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Array read by antidiagonals: A(n,k) = A(n-1,k+2) + Sum_{j=0..k} binomial(k,j)*A(n-1,j) with A(0,k) = 1, n >= 0, k >= 0.
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%I #12 Oct 27 2024 11:28:16

%S 1,1,2,1,3,7,1,5,14,37,1,9,30,89,264,1,17,68,227,737,2433,1,33,162,

%T 611,2169,7696,27913,1,65,404,1727,6695,25480,98093,386906,1,129,1050,

%U 5099,21573,87964,358993,1490687,6346119,1,257,2828,15647,72287,315688

%N Array read by antidiagonals: A(n,k) = A(n-1,k+2) + Sum_{j=0..k} binomial(k,j)*A(n-1,j) with A(0,k) = 1, n >= 0, k >= 0.

%F Conjecture: A(n, 0) = A135920(n+1). - _Mikhail Kurkov_, Oct 27 2024

%e Array begins:

%e =============================================================

%e n\k| 0 1 2 3 4 5 6 ...

%e ---+---------------------------------------------------------

%e 0 | 1 1 1 1 1 1 1 ...

%e 1 | 2 3 5 9 17 33 65 ...

%e 2 | 7 14 30 68 162 404 1050 ...

%e 3 | 37 89 227 611 1727 5099 15647 ...

%e 4 | 264 737 2169 6695 21573 72287 251109 ...

%e 5 | 2433 7696 25480 87964 315688 1174756 4522480 ...

%e 6 | 27913 98093 358993 1364681 5376121 21901073 92076673 ...

%e ...

%o (PARI)

%o A(m, n=m)={my(r=vectorv(m+1), v=vector(n+2*m+1, k, 1)); r[1] = v[1..n+1];

%o for(i=1, m, v=vector(#v-2, k, v[k+2] + sum(j=1, k, binomial(k-1, j-1)*v[j])); r[1+i] = v[1..n+1]); Mat(r)}

%o { A(6) }

%K nonn,tabl

%O 0,3

%A _Mikhail Kurkov_, Jan 27 2024