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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A094436 Triangular array T(n,k) = Fibonacci(k+1)*binomial(n,k) for k = 0..n; n >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 3, 1, 4, 12, 12, 5, 1, 5, 20, 30, 25, 8, 1, 6, 30, 60, 75, 48, 13, 1, 7, 42, 105, 175, 168, 91, 21, 1, 8, 56, 168, 350, 448, 364, 168, 34, 1, 9, 72, 252, 630, 1008, 1092, 756, 306, 55, 1, 10, 90, 360, 1050, 2016, 2730, 2520, 1530, 550, 89
Offset: 0

Views

Author

Clark Kimberling, May 03 2004

Keywords

Comments

Let F(n) denote the n-th Fibonacci number (A000045). Then n-th row sum of T is F(2n+1) and n-th alternating row sum is F(n-1).
A094436 is jointly generated with A094437 as a triangular array of coefficients of polynomials u(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x) = u(n-1,x) + x*v(n-1,x) and v(n,x) = x*u(n-1,x) + (x+1)*v(n-1,x). See the Mathematica section. - Clark Kimberling, Feb 26 2012
Subtriangle of the triangle given by (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 26 2012
This sequence gives the coefficients of the Jensen polynomials (increasing powers of x) for the sequence {A000045(k)}{k >= 0} of degree n with shift 1. Here the definition of Jensen polynomials of degree n and shift m of an arbitrary real sequence {s(k)}{k >= 0} is used: J(s,m;n,x) := Sum_{j=0..n} binomial(n,j)*s(j + m)*x^j, This definition is used by Griffin et al. with a different notation. - Wolfdieter Lang, Jun 25 2019

Examples

			First four rows:
  1
  1 1
  1 2 2
  1 3 6 3
Sum = 1+3+6+3=13=F(7); alt.Sum = 1-3+6-3=1=F(2).
T(3,2)=F(3)C(3,2)=2*3=6.
From _Philippe Deléham_, Mar 26 2012: (Start)
(1, 0, 0, 1, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, ...) begins :
  1
  1, 0
  1, 1, 0
  1, 2, 2, 0
  1, 3, 6, 3, 0
  1, 4, 12, 12, 5, 0
  1, 5, 20, 30, 25, 8, 0
  1, 6, 30, 60, 75, 48, 13, 0 . (End)

Links

Crossrefs

Cf. A000045.
Cf. A094435, A094437, A094438, A094439, A094440, A094441, A094442, A094443, A094444.

Programs

  • GAP
    Flat(List([0..12], n-> List([0..n], k-> Fibonacci(k+1)* Binomial(n,k) ))); # G. C. Greubel, Jul 11 2019
  • Magma
    [Fibonacci(k+1)*Binomial(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 11 2019
  • Maple
    with(combinat); seq(seq(fibonacci(k+1)*binomial(n,k), k=0..n), n=0..12); # G. C. Greubel, Oct 30 2019
  • Mathematica
    (* First program *)
u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A094436 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A094437 *)
(* Second program *)
Table[Fibonacci[k+1]*Binomial[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 11 2019 *)
  • PARI
    T(n,k) = fibonacci(k+1)*binomial(n,k); \\ G. C. Greubel, Jul 11 2019
  • Sage
    [[fibonacci(k+1)*binomial(n,k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jul 11 2019

Formula

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(1,1) = 1 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 26 2012
G.f. (-1+x)/(-1+2*x+x*y-x^2*y+x^2*y^2-x^2). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Oct 30 2019: (Start)
T(n, k) = binomial(n, k)*Fibonacci(k+1).
Sum_{k=0..n} T(n,k) = Fibonacci(2*n+1).
Sum_{k=0..n} (-1)^k*T(n,k) = Fibonacci(n-1). (End)

Extensions

Offset set to 0 by Alois P. Heinz, Aug 11 2015

A094435 Triangular array read by rows: T(n,k) = Fibonacci(k)*C(n,k), k = 1...n; n>=1.

Original entry on oeis.org

1, 2, 1, 3, 3, 2, 4, 6, 8, 3, 5, 10, 20, 15, 5, 6, 15, 40, 45, 30, 8, 7, 21, 70, 105, 105, 56, 13, 8, 28, 112, 210, 280, 224, 104, 21, 9, 36, 168, 378, 630, 672, 468, 189, 34, 10, 45, 240, 630, 1260, 1680, 1560, 945, 340, 55, 11, 55, 330, 990, 2310, 3696, 4290, 3465, 1870, 605, 89
Offset: 1

Views

Author

Clark Kimberling, May 03 2004

Keywords

Comments

Let F(n) denote the n-th Fibonacci number (A000045). Then n-th row sum of T is F(2n) and n-th alternating row sum is F(n).

