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.

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

Offset

0,8

Comments

See definition of pendular triangle and pendular sums at A118340.

Links

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

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. A118351, A118352, A118353, A118354.

Cf. A167763 (p=0), A118340 (p=1), A118345 (p=2), this sequence (p=3).

Sequence in context: A295860 A118345 A292804 * A361950 A183135 A294042

Adjacent sequences: A118347 A118348 A118349 * A118351 A118352 A118353

Keyword

nonn,tabl

Author

Paul D. Hanna, Apr 26 2006

Status

approved