A118343 Triangle, read by rows, where diagonals are successive self-convolutions of A108447.

1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 9, 20, 0, 1, 4, 15, 48, 113, 0, 1, 5, 22, 85, 282, 688, 0, 1, 6, 30, 132, 519, 1762, 4404, 0, 1, 7, 39, 190, 837, 3330, 11488, 29219, 0, 1, 8, 49, 260, 1250, 5516, 22135, 77270, 199140, 0, 1, 9, 60, 343, 1773, 8461, 37404, 151089, 532239, 1385904, 0

Offset

0,8

Comments

A108447 equals the central terms of pendular triangle A118340 and the diagonals of this triangle form the semi-diagonals of the triangle A118340. Row sums equal A054727, the number of forests of rooted trees with n nodes on a circle without crossing edges.

Links

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

Formula

Since g.f. G=G(x) of A108447 satisfies: G = 1 - x*G + x*G^2 + x*G^3 then T(n,k) = T(n-1,k) - T(n-1,k-1) + 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} [2*T(n-1+j,k-j) - T(n-2+j,k-j)] for n>k>=0.

Sum_{k=0..n} T(n,k) = [n=0] + A054727(n) = [n=0] + Sum_{j=1..n} binomial(n, j-1)*binomial(3*n-2*j-1, n-j)/(2*n-j). - G. C. Greubel, Mar 17 2021

Example

Show: T(n,k) = T(n-1,k) - T(n-1,k-1) + T(n,k-1) + T(n+1,k-1)
at n=8,k=4: T(8,4) = T(7,4) - T(7,3) + T(8,3) + T(9,3)
or 837 = 519 - 132 + 190 + 260.
Triangle begins:
  1;
  1, 0;
  1, 1,  0;
  1, 2,  4,   0;
  1, 3,  9,  20,    0;
  1, 4, 15,  48,  113,    0;
  1, 5, 22,  85,  282,  688,     0;
  1, 6, 30, 132,  519, 1762,  4404,      0;
  1, 7, 39, 190,  837, 3330, 11488,  29219,      0;
  1, 8, 49, 260, 1250, 5516, 22135,  77270, 199140,       0;
  1, 9, 60, 343, 1773, 8461, 37404, 151089, 532239, 1385904, 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) -T(n-1, k-1) +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] -T[n-1, k-1] +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-x+sqrt((1-x)*(1-5*x)+x*O(x^k)))/2/(1-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) -T(n-1, k-1) +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. A054727 (row sums), A108447, A118340.

Sequence in context: A173004 A378323 A378290 * A309148 A351761 A226031

Adjacent sequences: A118340 A118341 A118342 * A118344 A118345 A118346

Keyword

nonn,tabl

Author

Paul D. Hanna, Apr 26 2006

Status

approved