Examples

			First few rows:
  1;
  2   1;
  3   3   2;
  4   6   8   3;
  5, 10, 20, 15,  5;
  6, 15, 40, 45, 30, 8;

Links

Crossrefs

Cf. A000045.
Cf. A094436, A094437, A094438, A094439, A094440, A094441, A094442, A094443, A094444.

Programs

  • GAP
    Flat(List([1..12], n-> List([1..n], k-> Binomial(n,k)*Fibonacci(k) ))); # G. C. Greubel, Oct 30 2019
  • Magma
    [Binomial(n,k)*Fibonacci(k): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 30 2019
  • Maple
    with(combinat); seq(seq(binomial(n,k)*fibonacci(k), k=1..n), n=1..12); # G. C. Greubel, Oct 30 2019
  • Mathematica
    Table[Fibonacci[k]*Binomial[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Oct 30 2019 *)
  • PARI
    T(n,k) = binomial(n,k)*fibonacci(k);
for(n=1,12, for(k=1,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Oct 30 2019
  • Sage
    [[binomial(n,k)*fibonacci(k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Oct 30 2019

Formula

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

A094437 Triangular array T(n,k) = Fibonacci(k+2)*C(n,k), k=0..n, n>=0.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 1, 6, 9, 5, 1, 8, 18, 20, 8, 1, 10, 30, 50, 40, 13, 1, 12, 45, 100, 120, 78, 21, 1, 14, 63, 175, 280, 273, 147, 34, 1, 16, 84, 280, 560, 728, 588, 272, 55, 1, 18, 108, 420, 1008, 1638, 1764, 1224, 495, 89, 1, 20, 135, 600, 1680, 3276, 4410, 4080, 2475, 890
Offset: 0

Views

Author

Clark Kimberling, May 03 2004

Keywords

Comments

Let F(n) denote the n-th Fibonacci number (A000045). Then n-th row sum of T is F(2n+2) and n-th alternating row sum is -F(n-2).
A094437 is jointly generated with A094436 as a triangular array of coefficients of polynomials v(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+x*v(n-1)x and v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x). See the Mathematica section. [Clark Kimberling, Feb 26 2012]
Subtriangle of the triangle given by (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 28 2012

Examples

			First four rows:
  1;
  1 2;
  1 4 3;
  1 6 9 5;
sum = 1+6+9+5=21=F(8); alt.sum = 1-6+9-5=-1=-F(1).
T(3,2)=F(4)*C(3,2)=3*3=9.
From _Philippe Deléham_, Apr 28 2012: (Start)
(1, 0, 0, 1, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, ...) begins :
  1;
  1, 0;
  1, 2,  0;
  1, 4,  3,  0;
  1, 6,  9,  5, 0;
  1, 8, 18, 20, 8, 0; . (End)

Links

Crossrefs

Cf. A000045.
Cf. A094435, A094436, A094438, A094439, A094440, A094441, A094442, A094443, A094444.

Programs

  • GAP
    Flat(List([0..12], n-> List([0..n], k-> Binomial(n,k)*Fibonacci(k+2) ))); # G. C. Greubel, Oct 30 2019
  • Magma
    [Binomial(n,k)*Fibonacci(k+2): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 30 2019
  • Maple
    with(combinat); seq(seq(fibonacci(k+2)*binomial(n,k), k=0..n), n=0..12); # G. C. Greubel, Oct 30 2019
  • Mathematica
    (* First program *)
u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A094436 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A094437 *)
(* Second program *)
Table[Fibonacci[k+2]*Binomial[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 30 2019 *)
  • PARI
    T(n,k) = binomial(n,k)*fibonacci(k+2);
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Oct 30 2019
  • Sage
    [[binomial(n,k)*fibonacci(k+2) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Oct 30 2019

Formula

From Philippe Deléham, Apr 28 2012: (Start)
As DELTA-triangle T(n,k):
G.f.: (1-x-y*x+2*y*x^2-y^2*x^2)/(1-2*x-y*x+x^2+y*x^2-y^2*x^2).
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(2,1) = 2, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k<0 or if k>n. (End)
From G. C. Greubel, Oct 30 2019: (Start)
T(n, k) = binomial(n, k)*Fibonacci(k+2).
Sum_{k=0..n} T(n,k) = Fibonacci(2*n+2).
Sum_{k=0..n} (-1)^(k+1) * T(n,k) = Fibonacci(n-2). (End)

