A103415 Triangle, read by rows, T(n,k) = A000129(n+1) - Sum_{j=1..k} t(n+1, j), where t(n, k) is defined in the formula section.

1, 2, 1, 5, 4, 1, 12, 11, 6, 1, 29, 28, 21, 8, 1, 70, 69, 60, 35, 10, 1, 169, 168, 157, 116, 53, 12, 1, 408, 407, 394, 333, 204, 75, 14, 1, 985, 984, 969, 884, 653, 332, 101, 16, 1, 2378, 2377, 2360, 2247, 1870, 1189, 508, 131, 18, 1, 5741, 5740, 5721, 5576, 5001, 3712, 2029, 740, 165, 20, 1

Offset: 0

Author

Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson, Feb 04 2005

Keywords

• nonn
• tabl

Comments

Triangle is generated from the product A*B of the infinite lower triangular matrices A = A008288(n,k) and B =

1;

1 1;

1 1 1;

1 1 1 1; ...

Determinant(A*B) = 1 for all n.

Absolute values of coefficients of characteristic polynomials of n-th matrix are the (n+1)-th row of A007318 (Pascal's triangle). As they are:

x^1 - 1;

x^2 - 2*x^1 + 1;

x^3 - 3*x^2 + 3*x^1 - 1;

x^4 - 4*x^3 + 6*x^2 - 4*x^1 + 1;

x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x^1 - 1.

Examples

			Triangle begins as:
    1;
    2,   1;
    5,   4,   1;
   12,  11,   6,   1;
   29,  28,  21,   8,   1;
   70,  69,  60,  35,  10,  1;
  169, 168, 157, 116,  53, 12,  1;
  408, 407, 394, 333, 204, 75, 14, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A000129, A002203, A005409, A007318, A008288, A026937, A093328, A103416.

Programs

Mathematica

t[n_, k_]:= If[k==0, (2*Boole[n<2] + LucasL[n-1, 2]*Boole[n>1])/2, Binomial[n-1, k-1]*Hypergeometric2F1[1-k, k-n, 1-n, -1]];
st[n_, k_]:= Sum[t[n+1, j], {j,k}];
T[n_, k_]:= Fibonacci[n+1, 2] - st[n, k];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 25 2021 *)

PARI

Pell(n) = if( n<2, n, 2*Pell(n-1) + Pell(n-2) );
t(n, k) = if(n<3, 1, if(k==1||k==n, 1, t(n-1,k) + t(n-1,k-1) + t(n-2,k-1) ));
st(n, k) = sum(i=1, k, t(n+1,i));
T(n, k) = Pell(n+1) - st(n,k);
for(n=0, 10, for(k=0, n, print1(T(n,k), ",")); print()) \\ modified by G. C. Greubel, May 25 2021

Sage

@CachedFunction
def t(n,k): return 1 if (n<3) else 1 if (k==1 or k==n) else t(n-1,k) + t(n-1,k-1) + t(n-2,k-1)
def st(n,k): return sum(t(n+1, j) for j in (1..k))
def T(n,k): return lucas_number1(n+1,2,-1) - st(n,k)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 25 2021

Formula

T(n, k) = Pell(n+1) - ST(n, k), where ST(n, k) = Sum_{j=1..k} t(n+1, j), t(n, k) = t(n-1,k) + t(n-1,k-1) + t(n-2,k-1), t(n, 1) = t(n, n) = 1 and t(0, k) = t(1, k) = t(2, k) = 1.

T(n, 0) = A000129(n+1).

T(n, 1) = A005409(n) = A000129(n) - 1.

Sum_{k=0..n} T(n, k) = A026937(n).

From G. C. Greubel, May 25 2021: (Start)

T(n, k) = A000129(n+1) - st(n,k), where st(n, k) = Sum_{j=1..k} t(n+1, j), t(n, k) = A008288(n-1, k-1) for n >= 1 and k >= 1, and t(n, 0) = (1/2)*(2*[n<2] + A002203(n-1)*[n>1]).

T(n, n) = A000012(n).

T(n, n-1) = A005843(n+1).

T(n, n-2) = A093328(n-1).

T(n, n-3) = (4/3)*((n-3)^3 + 5*(n-3) + 3).

T(n, n-4) = (1/3)*(2*(n-4)^2 + 22*(n-4)^2 + 22*(n-4) + 39). (End)