A131641 Irregular triangle, read by rows: T(n, k) = [x^k]( y(n, x) ), where y(n, x) = - 2*y(3, x) - x*y(n-1, x) + 2*x^2*y(n-1, x) + x^2*y(n-2, x), and y(1, x) = -8 - 3*x + 8*x^2, y(2, x) = 4 - 4*x - 10*x^2 + 4*x^3 + 4*x^4, y(3, x) = -8 + 4*x + 24*x^2 - 9*x^3 - 24*x^4 + 4*x^5 + 8*x^6.

-8, -3, 8, 4, -4, -10, 4, 4, -8, 4, 24, -9, -24, 4, 8, 16, 0, -64, -2, 95, 2, -64, 0, 16, 16, -24, -24, 86, -54, -116, 148, 72, -120, -16, 32, 16, -24, 24, -6, -150, 216, 87, -378, 160, 264, -208, -64, 64, 16, -24, 24, -78, 78, 216, -586, 229, 700, -844, -64, 720, -320, -192, 128, 16, -24, 24, -78, 198, -248, -226, 1234, -1314, -620, 2404, -1360, -1056, 1696, -384, -512, 256

Offset

1,1

Links

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

Formula

T(n, k) = [x^k]( y(n, x) ), where y(n, x) = - 2*y(3, x) - x*y(n-1, x) + 2*x^2*y(n-1, x) + x^2*y(n-2, x), and y(1, x) = -8 - 3*x + 8*x^2, y(2, x) = 4 - 4*x - 10*x^2 + 4*x^3 + 4*x^4, y(3, x) = -8 + 4*x + 24*x^2 - 9*x^3 - 24*x^4 + 4*x^5 + 8*x^6, y(4, x) = 16 - 64*x^2 - 2*x^3 + 95*x^4 + 2*x^5 - 64*x^6 + 16*x^8.

From G. C. Greubel, Sep 10 2025: (Start)

T(n, k) = [x^k]( x^(n-4)*((y(4,x) - p(x))*Fibonacci(n-3, 2*x-1) +x*(y(3,x) -p(x))*Fibonacci(n-4, 2*x-1)) + p(x) ), for n > 1, otherwise T(1, k) = [x^k]( y(1, x) ), where p(x) = 2*y(3, x)/(3*x^2 - x - 1), and Fibonacci(n, x) are the Fibonacci polynomials.

Sum_{k=0..2*n} T(n, k) = A001911(n-5) - 2 - 2*[n==1]. (End)

Example

Irregular triangle begins as:
  -8,  -3,   8;
   4,  -4, -10,  4,    4;
  -8,   4,  24, -9,  -24,    4,   8;
  16,   0, -64, -2,   95,    2, -64,    0,   16;
  16, -24, -24, 86,  -54, -116, 148,   72, -120, -16,   32;
  16, -24,  24, -6, -150,  216,  87, -378,  160, 264, -208, -64, 64;

Mathematica

y[1, x_] := -8 - 3*x + 8*x^2;
y[2, x_]:= 4 -4*x -10*x^2 +4*x^3 +4*x^4;
y[3, x_]:= -8 +4*x +24*x^2 -9*x^3 -24*x^4 +4*x^5 +8*x^6;
y[4, x_]:= 16 -64*x^2 -2*x^3 +95*x^4 +2*x^5 -64*x^6 +16*x^8;
p[x_]:= 2*y[3, x]/(3*x^2-x-1);
f[n_, x_]:= If[n==1, y[1, x], x^(n-4)*((y[4, x] -p[x])*Fibonacci[n-3, 2*x-1] +x*(y[3, x] -p[x])*Fibonacci[n-4, 2*x-1]) +p[x]];
A131641[n_, k_]:= Coefficient[f[n, x], x, k]
Table[A131641[n, k], {n, 15}, {k, 0, 2*n}]//Flatten (* G. C. Greubel, Sep 10 2025 *)

Prog

(Magma)
function y(n, x)
    if n eq 1 then return -8 -3*x +8*x^2;
    elif n eq 2 then return 4 -4*x -10*x^2 +4*x^3 +4*x^4;
    elif n eq 3 then return -8 +4*x +24*x^2 -9*x^3 -24*x^4 +4*x^5 +8*x^6;
    else return -2*y(3, x) -x*y(n-1, x) +x^2*(2*y(n-1, x) +y(n-2, x));
    end if;
end function;
R<x>:=PowerSeriesRing(Rationals(), 40);
A131641:= func< n, k | Coefficient(R!( y(n, x) ), k) >;
[A131641(n, k): k in [0..2*n], n in [1..12]]; // G. C. Greubel, Sep 10 2025
(SageMath)
@CachedFunction
def y(n, x):
    if n==1: return -8 -3*x +8*x^2
    elif n==2: return 4 -4*x -10*x^2 +4*x^3 +4*x^4
    elif n==3: return -8 +4*x +24*x^2 -9*x^3 -24*x^4 +4*x^5 +8*x^6
    else: return -2*y(3, x) -x*y(n-1, x) +x^2*(2*y(n-1, x) +y(n-2, x))
def A131641(n, k):
    P.<x>= PowerSeriesRing(QQ)
    return P( y(n, x) ).list()[k]
flatten([[A131641(n, k) for k in range(2*n+1)] for n in range(1, 12)]) # G. C. Greubel, Sep 10 2025

Crossrefs

Cf. A001911.

Sequence in context: A387773 A011106 A386734 * A190408 A343851 A085672

Adjacent sequences: A131638 A131639 A131640 * A131642 A131643 A131644

Keyword

tabf,sign,less

Author

Roger L. Bagula, Sep 08 2007

Extensions

Edited by G. C. Greubel, Sep 08 2025

Status

approved