A039717 -id:A039717 - cp's OEIS frontend

A046757 Triangle of coefficients of certain polynomials (exponents in decreasing order).

1, 2, 1, 5, 5, 1, 30, 30, 10, 1, 272, 272, 102, 17, 1, 3250, 3250, 1300, 260, 26, 1, 47952, 47952, 19980, 4440, 555, 37, 1, 840350, 840350, 360150, 85750, 12250, 1050, 50, 1, 17039360, 17039360, 7454720, 1863680, 291200, 29120, 1820, 65, 1, 392203458

Offset: 0

Wolfdieter Lang

			Triangle begins:
  {1};
  {2,1};
  {5,5,1};
  {30,30,10,1};
  {272,272,102,17,1};
  ....
E.g. third row {5,5,1} corresponds to polynomial q(3,x)= 5*x^2+5*x+1.

x*p(k-1, -x)/q(k, -x), with the row polynomials p(n, x) from triangle A033842(n, m) is for k=1..5 g.f. for A000079 (powers of two), A039717, A043553, A045624, A046088, respectively.

a(n, n) = 1, a(n, m) = (1+n^2)*binomial(n, m)*n^(n-m-2), n>m >= 0, else 0.

A093132 Third binomial transform of Fibonacci(3n+2).

Original entry on oeis.org

1, 8, 60, 440, 3200, 23200, 168000, 1216000, 8800000, 63680000, 460800000, 3334400000, 24128000000, 174592000000, 1263360000000, 9141760000000, 66150400000000, 478668800000000, 3463680000000000, 25063424000000000

Offset: 0

Paul Barry, Mar 23 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-20).

Cf. A000032, A000045, A039717.

a:=[1,8];; for n in [2..30] do a[n]:=10*(a[n-1]-2*a[n-2]); od; a; # G. C. Greubel, Dec 27 2019

I:=[1,8]; [n le 2 select I[n] else 10*(Self(n-1) - 2*Self(n-2)): n in [1..30]]; // G. C. Greubel, Dec 27 2019

seq(coeff(series((1-2*x)/(1-10*x+20*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Dec 27 2019

Table[If[EvenQ[n], 2^n*5^(n/2)*Fibonacci[n+2], 2^n*5^((n-1)/2)*LucasL[n+2]], {n, 0, 30}] (* G. C. Greubel, Dec 27 2019 *)

my(x='x+O('x^30)); Vec((1-2*x)/(1-10*x+20*x^2)) \\ G. C. Greubel, Dec 27 2019

def A093132_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-2*x)/(1-10*x+20*x^2) ).list()
A093132_list(30) # G. C. Greubel, Dec 27 2019

G.f.: (1-2*x)/(1-10*x+20*x^2).
a(n) = ( (5 + 3*sqrt(5))*(5 + sqrt(5))^n + (5 - 3*sqrt(5))*(5 - sqrt(5))^n)/10.
a(n) = 2^n*A039717(n).
a(2*n) = 4^n*5^n*Fibonacci(2*n+2), a(2*n+1) = 2^(2*n+1)*5^n*Lucas(2*n+3). - G. C. Greubel, Dec 27 2019

cp's OEIS Frontend

A046757 Triangle of coefficients of certain polynomials (exponents in decreasing order).

Views

Author

Keywords

Examples

Crossrefs

Formula

A093132 Third binomial transform of Fibonacci(3n+2).

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula