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A119369 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) + 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. 9
1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 3, 2, 1, 0, 0, 1, 3, 6, 9, 7, 3, 1, 0, 0, 1, 4, 10, 20, 30, 23, 11, 4, 1, 0, 0, 1, 5, 15, 36, 70, 104, 81, 40, 16, 5, 1, 0, 0, 1, 6, 21, 58, 133, 253, 374, 293, 149, 63, 22, 6, 1, 0, 0, 1, 7, 28, 87, 226, 501, 938, 1380, 1087, 564, 248, 93, 29, 7, 1, 0, 0, 1 (list; 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 + x^2*(B^2 - B); the g.f. C(x) of the central terms satisfies: C(x) = 1/(1+x - x*B(x)).

LINKS

Table of n, a(n) for n=0..81.

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) + T(3,6) = 1 + 0 = 1;

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

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

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

[1,3,6,_,7,3,1]: T(4,4) = T(4,2) + T(3,4) = 6 + 1 = 7;

[1,3,6,9,7,3,1]: T(4,3) = T(4,4) + T(3,3) = 7 + 2 = 9;

[1,3,6,9,7,3,1,0,0]: 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, 9, 7, 3, 1, 0, 0;

1, 4, 10, 20, 30, 23, 11, 4, 1, 0, 0;

1, 5, 15, 36, 70, 104, 81, 40, 16, 5, 1, 0, 0;

1, 6, 21, 58, 133, 253, 374, 293, 149, 63, 22, 6, 1, 0, 0;

1, 7, 28, 87, 226, 501, 938, 1380, 1087, 564, 248, 93, 29, 7, 1, 0,0;

Central terms are:

C = A119371 = [1, 0, 1, 2, 7, 23, 81, 293, 1087, 4110, ...].

Lower diagonals start:

D1 = A119372 = [1, 1, 3, 9, 30, 104, 374, 1380, 5197, ...];

D2 = A119373 = [1, 2, 6, 20, 70, 253, 938, 3546, 13617, ...].

Diagonals above central terms (ignoring leading zeros) start:

U1 = A119375 = [1, 3, 11, 40, 149, 564, 2166, 8420, ...];

U2 = A119376 = [1, 4, 16, 63, 248, 980, 3894, 15563, ...].

There exists the base sequence:

B = A119370 = [1, 1, 2, 6, 19, 64, 225, 816, 3031, 11473, ...]

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, 7, 23, 81, 293, 1087, ...]

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, T(n-1, k)+if(k<n, T(n, 2*n-1-k), T(n, 2*n-2-k)))))

CROSSREFS

Cf. A119370, A119371, A119372, A119373, A119374, A119375, A119376; variants: A118340, A118345, A118350.

Sequence in context: A178904 A017858 A167769 * A122445 A189511 A165592

Adjacent sequences:  A119366 A119367 A119368 * A119370 A119371 A119372

KEYWORD

nonn,tabf

AUTHOR

Paul D. Hanna, May 16 2006

STATUS

approved

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Last modified August 18 08:57 EDT 2019. Contains 326077 sequences. (Running on oeis4.)