A094442 Triangular array T(n,k) = Fibonacci(n+2-k)*C(n,k), 0 <= k <= n.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 5, 9, 6, 1, 8, 20, 18, 8, 1, 13, 40, 50, 30, 10, 1, 21, 78, 120, 100, 45, 12, 1, 34, 147, 273, 280, 175, 63, 14, 1, 55, 272, 588, 728, 560, 280, 84, 16, 1, 89, 495, 1224, 1764, 1638, 1008, 420, 108, 18, 1, 144, 890, 2475, 4080, 4410, 3276, 1680, 600, 135, 20, 1
Offset: 0

Views

Author

Clark Kimberling, May 03 2004

Keywords

Comments

Triangle of coefficients of polynomials v(n,x) jointly generated with A094441; see the Formula section.
Column 1: Fibonacci numbers, A000045
Row sums: even-indexed Fibonacci numbers
Alternating row sum: signed Fibonacci numbers
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (0, 2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 02 2012

Examples

			First five rows:
  1;
  2,  1;
  3,  4,  1;
  5,  9,  6, 1;
  8, 20, 18, 8, 1;
First three polynomials v(n,x): 1, 2 + x, 3 + 4x + x^2.
From _Philippe Deléham_, Apr 02 2012: (Start)
(0, 2, -1/2, -1/2, 0, 0, 0, ...) DELTA (1, 0, 0, 1, 0, 0, ...) begins:
  1;
  0, 1;
  0, 2,  1;
  0, 3,  4,  1;
  0, 5,  9,  6, 1;
  0, 8, 20, 18, 8, 1. (End)

Links

Crossrefs

Cf. A000045, A094435, A094436, A094437, A094438, A094439, A094440, A094441, A094443, A094444.

Programs

  • GAP
    Flat(List([0..12], n-> List([0..n], k-> Binomial(n,k)*Fibonacci(n-k+2) ))); # G. C. Greubel, Oct 30 2019
  • Magma
    [Binomial(n,k)*Fibonacci(n-k+2): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 30 2019
  • Maple
    with(combinat); seq(seq(fibonacci(n-k+2)*binomial(n,k), k=0..n), n=0..12); # G. C. Greubel, Oct 30 2019
  • Mathematica
    (* First program *)
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
v[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A094441 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A094442 *)
(* Second program *)
Table[Fibonacci[n-k+2]*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+2);
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+2) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Oct 30 2019

Formula

Let u(n,x) = x*u(n-1,x) + v(n-1,x) and v(n,x) = u(n-1,x) + (x+1)*v(n-1, x), where u(1,x)=1, v(1,x)=1 then the coefficients of the polynomials of v(n,x) produce this sequence.
T(n,k) = T(n-1, k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-1) - T(n-2,k-2), T(1,0) = T(2,1) = 1, T(2,0) = 2 and T(n,k) = 0 if k < 0 or if k >= n. - Philippe Deléham, Apr 02 2012
From G. C. Greubel, Oct 30 2019: (Start)
T(n,k) = binomial(n,k)*Fibonacci(n-k+2).
Sum_{k=0..n} T(n,k) = Fibonacci(2*n+2)
Sum_{k=0..n} (-1)^(k+1) * T(n,k) = (-1)^n * Fibonacci(n-2). (End)

A367301 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 3 + 3*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 2*x - x^2.

Original entry on oeis.org

1, 3, 3, 10, 16, 8, 33, 75, 63, 21, 109, 320, 380, 220, 55, 360, 1296, 1980, 1620, 720, 144, 1189, 5070, 9459, 9940, 6255, 2262, 377, 3927, 19353, 42615, 54561, 44085, 22635, 6909, 987, 12970, 72532, 184034, 277480, 272854, 179972, 78230, 20672, 2584
Offset: 1

Views

Author

Clark Kimberling, Dec 23 2023

Keywords

Comments

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

Examples

			First eight rows:
     1
     3      3
    10     16      8
    33     75     63     21
   109    320    380    220     55
   360   1296   1980   1620    720    144
  1189   5070   9459   9940   6255   2262   377
  3927  19353  42615  54561  44085  22635  6909  987
Row 4 represents the polynomial p(4,x) = 33 + 75*x + 63*x^2 + 21*x^3, so (T(4,k)) = (33,75,63,21), k=0..3.

Links

Crossrefs

Cf. A006190 (column 1); A001906 (p(n,n-1)); A154244 (row sums, p(n,1)); A077957 (alternating row sums, p(n,-1)); A190984 (p(n,2)); A006190 (signed, p(n,-2)); A154244 (p(n,-3)); A190984 (p(n,-4)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300.

