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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
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 (list; table; graph; refs; listen; history; text; internal format)
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: A299444 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

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Last modified September 20 23:04 EDT 2021. Contains 347596 sequences. (Running on oeis4.)