cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A203006 (n-1)-st elementary symmetric function of the first n Fibonacci numbers.

Original entry on oeis.org

1, 2, 5, 17, 91, 758, 10094, 215094, 7378716, 408057060, 36439600740, 5258207000160, 1226732478115680, 462844011818878560, 282472779283129656000, 278884771717353348456000, 445462025196173918554440000, 1151206495594319717393795136000
Offset: 1

Views

Author

Clark Kimberling, Dec 29 2011

Keywords

Comments

From R. J. Mathar, Oct 01 2016: (Start)
The k-th elementary symmetric functions of the integers F(j), j=1..n, form a triangle T(n,k), 0<=k<=n, n>=0:
1;
1, 1;
1, 2, 1;
1, 4, 5, 2;
1, 7, 17, 17, 6;
which is the unsigned version of A158472. This here is the first subdiagonal. The diagonal seems to be A003266. The 2nd column is A000071, the 3rd A190173, the 4th A213787. (End)

Examples

			0th elementary symmetric function: 1
1st e.s.f. of {1,1}: 1+1=2
2nd e.s.f. of {1,1,2}: 1*1+1*2+2*2=5

Links

Crossrefs

Cf. A000045.

Programs

  • Maple
    f:= proc(n) local x,P,i;
P:= mul(x+combinat:-fibonacci(i),i=1..n);
coeff(P,x,1)
end proc:
map(f, [$1..20]); # Robert Israel, Aug 18 2024
  • Mathematica
    f[k_] := Fibonacci[k]; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 18}]  (* A203006 *)

A213626 a(n) = Sum_{0<=iA020985(m).

Original entry on oeis.org

0, 0, 1, -2, -2, 0, -5, -4, 0, 8, 21, 2, -10, -16, -15, -20, -20, -16, -7, -22, -14, 0, -21, -8, -28, -40, -45, -46, -42, -32, -49, -40, -24, 0, 33, -10, 22, 64, 11, 52, 104, 168, 245, 154, 78, 16, 65, 4, -44, -80, -105, -90, -114, -128, -123, -136, -132, -120
Offset: 0

Views

Author

Alois P. Heinz, Jun 23 2012

Keywords

Links

Crossrefs

Cf. A020985, A190173, A213786, A213787.

Programs

  • Maple
    b:= proc(n) option remember; local r;
      `if`(n=0, 1, `if`(irem(n, 2, 'r')=0, b(r), b(r)*(-1)^r))
    end:
s:= proc(j) option remember; `if`(j<0, 0, s(j-1)+b(j)       ) end:
t:= proc(k) option remember; `if`(k<1, 0, t(k-1)+b(k)*s(k-1)) end:
a:= proc(n) option remember; `if`(n<2, 0, a(n-1)+b(n)*t(n-1)) end:
seq(a(n), n=0..100);
  • Mathematica
    b[n_] := b[n] = Module[{q, r}, If[n==0, 1, {q, r}=QuotientRemainder[n, 2]; If[r == 0, b[q], b[q]*(-1)^q]]];
s[j_] := s[j] = If[j < 0, 0, s[j-1] + b[j]];
t[k_] := t[k] = If[k < 1, 0, t[k-1] + b[k]*s[k-1]];
a[n_] := a[n] = If[n < 2, 0, a[n-1] + b[n]*t[n-1]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jun 01 2022, after Alois P. Heinz *)
  • Python
    def A213626(n): return sum((-1 if (i&(i>>1)).bit_count()&1 else 1)*sum((-1 if (j&(j>>1)).bit_count()&1 else 1)*sum(-1 if (k&(k>>1)).bit_count()&1 else 1 for k in range(j+1,n+1)) for j in range(i+1,n+1)) for i in range(n+1)) # Chai Wah Wu, Feb 12 2023

A213807 a(n)=Sum(L(i)*L(j)*L(k), 0<=iA000032(m).

Original entry on oeis.org

0, 0, 6, 50, 295, 1450, 6706, 29790, 129900, 559680, 2395701, 10212620, 43430140, 184412740, 782337466, 3317046390, 14059122315, 59576034630, 252422169726, 1069418901650, 4530501461200, 19192481509300, 81303194179081, 344412501233400, 1458972161656920
Offset: 0

Views

Author

Jonathan Vos Post, Jun 20 2012

Keywords

Comments

This is to Lucas numbers A000032 as A213787 is to Fibonacci numbers A000045.

Links

Crossrefs

Cf. A000032, A213787.

Programs

  • Maple
    a:= n-> (Matrix(10, (i, j)-> `if`(i=j-1, 1, `if`(i=10,
         [-1, -1, 16, -3, -51, 24, 45, -27, -8, 7][j], 0)))^(n+3).
         <<-20, 1, 0, 0, 0, 6, 50, 295, 1450, 6706>>)[1, 1]:
seq (a(n), n=0..30);  # Alois P. Heinz, Jun 21 2012

Formula

G.f.: -(x^6-19*x^5+4*x^4+53*x^3+7*x^2-8*x-6)*x^2 / ((x-1) * (x+1) * (x^2-x-1) * (x^2+x-1)*(x^2+4*x-1)*(x^2-3*x+1)). - Alois P. Heinz, Jun 21 2012

A359045 a(n) = Sum_{1<=iA020985(m).

Original entry on oeis.org

0, 0, 0, -1, -2, -2, -4, -5, -4, 0, 8, -5, -12, -14, -16, -17, -20, -20, -16, -25, -22, -14, -28, -21, -34, -40, -40, -45, -46, -42, -52, -49, -40, -24, 0, -33, -10, 22, -20, 11, 52, 104, 168, 91, 28, -22, 16, -33, -70, -96, -112, -105, -120, -126, -128, -133
Offset: 0

Views

Author

Chai Wah Wu, Feb 12 2023

Keywords

Crossrefs

Cf. A020985, A190173, A213626, A213786, A213787.

Programs

  • Python
    def A359045(n): return sum((-1 if (i&(i>>1)).bit_count()&1 else 1)*sum((-1 if (j&(j>>1)).bit_count()&1 else 1)*sum(-1 if (k&(k>>1)).bit_count()&1 else 1 for k in range(j+1,n+1)) for j in range(i+1,n+1)) for i in range(1,n+1))

Formula

a(n) = A213626(n)-A213786(n).
Showing 1-4 of 4 results.

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