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A167769
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Pendular trinomial triangle (p=0), read by rows of 2n+1 terms (n>=0), defined by the recurrence : if 0<k<n T(n,k)= T(n-1,k) + p*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.
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2
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1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 3, 2, 1, 0, 0, 1, 3, 6, 8, 6, 3, 1, 0, 0, 1, 4, 10, 18, 24, 18, 10, 4, 1, 0, 0, 1, 5, 15, 33, 57, 75, 57, 33, 15, 5, 1, 0, 0, 1, 6, 21, 54, 111, 186, 243, 186, 111, 54, 21, 6, 1, 0, 0, 1, 7, 28, 82, 193, 379, 622, 808, 622, 379, 193, 82, 28, 7, 1, 0, 0
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OFFSET
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0,11
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COMMENTS
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See A119369 for p=1 and A122445 for p=2. The diagonals may be generated by iterated convolutions of a base sequence B (A000108(n)) with the sequence C (A000957(n+1)) of central terms.
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REFERENCES
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Kim, Ki Hang; Rogers, Douglas G.; Roush, Fred W. Similarity relations and semiorders. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 577--594, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561081 (81i:05013) - From N. J. A. Sloane, Jun 05 2012
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LINKS
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FORMULA
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Sum_{k=0..2*n} T(n,k) = A071724(n) = [n=0] + 3*binomial(2n,n-1)/(n+2) = [n=0] + n*C(n)/(n+2), where C(n) are the Catalan numbers (A000108). - G. C. Greubel, Mar 17 2021
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EXAMPLE
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Triangle begins :
1;
1, 0, 0;
1, 1, 1, 0, 0;
1, 2, 3, 2, 1, 0, 0;
1, 3, 6, 8, 6, 3, 1, 0, 0;
1, 4, 10, 18, 24, 18, 10, 4, 1, 0, 0,
1, 5, 15, 33, 57, 75, 57, 33, 15, 5, 1, 0, 0; ...
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MAPLE
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T:= proc(n, k) option remember;
if k=0 and n=0 then 1;
elif k<0 or k>2*(n-1) then 0;
elif n=2 and k<3 then 1;
elif k<n then T(n, 2*n-k-1) +T(n-1, k);
else T(n, 2*n-k-2);
fi; end:
seq(seq(T(n, k), k=0..2*n), n=0..12); # G. C. Greubel, Mar 17 2021
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MATHEMATICA
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T[n_, k_]:= T[n, k]= If[k==0 && n==0, 1, If[k<0 || k>2*(n-1), 0, If[n==2 && k<3, 1, If[k<n, T[n, 2*n-k-1] +T[n-1, k], T[n, 2*n-k-2] ]]]];
Table[T[n, k], {n, 0, 10}, {k, 0, 2*n}]//Flatten (* G. C. Greubel, Mar 17 2021 *)
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PROG
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(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), T(n, 2*n-2-k))))) \\ Paul D. Hanna, Nov 12 2009
(Sage)
@CachedFunction
def T(n, k):
if (k==0 and n==0): return 1
elif (k<0 or k>2*(n-1)): return 0
elif (n==2 and k<3): return 1
elif (k<n): return T(n, 2*n-k-1) +T(n-1, k)
else: return T(n, 2*n-k-2)
flatten([[T(n, k) for k in (0..2*n)] for n in (0..12)]) # G. C. Greubel, Mar 17 2021
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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