A392058 Array read by downward antidiagonals: A(n,k) = A(n-1,k+1) + Sum_{j=0..k} binomial(k+1,j)*A(n-1,j)*A(k-j,0) with A(0,k) = 1.

1, 1, 2, 1, 5, 7, 1, 17, 31, 38, 1, 83, 178, 254, 292, 1, 578, 1330, 2099, 2683, 2975, 1, 5474, 12748, 21338, 29425, 35375, 38350, 1, 66932, 153136, 263870, 382966, 489383, 566083, 604433, 1, 1014002, 2249503, 3913556, 5865952, 7837136, 9538360, 10747226, 11351659

Offset

0,3

Links

Table of n, a(n) for n=0..44.

Formula

Conjecture: A(n,0) = A233335(n+1).

Example

Array begins:
============================================================================
n\k|     0      1       2         3          4           5             6 ...
---+------------------------------------------------------------------------
0  |     1      1       1         1          1           1             1 ...
1  |     2      5      17        83        578        5474         66932 ...
2  |     7     31     178      1330      12748      153136       2249503 ...
3  |    38    254    2099     21338     263870     3913556      68599898 ...
4  |   292   2683   29425    382966    5865952   104739142    2158835605 ...
5  |  2975  35375  489383   7837136  144383006  3039205823   72586944074 ...
6  | 38350 566083 9538360 182707696 3959710333 96600644167 2639032949860 ...
  ...

Prog

(PARI) antidiagonals(n) = {my(v = vector(n+1, i, vector(n-i+2, j, i==1)));
for(i=1, n, forstep(j=i-1, 0, -1, v[i-j+1][j+1] = v[i-j][j+2] + sum(k=0, j, binomial(j+1, k)*v[i-j][k+1]*v[j-k+1][1])));
v = vector(n+1, i, vector(i, j, v[j][i-j+1]))}

Crossrefs

Cf. A233335.

Sequence in context: A370382 A385583 A059039 * A332022 A109261 A085240

Adjacent sequences: A392055 A392056 A392057 * A392059 A392060 A392061

Keyword

nonn,tabl

Author

Mikhail Kurkov, Dec 29 2025

Status

approved