OFFSET
0,2
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.
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
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).
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)
EXAMPLE
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;
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson, Feb 04 2005
STATUS
approved