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.

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A094443 Triangular array T(n,k) = Fibonacci(n+3-k)*C(n,k), k=0..n, n>=0.

Original entry on oeis.org

2, 3, 2, 5, 6, 2, 8, 15, 9, 2, 13, 32, 30, 12, 2, 21, 65, 80, 50, 15, 2, 34, 126, 195, 160, 75, 18, 2, 55, 238, 441, 455, 280, 105, 21, 2, 89, 440, 952, 1176, 910, 448, 140, 24, 2, 144, 801, 1980, 2856, 2646, 1638, 672, 180, 27, 2, 233, 1440, 4005, 6600, 7140, 5292, 2730, 960, 225, 30, 2
Offset: 0

Views

Author

Clark Kimberling, May 03 2004

Keywords

Examples

			First few rows:
   2;
   3,  2;
   5,  6,  2;
   8, 15,  9,  2;
  13, 32, 30, 12,  2;
  21, 65, 80, 50, 15, 2;

Links

Crossrefs

Cf. A000045, A094435, A094436, A094437, A094438, A094439, A094440, A094441, A094442, A094444.

Programs

  • GAP
    Flat(List([0..12], n-> List([0..n], k-> Binomial(n,k)*Fibonacci(n-k+3) ))); # G. C. Greubel, Oct 30 2019
  • Magma
    [Binomial(n,k)*Fibonacci(n-k+3): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 30 2019
  • Maple
    with(combinat): seq(seq(fibonacci(n-k+3)*binomial(n,k), k=0..n), n=0..12); # G. C. Greubel, Oct 30 2019
  • Mathematica
    Table[Fibonacci[n-k+3]*Binomial[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 30 2019 *)
  • PARI
    T(n,k) = binomial(n,k)*fibonacci(n-k+3);
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Oct 30 2019
  • Sage
    [[binomial(n,k)*fibonacci(n-k+3) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Oct 30 2019

Formula

From G. C. Greubel, Oct 30 2019: (Start)
T(n,k) = binomial(n,k)*Fibonacci(n-k+3).
Sum_{k=0..n} T(n,k) = Fibonacci(2*n+3).
Sum_{k=0..n} (-1)^k * T(n,k) = (-1)^n * Fibonacci(n-3). (End)

A094444 Triangular array T(n,k) = Fibonacci(n+4-k)*C(n,k), k=0..n, n>=0.

Original entry on oeis.org

3, 5, 3, 8, 10, 3, 13, 24, 15, 3, 21, 52, 48, 20, 3, 34, 105, 130, 80, 25, 3, 55, 204, 315, 260, 120, 30, 3, 89, 385, 714, 735, 455, 168, 35, 3, 144, 712, 1540, 1904, 1470, 728, 224, 40, 3, 233, 1296, 3204, 4620, 4284, 2646, 1092, 288, 45, 3, 377, 2330, 6480, 10680, 11550, 8568, 4410, 1560, 360, 50, 3
Offset: 0

Views

Author

Clark Kimberling, May 03 2004

Keywords

Comments

Row sums are Fibonacci numbers.
Row sums with alternating signs are Fibonacci numbers or their negatives.

Examples

			First few rows:
   3;
   5,   3;
   8,  10,   3;
  13,  24,  15,  3;
  21,  52,  48, 20,  3;
  34, 105, 130, 80, 25, 3;

Links

Crossrefs

Cf. A000045, A094435, A094436, A094437, A094438, A094439, A094440, A094441, A094442, A094443.

Programs

  • GAP
    Flat(List([0..12], n-> List([0..n], k-> Binomial(n,k)*Fibonacci(n-k+4) ))); # G. C. Greubel, Oct 30 2019
  • Magma
    [Binomial(n,k)*Fibonacci(n-k+4): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 30 2019
  • Maple
    with(combinat); seq(seq(fibonacci(n-k+4)*binomial(n,k), k=0..n), n=0..12); # G. C. Greubel, Oct 30 2019
  • Mathematica
    Table[Fibonacci[n-k+4]*Binomial[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 30 2019 *)
  • PARI
    T(n,k) = binomial(n,k)*fibonacci(n-k+4);
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Oct 30 2019
  • Sage
    [[binomial(n,k)*fibonacci(n-k+4) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Oct 30 2019

Formula

From G. C. Greubel, Oct 30 2019: (Start)
T(n,k) = binomial(n,k)*Fibonacci(n-k+4).
Sum_{k=0..n} T(n,k) = Fibonacci(2*n+4).
Sum_{k=0..n} (-1)^(k+1) * T(n,k) = (-1)^n * Fibonacci(n-4). (End)

A098473 Triangle T(n,k) read by rows, T(n, k) = binomial(2*k, k)*binomial(n, k), 0<=k<=n.

