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 1-7 of 7 results.

A060920 Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting leading zeros).

Original entry on oeis.org

1, 2, 1, 5, 5, 1, 13, 20, 9, 1, 34, 71, 51, 14, 1, 89, 235, 233, 105, 20, 1, 233, 744, 942, 594, 190, 27, 1, 610, 2285, 3522, 2860, 1295, 315, 35, 1, 1597, 6865, 12473, 12402, 7285, 2534, 490, 44, 1, 4181, 20284, 42447, 49963, 36122, 16407, 4578, 726, 54, 1
Offset: 0

Views

Author

Wolfdieter Lang, Apr 20 2001

Keywords

Comments

Companion triangle (odd-indexed members) A060921.

Examples

			Triangle begins as:
      1;
      2,     1;
      5,     5,      1;
     13,    20,      9,      1;
     34,    71,     51,     14,      1;
     89,   235,    233,    105,     20,     1;
    233,   744,    942,    594,    190,    27,     1;
    610,  2285,   3522,   2860,   1295,   315,    35,    1;
   1597,  6865,  12473,  12402,   7285,  2534,   490,   44,    1;
   4181, 20284,  42447,  49963,  36122, 16407,  4578,  726,   54,  1;
  10946, 59155, 140109, 190570, 163730, 91959, 33705, 7776, 1035, 65, 1;

Links

Crossrefs

Column sequences: A001519 (k=0), A054444 (k=1), A061178 (k=2), A061179 (k=3), A061180 (k=4), A061181 (k=5).
Cf. A000045, A001519, A007583, A037027, A060921, A061176.

Programs

  • Magma
    A060920:= func< n,k | (&+[Binomial(2*n-k-j, j)*Binomial(2*n-k-2*j, k): j in [0..2*n-k]]) >;
[A060920(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 06 2021
  • Mathematica
    A060920[n_, k_]:= Sum[Binomial[2*n-k-j, j]*Binomial[2*n-k-2*j, k], {j,0,2*n-k}];
Table[A060920[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 06 2021 *)
  • Sage
    def A060920(n,k): return sum(binomial(2*n-k-j, j)*binomial(2*n-k-2*j, k) for j in (0..2*n-k))
flatten([[A060920(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 06 2021

Formula

T(n, k) = A037027(2*n-k, k).
T(n, k) = ((2*(n-k) + 1)*A060921(n-1, k-1) + 4*n*T(n-1, k-1))/(5*k), n >= k >= 1.
T(n, 0) = F(n)^2 + F(n+1)^2 = A001519(n), with the Fibonacci numbers F(n) = A000045(n).
Sum_{k=0..n} T(n, k) = (2^(2*n + 1) + 1)/3 = A007583(n).
G.f. for column m >= 0: x^m*pFe(m+1, x)/(1-3*x+x^2)^(m+1), where pFe(n, x) := Sum_{m=0..n} A061176(n, m)*x^m (row polynomials of signed triangle A061176).
G.f.: (1-x*(1+y))/(1 - (3+2*y)*x + (1+y)^2*x^2). - Vladeta Jovovic, Oct 11 2003

A061177 Coefficients of polynomials ( (1 -x +sqrt(x))^(n+1) - (1 -x -sqrt(x))^(n+1) )/(2*sqrt(x)).

Original entry on oeis.org

1, 2, -2, 3, -5, 3, 4, -8, 8, -4, 5, -10, 11, -10, 5, 6, -10, 6, -6, 10, -6, 7, -7, -14, 29, -14, -7, 7, 8, 0, -56, 120, -120, 56, 0, -8, 9, 12, -126, 288, -365, 288, -126, 12, 9, 10, 30, -228, 540, -770, 770, -540, 228, -30, -10, 11, 55, -363, 858
Offset: 0

Views

Author

Wolfdieter Lang, Apr 20 2001

Keywords

Comments

The row polynomial pFo(m,x) = Sum_{j=0..m} T(m, j)*x^j is the numerator of the g.f. for the m-th column sequence of A060921, the odd part of the bisected Fibonacci triangle.

