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A371567 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.

%I A371567 #19 Nov 24 2024 03:32:56
%S A371567 1,2,3,3,9,12,4,22,46,58,5,45,147,263,321,6,81,397,1012,1654,1975,7,
%T A371567 133,933,3341,7340,11290,13265,8,204,1962,9637,28333,56278,82808,
%U A371567 96073,9,297,3776,24758,96313,246905,455534,647680,743753,10,415,6767,57678,292092,961897,2227689,3882510,5370016,6113769
%N A371567 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 A371567 Conjecture: A(n,0) = A258173(n+1). - _Mikhail Kurkov_, Oct 27 2024
%F A371567 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 A371567 Array begins:
%e A371567 ==============================================================
%e A371567 n\k|     0     1      2       3       4        5         6 ...
%e A371567 ---+----------------------------------------------------------
%e A371567 0  |     1     2      3       4       5        6         7 ...
%e A371567 1  |     3     9     22      45      81      133       204 ...
%e A371567 2  |    12    46    147     397     933     1962      3776 ...
%e A371567 3  |    58   263   1012    3341    9637    24758     57678 ...
%e A371567 4  |   321  1654   7340   28333   96313   292092    800991 ...
%e A371567 5  |  1975 11290  56278  246905  961897  3357309  10601156 ...
%e A371567 6  | 13265 82808 455534 2227689 9749034 38415080 137251108 ...
%e A371567   ...
%o A371567 (PARI)
%o A371567 A(m, n=m)={my(r=vectorv(m+1), v=vector(n+m+1, k, k)); r[1] = v[1..n+1];
%o A371567 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 A371567 { A(6) }
%Y A371567 Cf. A258173.
%K A371567 nonn,tabl
%O A371567 0,2
%A A371567 _Mikhail Kurkov_, Mar 28 2024