A167769 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.

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

Offset

0,11

Comments

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.

References

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

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

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

Example

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; ...

Maple

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

Mathematica

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 *)

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), 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

Crossrefs

Cf. A000108, A000957, A000958, A001558, A001559, A071724, A104629.

Sequence in context: A319573 A178904 A017858 * A119369 A122445 A189511

Adjacent sequences: A167766 A167767 A167768 * A167770 A167771 A167772

Keyword

nonn,tabf

Author

Philippe Deléham, Nov 11 2009

Status

approved