Examples

			The first few polynomials are:
pFo(0, x) = 1.
pFo(1, x) = 2 -  2*x.
pFo(2, x) = 3 -  5*x +  3*x^2.
pFo(3, x) = 4 -  8*x +  8*x^2 -  4*x^3.
pFo(4, x) = 5 - 10*x + 11*x^2 - 10*x^3 +  5*x^4.
pFo(5, x) = 6 - 10*x +  6*x^2 -  6*x^3 + 10*x^4 - 6*x^5.
Number triangle begins as:
   1;
   2,  -2;
   3,  -5,    3;
   4,  -8,    8,  -4;
   5, -10,   11, -10,    5;
   6, -10,    6,  -6,   10,  -6;
   7,  -7,  -14,  29,  -14,  -7,    7;
   8,   0,  -56, 120, -120,  56,    0,  -8;
   9,  12, -126, 288, -365, 288, -126,  12,   9;
  10,  30, -228, 540, -770, 770, -540, 228, -30, -10;

Links

Crossrefs

Cf. A059841, A060921, A061176 (companion triangle).

Programs

  • Magma
    A061177:= func< n,k | (&+[(-1)^(k+j)*Binomial(n+1,2*j+1)*Binomial(n-2*j,k-j): j in [0..k]]) >;
[A061177(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Apr 06 2021
  • Mathematica
    T[n_, k_]:= Sum[(-1)^(k-j)*Binomial[n+1, 2*j+1]*Binomial[n-2*j, k-j], {j,0,k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 06 2021 *)
  • Sage
    def A061177(n,k): return sum((-1)^(k+j)*binomial(n+1,2*j+1)*binomial(n-2*j,k-j) for j in (0..k))
flatten([[A061177(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Apr 06 2021

Formula

T(n, k) = coefficient of x^k of ( (1 -x +sqrt(x))^(n+1) - (1 -x -sqrt(x))^(n+1) )/(2*sqrt(x)).
T(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(n+1, 2*j+1)*binomial(n-2*j, k-j), if 0 <= k <= floor(n/2), T(n, k) = (-1)^n*T(n, n-k) if floor(n/2) < k <= n else 0.
Sum_{k=0..n} T(n, k) = (1 + (-1)^n)/2 = A059841(n). - G. C. Greubel, Apr 06 2021

A180957 Generalized Narayana triangle for (-1)^n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, -2, -5, -2, 1, 1, -5, -15, -15, -5, 1, 1, -9, -30, -41, -30, -9, 1, 1, -14, -49, -77, -77, -49, -14, 1, 1, -20, -70, -112, -125, -112, -70, -20, 1, 1, -27, -90, -126, -117, -117, -126, -90, -27, 1, 1, -35, -105, -90, 45, 131, 45, -90, -105, -35, 1
Offset: 0

Views

Author

Paul Barry, Sep 28 2010

Keywords

Examples

			Triangle begins
  1;
  1,   1;
  1,   1,    1;
  1,   0,    0,    1;
  1,  -2,   -5,   -2,    1;
  1,  -5,  -15,  -15,   -5,    1;
  1,  -9,  -30,  -41,  -30,   -9,    1;
  1, -14,  -49,  -77,  -77,  -49,  -14,   1;
  1, -20,  -70, -112, -125, -112,  -70, -20,    1;
  1, -27,  -90, -126, -117, -117, -126, -90,  -27,   1;
  1, -35, -105,  -90,   45,  131,   45, -90, -105, -35, 1;

Links

Crossrefs

Cf. A056241, A139011.
Variant: A061176.

Programs

  • Magma
    A180957:= func< n,k | (&+[ (-1)^(k-j)*Binomial(n, j)*Binomial(n-j, 2*(k-j)) : j in [0..n]]) >;
[A180957(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Apr 06 2021
  • Mathematica
    T[n_, k_]:= Sum[(-1)^(k-j)*Binomial[n, j]*Binomial[n-j, 2*(k-j)], {j,0,n}];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 06 2021 *)
  • Sage
    def A180957(n,k): return sum( (-1)^(k+j)*binomial(n,j)*binomial(n-j, 2*(k-j)) for j in (0..n))
flatten([[A180957(n,k) for k in (0..n)] for n in [0..15]]) # G. C. Greubel, Apr 06 2021

Formula

G.f.: 1/(1 -x -x*y + x/(1 -x -x*y)) = (1 -x*(1+y))/(1 -2*x*(1+y) +x^2*(1 +3*y +y^2)).
E.g.f.: exp((1+y)*x) * cos(sqrt(y)*x).
T(n, k) = Sum_{j=0..n} (-1)^(k-j)*binomial(n,j)*binomial(n-j, 2*(k-j)).
Sum_{k=0..n} T(n, k) = A139011(n) (row sums).
Sum_{k=0..floor(n/2)} T(n-k, k) = A180958(n) (diagonal sums).

