A100324 Square array, read by antidiagonals, where rows are successive self-convolutions of the top row, which equals A003169 shifted one place right.

1, 1, 1, 1, 2, 3, 1, 3, 7, 14, 1, 4, 12, 34, 79, 1, 5, 18, 61, 195, 494, 1, 6, 25, 96, 357, 1230, 3294, 1, 7, 33, 140, 575, 2277, 8246, 22952, 1, 8, 42, 194, 860, 3716, 15372, 57668, 165127, 1, 9, 52, 259, 1224, 5641, 25298, 108018, 415995, 1217270

Offset: 0

Author

Paul D. Hanna, Nov 16 2004

Keywords

• nonn
• tabl

Comments

Column k forms the binomial transform of row k in triangle A100326 for k>=0.

Examples

			Array, A(n,k), begins as:
  1, 1,  3,  14,   79,   494,  3294, ...;
  1, 2,  7,  34,  195,  1230,  8246, ...;
  1, 3, 12,  61,  357,  2277, 15372, ...;
  1, 4, 18,  96,  575,  3716, 25298, ...;
  1, 5, 25, 140,  860,  5641, 38775, ...;
  1, 6, 33, 194, 1224,  8160, 56695, ...;
  1, 7, 42, 259, 1680, 11396, 80108, ...;
Antidiagonal triangle, T(n,k), begins as:
  1;
  1, 1;
  1, 2,  3;
  1, 3,  7,  14;
  1, 4, 12,  34,  79;
  1, 5, 18,  61, 195,  494;
  1, 6, 25,  96, 357, 1230, 3294;
  1, 7, 33, 140, 575, 2277, 8246, 22952;
		

Links

• G. C. Greubel, Antidiagonals n = 0..50, flattened

Crossrefs

Cf. A003169, A032349, A100325, A100326.

Programs

Mathematica

f[n_]:= f[n]= If[n<2, 1, If[n==2, 3, ((324*n^2-708*n+360)*f[n-1] - (371*n^2-1831*n+2250)*f[n-2] +(20*n^2-130*n+210)*f[n-3])/(16*n*(2*n -1)) ]]; (* f = A003169 *)
A[n_, k_]:= A[n, k]= If[n==0, f[k], If[k==0, 1, Sum[A[0,k-j]*A[n-1,j], {j,0,k}]]]; (* A = A100324 *)
T[n_, k_]:= A[n-k, k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 31 2023 *)

PARI

{A(n,k)=if(k==0,1,if(n>0,sum(i=0,k,A(0,k-i)*A(n-1,i)), if(k==1,1,if(k==2,3,( (324*k^2-708*k+360)*A(0,k-1)-(371*k^2-1831*k+2250)*A(0,k-2)+(20*k^2-130*k+210)*A(0,k-3))/(16*k*(2*k-1)) )));)}

SageMath

def f(n): # f = A003169
    if (n<2): return 1
    elif (n==2): return 3
    else: return ((324*n^2-708*n+360)*f(n-1) - (371*n^2-1831*n+2250)*f(n-2) + (20*n^2-130*n+210)*f(n-3))/(16*n*(2*n-1))
@CachedFunction
def A(n, k): # A = 100324
    if (n==0): return f(k)
    elif (k==0): return 1
    else: return sum( A(0,k-j)*A(n-1, j) for j in range(k+1) )
def T(n,k): return A(n-k,k)
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 31 2023

Formula

A(n, k) = Sum_{i=0..k} A(0, k-i)*A(n-1, i) for n>0.

A(0, k) = A003169(k+1) = ( (324*k^2-708*k+360)*A(0, k-1) - (371*k^2-1831*k+2250)*A(0, k-2) +(20*k^2-130*k+210)*A(0, k-3) )/(16*k*(2*k-1)) for k>2, with A(0, 0) = A(0, 1)=1, A(0, 2)=3.

A(n, n) = (n+1)*A032349(n+1).

T(n, k) = A(n-k, k) (Antidiagonal triangle).

T(n, n) = A003169(n+1).

Sum_{k=0..n} T(n, k) = A100325(n) (Antidiagonal row sums).