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A118350 Pendular triangle, read by rows, where row n is formed from row n-1 by the recurrence: if n > 2k, T(n,k) = T(n,n-k) + T(n-1,k), else T(n,k) = T(n,n-1-k) + 3*T(n-1,k), for n>=k>=0, with T(n,0)=1 and T(n,n)=0^n. 8
1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 13, 7, 1, 0, 1, 5, 21, 42, 8, 1, 0, 1, 6, 30, 96, 54, 9, 1, 0, 1, 7, 40, 163, 325, 67, 10, 1, 0, 1, 8, 51, 244, 770, 445, 81, 11, 1, 0, 1, 9, 63, 340, 1353, 2688, 583, 96, 12, 1, 0, 1, 10, 76, 452, 2093, 6530, 3842, 740, 112, 13, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

See definition of pendular triangle and pendular sums at A118340.

LINKS

Table of n, a(n) for n=0..77.

FORMULA

T(2*n+m,n) = [A118351^(m+1)](n), i.e., the m-th lower semi-diagonal forms the self-convolution (m+1)-power of the central terms A118351.

EXAMPLE

Row 6 equals the pendular sums of row 5,

[1, 4,13, 7, 1, 0], where the sums proceed as follows:

[1,__,__,__,__,__]: T(6,0) = T(5,0) = 1;

[1,__,__,__,__, 1]: T(6,5) = T(6,0) + 3*T(5,5) = 1 + 3*0 = 1;

[1, 5,__,__,__, 1]: T(6,1) = T(6,5) + T(5,1) = 1 + 4 = 5;

[1, 5,__,__, 8, 1]: T(6,4) = T(6,1) + 3*T(5,4) = 5 + 3*1 = 8;

[1, 5,21,__, 8, 1]: T(6,2) = T(6,4) + T(5,2) = 8 + 13 = 21;

[1, 5,21,42, 8, 1]: T(6,3) = T(6,2) + 3*T(5,3) = 21 + 3*7 = 42;

[1, 5,21,42, 8, 1, 0] finally, append a zero to obtain row 6.

Triangle begins:

1;

1, 0;

1, 1, 0;

1, 2, 1, 0;

1, 3, 6, 1, 0;

1, 4, 13, 7, 1, 0;

1, 5, 21, 42, 8, 1, 0;

1, 6, 30, 96, 54, 9, 1, 0;

1, 7, 40, 163, 325, 67, 10, 1, 0;

1, 8, 51, 244, 770, 445, 81, 11, 1, 0;

1, 9, 63, 340, 1353, 2688, 583, 96, 12, 1, 0;

1, 10, 76, 452, 2093, 6530, 3842, 740, 112, 13, 1, 0;

1, 11, 90, 581, 3010, 11760, 23286, 5230, 917, 129, 14, 1, 0; ...

Central terms are T(2*n,n) = A118351(n);

semi-diagonals form successive self-convolutions of the central terms:

T(2*n+1,n) = A118352(n) = [A118351^2](n),

T(2*n+2,n) = A118353(n) = [A118351^3](n).

PROG

(PARI) T(n, k)=if(n<k || k<0, 0, if(k==0, 1, if(n==k, 0, if(n>2*k, T(n, n-k)+T(n-1, k), T(n, n-1-k)+3*T(n-1, k)))))

CROSSREFS

Cf. A118351, A118352, A118353, A118354.

Sequence in context: A295860 A118345 A292804 * A183135 A294042 A287316

Adjacent sequences:  A118347 A118348 A118349 * A118351 A118352 A118353

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Apr 26 2006

STATUS

approved

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Last modified January 21 16:47 EST 2020. Contains 331114 sequences. (Running on oeis4.)