A370381 - OEIS

%I #15 Nov 25 2024 08:31:34

%S 1,1,3,1,7,13,1,15,45,71,1,31,145,319,461,1,63,453,1355,2525,3447,1,

%T 127,1393,5623,13241,22199,29093,1,255,4245,23051,68261,138219,215157,

%U 273343,1,511,12865,93799,348761,850031,1549889,2282639,2829325,1,1023,38853,379835,1771925,5193867,11065437,18672307,26340253,31998903

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

%F Conjecture: A(n,0) = A003319(n+2). - _Mikhail Kurkov_, Oct 27 2024

%e Array begins:

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

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

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

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

%e 1  |    3     7     15     31      63      127 ...

%e 2  |   13    45    145    453    1393     4245 ...

%e 3  |   71   319   1355   5623   23051    93799 ...

%e 4  |  461  2525  13241  68261  348761  1771925 ...

%e 5  | 3447 22199 138219 850031 5193867 31604159 ...

%e   ...

%o (PARI)

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

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

%o { A(5) }

%Y Cf. A003319.

%K nonn,tabl,changed

%O 0,3

%A _Mikhail Kurkov_, Feb 17 2024