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A122445 Pendular trinomial triangle, read by rows of 2n+1 terms (n>=0), defined by the recurrence: if 0<k<n, T(n,k) = T(n-1,k) + 2*T(n,2n-1-k); else if n-1<k<2n-1, T(n,k) = T(n-1,k) + T(n,2n-2-k); with T(n,0)=T(n+1,2n)=1 and T(n+1,2n+1)=T(n+1,2n+2)=0. 8
1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 3, 2, 1, 0, 0, 1, 3, 6, 10, 8, 3, 1, 0, 0, 1, 4, 10, 22, 36, 28, 12, 4, 1, 0, 0, 1, 5, 15, 39, 83, 135, 107, 47, 17, 5, 1, 0, 0, 1, 6, 21, 62, 155, 324, 525, 418, 189, 72, 23, 6, 1, 0, 0, 1, 7, 28, 92, 259, 629, 1298, 2094, 1676, 773, 305, 104, 30, 7, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,11

COMMENTS

The diagonals may be generated by iterated convolutions of a base sequence B with the sequence C of central terms. The g.f. B(x) of the base sequence satisfies: B = 1 + x*B^2 + 2x^2*(B^2 - B); the g.f. C(x) of the central terms satisfies: C(x) = 1/(1+x - xB(x)).

LINKS

Table of n, a(n) for n=0..78.

EXAMPLE

To obtain row 4, pendular sums of row 3 are carried out as follows.

[1,2,3, 2,1,0,0]: given row 3;

[1,_,_,__,_,_,_]: start with T(4,0) = T(3,0) = 1;

[1,_,_,__,_,_,1]: T(4,6) = T(4,0) + 2*T(3,6) = 1 + 2*0 = 1;

[1,3,_,__,_,_,1]: T(4,1) = T(4,6) + 1*T(3,1) = 1 + 1*2 = 3;

[1,3,_,__,_,3,1]: T(4,5) = T(4,1) + 2*T(3,5) = 3 + 2*0 = 3;

[1,3,6,__,_,3,1]: T(4,2) = T(4,5) + 1*T(3,2) = 3 + 1*3 = 6;

[1,3,6,__,8,3,1]: T(4,4) = T(4,2) + 2*T(3,4) = 6 + 2*1 = 8;

[1,3,6,10,8,3,1]: T(4,3) = T(4,4) + 1*T(3,3) = 8 + 1*2 = 10;

[1,3,6,10,8,3,1,0,0]: complete row 4 by appending two zeros.

Triangle begins:

1;

1, 0, 0;

1, 1, 1, 0, 0;

1, 2, 3, 2, 1, 0, 0;

1, 3, 6, 10, 8, 3, 1, 0, 0;

1, 4, 10, 22, 36, 28, 12, 4, 1, 0, 0;

1, 5, 15, 39, 83, 135, 107, 47, 17, 5, 1, 0, 0;

1, 6, 21, 62, 155, 324, 525, 418, 189, 72, 23, 6, 1, 0, 0; ...

Central terms are:

C = A122447 = [1, 0, 1, 2, 8, 28, 107, 418, 1676, 6848, ...].

Lower diagonals start:

D1 = A122448 = [1, 1, 3, 10, 36, 135, 525, 2094, 8524, ...];

D2 = A122449 = [1, 2, 6, 22, 83, 324, 1298, 5302, 22002, ...].

Diagonals above central terms (ignoring leading zeros) start:

U1 = A122450 = [1, 3, 12, 47, 189, 773, 3208, 13478, 57222,...];

U2 = A122451 = [1, 4, 17, 72, 305, 1300, 5576, 24068, 104510,...].

There exists the base sequence:

B = A122446 = [1, 1, 2, 7, 24, 88, 336, 1321, 5316, 21788, ...]

which generates all diagonals by convolutions with central terms:

D2 = B * D1 = B^2 * C

U2 = B * U1 = B^2 * C"

where C" = [1, 2, 8, 28, 107, 418, 1676, 6848, 28418, ...]

are central terms not including the initial [1,0].

PROG

(PARI) {T(n, k)=if(k==0&n==0, 1, if(k>2*n-2|k<0, 0, if(n==2&k<=2, 1, if(k<n, T(n-1, k)+T(n, 2*n-1-k), 2*T(n-1, k)+T(n, 2*n-2-k)))))}

CROSSREFS

Cf. A122446, A122447 (central terms); diagonals: A122448, A122449, A122450, A122451; A122452 (row sums); variants: A118340, A118345, A118350, A119369.

Sequence in context: A017858 A167769 A119369 * A189511 A165592 A059285

Adjacent sequences:  A122442 A122443 A122444 * A122446 A122447 A122448

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Sep 07 2006

STATUS

approved

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Last modified August 20 01:14 EDT 2019. Contains 326136 sequences. (Running on oeis4.)