0,4

The left-most column equals A003169 shift one place right. Each column k>0 equals the convolution of the prior column and A003169. Row sums form A100327.

T(n, 0) = A003169(n) = Sum_{k=0..n-1} (k+1)*T(n-1, k) for n>0, with T(0, 0)=1. T(n, k) = Sum_{i=0..n-k} T(i+1, 0)*T(n-i-1, k-1) for n>0. G.f. A(x, y) = (1+G003169(x))/(1-y*G003169(x)) where G003169(x) is the g.f. of A003169.

Left-most column equals Sum_{k=0..n-1} (k+1)*T(n-1,k):

T(4,0) = 79 = 1*(14)+2*(20)+3*(7)+4*(1) = 1*T(3,0)+2*T(3,1)+3*T(3,2)+4*T(3,3).

All other elements are from the convolution of prior column and A003169:

T(4,2) = 46 = 1*(20)+3*(4)+14*(1) = T(1,0)*T(3,1)+T(2,0)*T(2,1)+T(3,0)*T(1,1).

Rows begin:

[1],

[1,1],

[3,4,1],

[14,20,7,1],

[79,116,46,10,1],

[494,736,311,81,13,1],

[3294,4952,2174,626,125,16,1],

[22952,34716,15634,4798,1088,178,19,1],...

First column forms A003169 shift right.

Binomial transform of row 3 forms column 3 of square A100324:

BINOMIAL([14,20,7,1]) = [14,34,61,96,140,194,259,...].

Binomial transform of row 4 forms column 4 of square A100324:

BINOMIAL([79,116,46,10,1]) = [79,195,357,575,860,1224,...].

(PARI) {T(n, k)=if(n<k|k<0, 0, if(n==0, 1, if(k==0, sum(i=0, n-1, (i+1)*T(n-1, i)), sum(i=0, n-k, T(i+1, 0)*T(n-i-1, k-1))); ))}

Cf. A003169, A100324, A100327.

Sequence in context: A110506 A114189 A059110 this_sequence A028338 A039757 A136228

Adjacent sequences: A100323 A100324 A100325 this_sequence A100327 A100328 A100329

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Nov 17 2004