Programs

  • Mathematica
    p[1, x_] := 1; p[2, x_] := 3 + 3 x; u[x_] := p[2, x]; v[x_] := 1 - 2 x - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

Formula

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 3 + 3*x, u = p(2,x), and v = 1 - 2*x - x^2.
p(n,x) = k*(b^n - c^n), where k = -(1/sqrt(13 + 10*x + 5*x^2)), b = (1/2) (3*x + 3 + 1/k), c = (1/2) (3*x + 3 - 1/k).

A368518 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 + 3*x^2.

Original entry on oeis.org

1, 1, 2, 2, 4, 7, 3, 10, 18, 20, 5, 20, 51, 68, 61, 8, 40, 118, 220, 251, 182, 13, 76, 264, 584, 905, 888, 547, 21, 142, 558, 1452, 2678, 3540, 3076, 1640, 34, 260, 1145, 3380, 7279, 11536, 13418, 10456, 4921, 55, 470, 2286, 7548, 18391, 33990, 47600, 49552
Offset: 1

Views

Author

Clark Kimberling, Jan 22 2024

Keywords

Comments

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

Examples

			First eight rows:
   1
   1    2
   2    4    7
   3   10   18    20
   5   20   51    68    61
   8   40  118   220   251   182
  13   76  264   584   905   888   547
  21  142  558  1452  2678  3540  3076  1640
Row 4 represents the polynomial p(4,x) = 3 + 10*x + 18*x^2 + 20*x^3, so (T(4,k)) = (3,10,18,20), k=0..3.

Links

Crossrefs

Cf. A000045 (column 1); A002605, (p(n,n-1)); A030195 (row sums), (p(n,1)); A182228 (alternating row sums), (p(n,-1)); A015545, (p(n,2)); A099012, (p(n,-2)); A087567, (p(n,3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150, A368151, A368152, A368153, A368154, A368155, A368156.

Programs

  • Mathematica
    p[1, x_] := 1; p[2, x_] := 1 + 2 x; u[x_] := p[2, x]; v[x_] := 1 + 3x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

Formula

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 1 + 2*x, u = p(2,x), and v = 1 + 32*x^2.
p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 + 4*x + 16*x^2), b = (1/2)*(2*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).

A094438 Triangular array T(n,k) = Fibonacci(k+3)*C(n,k), k=0..n, n>=0.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, May 03 2004

Keywords

Comments

Let F(n) denote the n-th Fibonacci number (A000045). Then n-th row sum of T is F(2n+3) and n-th alternating row sum is F(n-3).

Examples

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

Links

Crossrefs

Cf. A000045, A094435, A094436, A094437, A094439.
Cf. A094440, A094441, A094442, A094443, A094444.

Programs

  • GAP
    Flat(List([0..12], n-> List([0..n], k-> Binomial(n,k)*Fibonacci(k+3) ))); # G. C. Greubel, Oct 30 2019
  • Magma
    [Binomial(n,k)*Fibonacci(k+3): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 30 2019
  • Maple
    with(combinat); seq(seq(fibonacci(k+3)*binomial(n,k), k=0..n), n=0..12); # G. C. Greubel, Oct 30 2019
  • Mathematica
    Table[Fibonacci[k+3]Binomial[n,k],{n,0,12},{k,0,n}]//Flatten (* Harvey P. Dale, Dec 16 2017 *)
  • PARI
    T(n,k) = binomial(n,k)*fibonacci(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(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(k+3).
Sum_{k=0..n} T(n,k) = Fibonacci(2*n+3).
Sum_{k=0..n} (-1)^k * T(n,k) = Fibonacci(n-3). (End)

A094439 Triangular array T(n,k) = Fibonacci(k+4)*C(n,k), k=0..n, n>=0.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, May 03 2004

Keywords

Comments

Let F(n) denote the n-th Fibonacci number (A000045). Then n-th row sum of T is F(2n+4) and n-th alternating row sum is -F(n-4).

Examples

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

Links

Crossrefs

Cf. A000045, A007318, A094435, A094436, A094437, A094438, A094440, A094441, A094442, A094443, A094444.

Programs

  • GAP
    Flat(List([0..12], n-> List([0..n], k-> Binomial(n,k)* Fibonacci(k+4) ))); # G. C. Greubel, Oct 30 2019
  • Magma
    [Binomial(n,k)*Fibonacci(k+4): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 30 2019
  • Maple
    with(combinat); seq(seq(fibonacci(k+4)*binomial(n,k), k=0..n), n=0..12); # G. C. Greubel, Oct 30 2019
  • Mathematica
    Table[Fibonacci[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(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(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(k+4).
Sum_{k=0..n} T(n,k) = Fibonacci(2*n+4).
Sum_{k=0..n} (-1)^(k+1) * T(n,k) = Fibonacci(n-4). (End)

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)
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