = 0..n, read by rows.

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A368892 a(n) = Sum_{k=0..floor(n/3)} n^(n-3k) binomial(n-2*k,k).

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cp's OEIS Frontend

A117715 Triangle, read by rows, T(n, k) = Fibonacci(n, k), where Fibonacci(n, x) is the Fibonacci polynomial.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Python

SageMath

Formula

Extensions

A117724 Triangle T(n,k) = coefficient [x^n] of x^2/(1-(k+1)*x^2-x^3) for row n, and columns k = 0..n, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

Extensions

A368892 a(n) = Sum_{k=0..floor(n/3)} n^(n-3*k) * binomial(n-2*k,k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A368892 a(n) = Sum_{k=0..floor(n/3)} n^(n-3k) binomial(n-2*k,k).