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

0,2

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).

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)

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

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

Sequence in context: A371300 A110552 A129161 * A054456 A096164 A201166

Adjacent sequences: A103412 A103413 A103414 * A103416 A103417 A103418

Keyword

nonn,tabl

Author

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

Status

approved