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A100324 Square array, read by antidiagonals, where rows are successive self-convolutions of the top row, which equals A003169 shifted one place right. 5

%I

%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 57668,165127,1,9,52,259,1224,5641,25298,108018,415995,1217270,1,10,63,336

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

%C Main diagonal: T(n,n) = (n+1)*A032349(n+1) for n>0 with T(0,0) = 1. Antidiagonal sums form A100325. Also, column k forms the binomial transform of row k in triangle A100326 for k>=0.

%F T(n, k) = Sum_{i=0..k} T(0, k-i)*T(n-1, i) for n>0. T(0, k) = A003169(k+1) = ( (324*k^2-708*k+360)*T(0, k-1) - (371*k^2-1831*k+2250)*T(0, k-2) +(20*k^2-130*k+210)*T(0, k-3) )/(16*k*(2*k-1)) for k>2, with T(0, 0)=T(0, 1)=1, T(0, 2)=3.

%e Rows begin:

%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 Main diagonal = [1,2*1,3*4,4*24,5*172,6*1360,...,(n+1)*A032349(n+1),...].

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

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

%K nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Nov 16 2004

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Last modified September 26 08:11 EDT 2021. Contains 347664 sequences. (Running on oeis4.)