A133601 - OEIS

A133601

Triangle read by rows: A007318 * (A097806 + A133080 - I) as infinite lower triangular matrices, where I is the identity matrix.

1, 3, 1, 5, 3, 1, 7, 6, 5, 1, 9, 10, 14, 5, 1, 11, 15, 30, 15, 7, 1, 13, 21, 55, 35, 27, 7, 1, 15, 28, 91, 70, 77, 28, 9, 1, 17, 36, 140, 126, 182, 84, 44, 9, 1, 19, 45, 204, 210, 378, 210, 156, 45, 11, 1, 21, 55, 285, 330, 714, 462, 450, 165, 65, 11, 1, 23, 66, 385, 495, 1254, 924, 1122, 495, 275, 66, 13, 1

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OFFSET

0,2

COMMENTS

The matrix (A097806 + A133080 - I) is the infinite lower triangular matrix with (1,1,1,...) in the main diagonal and (2,1,2,1,2,...) in the subdiagonal; and the rest zeros.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

FORMULA

From Andrew Howroyd, Sep 25 2025: (Start)

T(n,k) = binomial(n,k) + (3*binomial(n,k+1) + (-1)^k*binomial(n,k+1))/2.

G.f.: (1 + y*x - x^2)/((1 - x)*(1 - (1 + y)*x)*(1 - (1 - y)*x)). (End)

EXAMPLE

Triangle begins:

3, 1;

5, 3, 1;

7, 6, 5, 1;

9, 10, 14, 5, 1;

11, 15, 30, 15, 7, 1;

13, 21, 55, 35, 27, 7, 1;

15, 28, 91, 70, 77, 28, 9, 1;

...

MAPLE

A133601aux := proc(n, k)

if n <> k then

A097806(n, k)+A133080(n, k) ;

else

A097806(n, k)+A133080(n, k)-1 ;

end if;

end proc:

A133601 := proc(n, k)

add( A007318(n, j)*A133601aux(j+1, k+1), j=k..n) ;

end proc: # R. J. Mathar, Jun 20 2015

PROG

(PARI) T(n, k) = binomial(n, k) + (3*binomial(n, k+1) + (-1)^k*binomial(n, k+1))/2 \\ Andrew Howroyd, Sep 25 2025

CROSSREFS

Row sums are A052549.

Cf. A007318, A097806, A133080, A133807.

Sequence in context: A348835 A391735 A130301 * A258207 A133094 A300437

Adjacent sequences: A133598 A133599 A133600 * A133602 A133603 A133604

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Sep 18 2007

STATUS

approved