A369527 Array read by downward antidiagonals: A(n,k) = (k+1)^2*A(n-1,k) + A(n-1,k+1) with A(0,k) = 1, n >= 0, k >= 0.

1, 1, 2, 1, 5, 7, 1, 10, 30, 37, 1, 17, 107, 227, 264, 1, 26, 298, 1261, 2169, 2433, 1, 37, 687, 5455, 16804, 25480, 27913, 1, 50, 1382, 18557, 105837, 257073, 358993, 386906, 1, 65, 2515, 52267, 516192, 2209584, 4523241, 5959213, 6346119, 1, 82, 4242, 127477, 2009089, 14913889, 50267233, 90976402, 114813254, 121159373

Offset

0,3

Links

Table of n, a(n) for n=0..54.

Ira M. Gessel, General case of the some R-recursions, answer to question on MathOverflow (2024).

Example

Array begins:
====================================================
n\k|    0     1      2       3        4        5 ...
---+------------------------------------------------
0  |    1     1      1       1        1        1 ...
1  |    2     5     10      17       26       37 ...
2  |    7    30    107     298      687     1382 ...
3  |   37   227   1261    5455    18557    52267 ...
4  |  264  2169  16804  105837   516192  2009089 ...
5  | 2433 25480 257073 2209584 14913889 78851808 ...
  ...

Prog

(PARI)
A(m, n=m)={my(r=vectorv(m+1), v=vector(n+m+1, k, 1)); r[1] = v[1..n+1];
for(i=1, m, v=vector(#v-1, k, k^2*v[k] + v[k+1]); r[1+i] = v[1..n+1]); Mat(r)}
{ A(5) }

Crossrefs

Column k=0 is A135920 (without initial term and with different offset).

Sequence in context: A064642 A193630 A144505 * A370382 A059039 A332022

Adjacent sequences: A369524 A369525 A369526 * A369528 A369529 A369530

Keyword

nonn,tabl

Author

Mikhail Kurkov, Jan 25 2024

Status

approved