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

%I #19 Nov 24 2024 03:32:56

%S 1,2,3,3,9,12,4,22,46,58,5,45,147,263,321,6,81,397,1012,1654,1975,7,

%T 133,933,3341,7340,11290,13265,8,204,1962,9637,28333,56278,82808,

%U 96073,9,297,3776,24758,96313,246905,455534,647680,743753,10,415,6767,57678,292092,961897,2227689,3882510,5370016,6113769

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

%F Conjecture: A(n,0) = A258173(n+1). - _Mikhail Kurkov_, Oct 27 2024

%F A(n,k) = A(n,k-1) + (A(n,k-1) - A(n-1,k))/k + k*A(n-1,k) + A(n-1,k+1) with A(n,0) = A(n-1,0) + A(n-1,1), A(0,k) = k+1. - _Mikhail Kurkov_, Nov 24 2024

%e Array begins:

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

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

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

%e 0 | 1 2 3 4 5 6 7 ...

%e 1 | 3 9 22 45 81 133 204 ...

%e 2 | 12 46 147 397 933 1962 3776 ...

%e 3 | 58 263 1012 3341 9637 24758 57678 ...

%e 4 | 321 1654 7340 28333 96313 292092 800991 ...

%e 5 | 1975 11290 56278 246905 961897 3357309 10601156 ...

%e 6 | 13265 82808 455534 2227689 9749034 38415080 137251108 ...

%e ...

%o (PARI)

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

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

%o { A(6) }

%Y Cf. A258173.

%K nonn,tabl,changed

%O 0,2

%A _Mikhail Kurkov_, Mar 28 2024