A061178 Third column (m=2) of triangle A060920 (bisection of Fibonacci triangle, even part).

Original entry on oeis.org

1, 9, 51, 233, 942, 3522, 12473, 42447, 140109, 451441, 1426380, 4434420, 13599505, 41225349, 123723351, 368080793, 1086665562, 3186317718, 9286256393, 26916587307, 77634928209, 222920650081
Offset: 0

Views

Author

Wolfdieter Lang, Apr 20 2001

Keywords

Comments

Numerator polynomial of g.f. is sum(A061176(3,m)*x^m,m=0..3).

Crossrefs

A001519, A054444.

Formula

a(n)= A060920(n+2, 2).
G.f.: (1-x^3)/(1-3*x+x^2)^3.

A061179 Fourth column (m=3) of triangle A060920 (bisection of Fibonacci triangle, even part).

Original entry on oeis.org

1, 14, 105, 594, 2860, 12402, 49963, 190570, 696787, 2463300, 8472280, 28481220, 93914325, 304597382, 973877245, 3075011478, 9602753412, 29695165110, 91026167999, 276833858530, 835933445799, 2507876305416
Offset: 0

Views

Author

Wolfdieter Lang, Apr 20 2001

Keywords

Comments

Numerator polynomial of g.f. is sum(A061176(4,m)*x^m,m=0..4).

Crossrefs

A061178.

Formula

a(n)= A060920(n+3, 3).
G.f.: (1+2*x-5*x^2+2*x^3+x^4)/(1-3*x+x^2)^4.

A061180 Fifth column (m=4) of triangle A060920 (bisection of Fibonacci triangle, even part).

Original entry on oeis.org

1, 20, 190, 1295, 7285, 36122, 163730, 693835, 2790100, 10758050, 40075630, 145052300, 512347975, 1772132390, 6018885570, 20118711993, 66306068715, 215797999830, 694463680160, 2212291834405, 6982976069384
Offset: 0

Views

Author

Wolfdieter Lang, Apr 20 2001

Keywords

Comments

Numerator polynomial of g.f. is sum(A061176(5,m)*x^m, m=0..5).

Links

Crossrefs

Cf. A061179.

Programs

  • Mathematica
    CoefficientList[Series[((1-x^5)+5(x-x^4)-15(x^2-x^3))/(1-3x+x^2)^5,{x,0,40}],x] (* or *) LinearRecurrence[{15,-95,330,-685,873,-685,330,-95,15,-1},{1,20,190,1295,7285,36122,163730,693835,2790100,10758050},30] (* Harvey P. Dale, Sep 01 2022 *)

Formula

a(n) = A060920(n+4,4).
G.f.: ((1-x^5)+5*(x-x^4)-15*(x^2-x^3))/(1-3*x+x^2)^5.

A061181 Sixth column (m=5) of triangle A060920 (bisection of Fibonacci triangle, even part).

Original entry on oeis.org

1, 27, 315, 2534, 16407, 91959, 464723, 2171850, 9546570, 39940460, 160437690, 622844730, 2348773525, 8638447293, 31086197469, 109744786482, 380920122009, 1302304276665, 4392297900647
Offset: 0

Views

Author

Wolfdieter Lang, Apr 20 2001

Keywords

Comments

Numerator polynomial of g.f. is sum(A061176(6,m)*x^m,m=0..6).

Crossrefs

A061180.

Formula

a(n)= A060920(n+5, 5).
G.f.: ((1+x^6)+9*(x+x^5)-30*(x^2+x^4)+41*x^3)/(1-3*x+x^2)^6.
Showing 1-7 of 7 results.

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