A167763 Pendular triangle (p=0), 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), otherwise T(n,k) = T(n,n-1-k) + p*T(n-1,k), for n >= k <= 0, with T(n,0) = 1 and T(n,n) = 0^n.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 7, 4, 1, 0, 1, 5, 12, 12, 5, 1, 0, 1, 6, 18, 30, 18, 6, 1, 0, 1, 7, 25, 55, 55, 25, 7, 1, 0, 1, 8, 33, 88, 143, 88, 33, 8, 1, 0, 1, 9, 42, 130, 273, 273, 130, 42, 9, 1, 0, 1, 10, 52, 182, 455, 728, 455, 182, 52, 10, 1, 0

Offset

0,8

Comments

See A118340 for definition of pendular triangles and pendular sums.

The last five rows in the example section appear in the table on p. 8 of Getzler. Cf. also A173075. - Tom Copeland, Jan 22 2020

Links

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

E. Getzler, The semi-classical approximation for modular operads, arXiv:alg-geom/9612005, 1996.

Formula

T(2n+m) = [A001764^(m+1)](n), i.e., the m-th lower semi-diagonal forms the self-convolution (m+1)-power of A001764.

If n > 2k, T(n,k) = binomial(n+k+1,k)*(n-2k+1)/(n+k+1), else T(n,k) = T(n,n-1-k), with conditions: T(n,0)=1 for n>=0 and T(n,n)=0 for n > 0. - Paul D. Hanna, Nov 12 2009

Sum_{k=0..n} T(n, k, p=0) = A093951(n). - G. C. Greubel, Feb 17 2021

Example

Triangle begins:
  1;
  1,  0;
  1,  1,  0;
  1,  2,  1,  0;
  1,  3,  3,  1,  0;
  1,  4,  7,  4,  1,  0;
  1,  5, 12, 12,  5,  1,  0; ...

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, 0], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 17 2021 *)

Prog

(PARI) {T(n, k)=if(k==0, 1, if(n==k, 0, if(n>2*k, binomial(n+k+1, k)*(n-2*k+1)/(n+k+1), T(n, n-1-k))))} \\ Paul D. Hanna, Nov 12 2009
(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, 0) 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, 0): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 17 2021

Crossrefs

Cf. this sequence (p=0), A118340 (p=1), A118345 (p=2), A118350 (p=3).

Cf. A001764, A006013, A006629, A093951, A102893, A173075.

Sequence in context: A373449 A373424 A277504 * A277666 A274581 A353279

Adjacent sequences: A167760 A167761 A167762 * A167764 A167765 A167766

Keyword

nonn,tabl

Author

Philippe Deléham, Nov 11 2009

Status

approved