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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. 10
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
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).
MATHEMATICA
T[n_, k_, p_]:= T[n, k, p] = If[n<k || k<0, 0, If[k==0, 1, If[k==n, 0, If[n<=2*k, T[n, n-k-1, p] + p*T[n-1, k, p], T[n, n-k, p] + T[n-1, k, p] ]]]];
Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 17 2021 *)
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)))))
(Sage)
@CachedFunction
def T(n, k, p):
if (k<0 or n<k): return 0
elif (k==0): return 1
elif (k==n): return 0
elif (n>2*k): return T(n, n-k, p) + T(n-1, k, p)
else: return T(n, n-k-1, p) + p*T(n-1, k, p)
flatten([[T(n, k, 3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 17 2021
(Magma)
function T(n, k, p)
if k lt 0 or n lt k then return 0;
elif k eq 0 then return 1;
elif k eq n then return 0;
elif n gt 2*k then return T(n, n-k, p) + T(n-1, k, p);
else return T(n, n-k-1, p) + p*T(n-1, k, p);
end if;
return T;
end function;
[T(n, k, 3): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 17 2021
CROSSREFS
Cf. A167763 (p=0), A118340 (p=1), A118345 (p=2), this sequence (p=3).
Sequence in context: A295860 A118345 A292804 * A361950 A183135 A294042
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Apr 26 2006
STATUS
approved

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Last modified March 19 03:33 EDT 2024. Contains 370952 sequences. (Running on oeis4.)