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 1, 6, 18, 20, 1, 8, 36, 80, 70, 1, 10, 60, 200, 350, 252, 1, 12, 90, 400, 1050, 1512, 924, 1, 14, 126, 700, 2450, 5292, 6468, 3432, 1, 16, 168, 1120, 4900, 14112, 25872, 27456, 12870, 1, 18, 216, 1680, 8820, 31752, 77616, 123552, 115830
Offset: 0

Views

Author

Paul Barry, Sep 09 2004

Keywords

Comments

This sequence gives the coefficients of the Jensen polynomials (increasing powers of x) of degree n and shift 0 for the central binomial sequence A000984. For a definition of Jensen polynomials see a comment in A094436. - Wolfdieter Lang, Jun 25 2019

Examples

			Rows begin
  1;
  1,  2;
  1,  4,  6;
  1,  6, 18,  20;
  1,  8, 36,  80,  70;
  1, 10, 60, 200, 350, 252;

Links

Crossrefs

Row sums are A026375.
Antidiagonal sums are A026569.
Principal diagonal is A000984.
Cf. A002426, A081671, A098409, A098410, A094436, A126869.

Programs

  • Maple
    A098473 := proc(n,k) binomial(2*k,k)*binomial(n,k) ; end proc:
  • Mathematica
    Table[Binomial[2k,k]Binomial[n,k],{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Aug 15 2020 *)
  • PARI
    T(n,k)=binomial(2*k, k)*binomial(n, k);
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print()); /* as triangle */

Formula

T(n, k) = binomial(2*k, k)*binomial(n, k).
Sum_{k=0..n} T(n,k)*x^(n-k) = A126869(n), A002426(n), A000984(n), A026375(n), A081671(n), A098409(n), A098410(n) for x = -2, -1, 0, 1, 2, 3, 4 respectively. - Philippe Deléham, Sep 28 2007
From Peter Bala, Jun 06 2011: (Start)
O.g.f.: 1/sqrt(1 - t)*1/sqrt(1 - t*(1 + 4*x)) = 1 + (2*x + 1)*t + (1 + 4*x + 6*x^2)*t^2 + ....
Let R_n(x) denote the row generating polynomials of this triangle, which begin
R_1(x) = 1 + 2*x, R_2(x) = 1 + 4*x + 6*x^2, R_3(x) = 1 + 6*x + 18*x^2 + 20*x^3.
Dasbach gives the following slowly converging series for the logarithm function:
log(x) = Sum_{n >= 1} 1/n*R_n(-1/x), valid for x >= 4.
The polynomials (1 - x)^n*R_n(x/(1 - x)) appear to be the row polynomials of A135091 (see also A171128). (End)

A374917 Inverse of the Fibonacci sequence beginning 1,1 with respect to binomial convolution.

Original entry on oeis.org

1, -1, 0, 3, -5, -18, 113, 35, -3044, 9755, 87999, -882894, -1155935, 69780087, -292042360, -5040306157, 64613044147, 197030202470, -10570955773551, 48865639709115, 1470783141900676, -21819085085811861, -123330624543827305, 6244177033369108298, -28216305335425392575, -1453926618188019546193
Offset: 0

Views

Author

Fernando Miranda, Maria Irene Falcao and Goncalo Carvalho, Jul 23 2024

Keywords

Comments

The binomial convolution of this sequence with the Fibonacci sequence beginning 1,1 gives the identity sequence with respect to convolution (A000007).

Links

Crossrefs

Cf. A000045, A001622, A094436.

Programs

  • Maple
    p:=(1-sqrt(5))/2: q:=(1+sqrt(5))/2:
egf := (1-2*q)/(p*exp(p*x)-q*exp(q*x)): ser := series(egf, x, 27):
seq(n!*simplify(coeff(ser, x, n)), n=0..25); # Peter Luschny, Aug 05 2024
  • Mathematica
    a[0] = 1; a[n_]:=a[n]= -Sum[Binomial[n, k] Fibonacci[k + 1] a[n - k], {k, 1, n}]
(* or, to generate the list L of the first n terms *)
phi = (1 + Sqrt[5])/2; psi = 1 - phi; L[n_] := CoefficientList[Series[(phi - psi)/(phi Exp[phi x] - psi Exp[psi x]), {x, 0, n}], x] Table[k!, {k, 0, n}]

Formula

a(0) = 1, a(n) = -Sum_{k=1..n} binomial(n, k)*a(n - k)*A000045(k+1).
E.g.f.: 1/G'(x) where G(x) is the e.g.f. of A000045.
The recursion P(0, x) = 1, P(n, x) = x^n - Sum_{k=0..n-1} binomial(n, k)*a(n-k)*P(k, x) defines the so-called Appell-Fibonacci polynomials P(n, x) = Sum_{k=0..n} T(n, k)*x^k, where T(n, k) is the triangular array A094436.
Previous Showing 11-14 of 14 results.

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