A078027 -id:A078027 - cp's OEIS frontend

A144400 Triangle read by rows: row n (n > 0) gives the coefficients of x^k (0 <= k <= n - 1) in the expansion of Sum_{j=0..n} A000931(j+4)binomial(n, j)x^(j - 1)*(1 - x)^(n - j).

1, 2, -1, 3, -3, 1, 4, -6, 4, 0, 5, -10, 10, 0, -3, 6, -15, 20, 0, -18, 10, 7, -21, 35, 0, -63, 70, -24, 8, -28, 56, 0, -168, 280, -192, 49, 9, -36, 84, 0, -378, 840, -864, 441, -89, 10, -45, 120, 0, -756, 2100, -2880, 2205, -890, 145, 11, -55, 165, 0

Offset: 1

Roger L. Bagula and Gary W. Adamson, Oct 03 2008

			Triangle begins:
    1;
    2,  -1;
    3,  -3,   1;
    4,  -6,   4, 0;
    5, -10,  10, 0,   -3;
    6, -15,  20, 0,  -18,   10;
    7, -21,  35, 0,  -63,   70,   -24;
    8, -28,  56, 0, -168,  280,  -192,   49;
    9, -36,  84, 0, -378,  840,  -864,  441,  -89;
   10, -45, 120, 0, -756, 2100, -2880, 2205, -890, 145;
     ... reformatted. - _Franck Maminirina Ramaharo_, Oct 22 2018

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Row sums: subsequence of A000931, A078027, A182097.

Cf. A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123217, A123221, A141720, A144387, A174128.

a[n_]:= a[n]= If[n<3, Fibonacci[n], a[n-2] + a[n-3]];
p[x_, n_]:= Sum[a[k]*Binomial[n, k]*x^(k-1)*(1-x)^(n-k), {k, 0, n}];
Table[Coefficient[p[x, n], x, k], {n, 12}, {k, 0, n-1}]//Flatten

@CachedFunction
def f(n): return fibonacci(n) if (n<3) else f(n-2) + f(n-3)
def p(n,x): return sum( binomial(n,j)*f(j)*x^(j-1)*(1-x)^(n-j) for j in (0..n) )
def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
[T(n) for n in (1..12)] # G. C. Greubel, Jul 14 2021

G.f.: (y - (1 - 2*x)*y^2)/(1 - 3*(1 - x)*y + (3 - 6*x + 2*x^2)*y^2 - (1 - 3*x + 2*x^2 + x^3)*y^3). - Franck Maminirina Ramaharo, Oct 22 2018

Edited, and new name by Franck Maminirina Ramaharo, Oct 22 2018

A141683 a(n) = Sum_{k=1..n} b(k)*a(n - k) for n >= 1, where b(n) = b(n-2) + b(n-3) for n >= 3 with b(0) = 0 and b(1) = b(2) = 1.

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 41, 88, 189, 406, 872, 1873, 4023, 8641, 18560, 39865, 85626, 183916, 395033, 848491, 1822473, 3914488, 8407925, 18059374, 38789712, 83316385, 178955183, 384377665, 825604416, 1773314929, 3808901426

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 07 2008

Essentially the same as A141015. - R. J. Mathar, Sep 14 2008

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,1).

Subsequence of A000930.

Cf. A000931, A001906, A078027, A134816, A141015, A182097.

m:=35; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1-x^2-x^3)/(1-x-2*x^2-x^3))); // G. C. Greubel, Jun 05 2018

(* b = A000931 *)
b[0]=0; b[1]=1; b[2]=1; b[n_]:= b[n]= b[n-2] + b[n-3];
a[1]=1; a[n_]:= a[n]= Sum[b[k]*a[n-k], {k,n-1}];
Table[a[n], {n, 35}]
(* or *)
LinearRecurrence[{1, 2, 1}, {1, 1, 2, 4}, 31] (* Georg Fischer, Mar 23 2019 *)

x='x+O('x^35); Vec(x*(1-x^2-x^3)/(1-x-2*x^2-x^3)) \\ G. C. Greubel, Jun 05 2018

a(n) = Sum_{k=1..n} b(k)*a(n - k) for n >= 1, where b(n) = b(n-2) + b(n-3) for n >= 3 with b(0) = 0 and b(1) = b(2) = 1. [That is, b(n) = A000931(n+4) = A078027(n+6) = A134816(n) = A182097(n+1). - Petros Hadjicostas, Aug 09 2020]
From Colin Barker, Feb 01 2012: (Start)
a(n) = a(n-1) + 2*a(n-2) + a(n-3), n > 4.
G.f.: x*(1 - x^2 - x^3)/(1 - x - 2*x^2 - x^3). (End)
a(n) = A000930(2*n - 3) for n >= 3. - Georg Fischer, Mar 23 2019

cp's OEIS Frontend

A144400 Triangle read by rows: row n (n > 0) gives the coefficients of x^k (0 <= k <= n - 1) in the expansion of Sum_{j=0..n} A000931(j+4)binomial(n, j)x^(j - 1)*(1 - x)^(n - j).

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A141683 a(n) = Sum_{k=1..n} b(k)*a(n - k) for n >= 1, where b(n) = b(n-2) + b(n-3) for n >= 3 with b(0) = 0 and b(1) = b(2) = 1.

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cp's OEIS Frontend

A144400 Triangle read by rows: row n (n > 0) gives the coefficients of x^k (0 <= k <= n - 1) in the expansion of Sum_{j=0..n} A000931(j+4)*binomial(n, j)*x^(j - 1)*(1 - x)^(n - j).

Views

Author

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Examples

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Crossrefs

Programs

Mathematica

Sage

Formula

Extensions

A141683 a(n) = Sum_{k=1..n} b(k)*a(n - k) for n >= 1, where b(n) = b(n-2) + b(n-3) for n >= 3 with b(0) = 0 and b(1) = b(2) = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A144400 Triangle read by rows: row n (n > 0) gives the coefficients of x^k (0 <= k <= n - 1) in the expansion of Sum_{j=0..n} A000931(j+4)binomial(n, j)x^(j - 1)*(1 - x)^(n - j).