A100324 - OEIS

%I #9 Jan 31 2023 08:29:44

%S 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,

%T 1230,3294,1,7,33,140,575,2277,8246,22952,1,8,42,194,860,3716,15372,

%U 57668,165127,1,9,52,259,1224,5641,25298,108018,415995,1217270

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

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

%H G. C. Greubel, <a href="/A100324/b100324.txt">Antidiagonals n = 0..50, flattened</a>

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

%F 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.

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

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

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

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

%e Array, A(n,k), begins as:

%e   1, 1,  3,  14,   79,   494,  3294, ...;

%e   1, 2,  7,  34,  195,  1230,  8246, ...;

%e   1, 3, 12,  61,  357,  2277, 15372, ...;

%e   1, 4, 18,  96,  575,  3716, 25298, ...;

%e   1, 5, 25, 140,  860,  5641, 38775, ...;

%e   1, 6, 33, 194, 1224,  8160, 56695, ...;

%e   1, 7, 42, 259, 1680, 11396, 80108, ...;

%e Antidiagonal triangle, T(n,k), begins as:

%e   1;

%e   1, 1;

%e   1, 2,  3;

%e   1, 3,  7,  14;

%e   1, 4, 12,  34,  79;

%e   1, 5, 18,  61, 195,  494;

%e   1, 6, 25,  96, 357, 1230, 3294;

%e   1, 7, 33, 140, 575, 2277, 8246, 22952;

%t 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 *)

%t 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 T[n_, k_]:= A[n-k, k];

%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jan 31 2023 *)

%o (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)) )));)}

%o (SageMath)

%o def f(n): # f = A003169

%o     if (n<2): return 1

%o     elif (n==2): return 3

%o     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))

%o @CachedFunction

%o def A(n, k): # A = 100324

%o     if (n==0): return f(k)

%o     elif (k==0): return 1

%o     else: return sum( A(0,k-j)*A(n-1, j) for j in range(k+1) )

%o def T(n,k): return A(n-k,k)

%o flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jan 31 2023

%Y Cf. A003169, A032349, A100325, A100326.

%K nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Nov 16 2004