A118349 Convolution triangle, read by rows, where diagonals are successive self-convolutions of A118346.

1, 1, 0, 1, 1, 0, 1, 2, 5, 0, 1, 3, 11, 30, 0, 1, 4, 18, 70, 201, 0, 1, 5, 26, 121, 487, 1445, 0, 1, 6, 35, 184, 873, 3592, 10900, 0, 1, 7, 45, 260, 1375, 6606, 27600, 85128, 0, 1, 8, 56, 350, 2010, 10672, 51728, 218566, 682505, 0, 1, 9, 68, 455, 2796, 15996, 85182, 415629, 1771367, 5585115, 0

Offset

0,8

Comments

A118346 equals the central terms of pendular triangle A118345 and the diagonals of this triangle form the semi-diagonals of the triangle A118345.

Links

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

Formula

Since g.f. G=G(x) of A118346 satisfies: G = 1 - 2*x*G + 2*x*G^2 + x*G^3 then T(n,k) = T(n-1,k) - 2*T(n-1,k-1) + 2*T(n,k-1) + T(n+1,k-1). Also, a recurrence involving antidiagonals is: T(n,k) = T(n-1,k) + Sum_{j=1..k} [3*T(n-1+j,k-j) - 2*T(n-2+j,k-j)] for n>k>=0.

Example

Show: T(n,k) = T(n-1,k) - 2*T(n-1,k-1) + 2*T(n,k-1) + T(n+1,k-1)
at n=8,k=4: T(8,4) = T(7,4) - 2*T(7,3) + 2*T(8,3) + T(9,3)
or: 1375 = 873 - 2*184 + 2*260 + 350.
Triangle begins:
  1;
  1, 0;
  1, 1,  0;
  1, 2,  5,   0;
  1, 3, 11,  30,    0;
  1, 4, 18,  70,  201,     0;
  1, 5, 26, 121,  487,  1445,     0;
  1, 6, 35, 184,  873,  3592, 10900,      0;
  1, 7, 45, 260, 1375,  6606, 27600,  85128,       0;
  1, 8, 56, 350, 2010, 10672, 51728, 218566,  682505,       0;
  1, 9, 68, 455, 2796, 15996, 85182, 415629, 1771367, 5585115, 0;

Maple

T:= proc(n, k) option remember;
      if k<0 or  k>n then 0;
    elif k=0 then 1;
    elif k=n then 0;
    else T(n-1, k) -2*T(n-1, k-1) +2*T(n, k-1) +T(n+1, k-1);
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 17 2021

Mathematica

T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, 1, If[k==n, 0, T[n-1, k] -2*T[n-1, k-1] +2*T[n, k-1] +T[n+1, k-1] ]]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 17 2021 *)

Prog

(PARI) {T(n, k)=polcoeff((serreverse(x*(1-2*x+sqrt((1-2*x)*(1-6*x)+x*O(x^k)))/2/(1-2*x))/x)^(n-k), k)}
(Sage)
@CachedFunction
def T(n, k):
    if (k<0 or k>n): return 0
    elif (k==0): return 1
    elif (k==n): return 0
    else: return T(n-1, k) -2*T(n-1, k-1) +2*T(n, k-1) +T(n+1, k-1)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 17 2021

Crossrefs

Cf. A118346, A118345.

Sequence in context: A091086 A336686 A371501 * A341268 A011183 A005671

Adjacent sequences: A118346 A118347 A118348 * A118350 A118351 A118352

Keyword

nonn,tabl

Author

Paul D. Hanna, Apr 26 2006

Status

approved