A370380 Array read by downward antidiagonals: A(n,k) = (k+2)*A(n-1,k+1) + Sum_{j=0..k} A(n-1,j) with A(0,k) = 1, n >= 0, k >= 0.

1, 1, 3, 1, 5, 13, 1, 7, 29, 71, 1, 9, 51, 195, 461, 1, 11, 79, 409, 1493, 3447, 1, 13, 113, 737, 3623, 12823, 29093, 1, 15, 153, 1203, 7427, 35285, 122125, 273343, 1, 17, 199, 1831, 13601, 81009, 375591, 1277991, 2829325, 1, 19, 251, 2645, 22961, 164371, 954419, 4344485, 14584789, 31998903

Offset

0,3

Links

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

Formula

Conjecture: A(n,0) = A003319(n+2). - Mikhail Kurkov, Oct 27 2024

A(n,k) = A(n,k-1) - k*A(n-1,k) + (k+2)*A(n-1,k+1) with A(n,0) = A(n-1,0) + 2*A(n-1,1), A(0,k) = 1. - Mikhail Kurkov, Nov 23 2024

Example

Array begins:
===========================================================
n\k|     0      1      2      3       4       5       6 ...
---+-------------------------------------------------------
0  |     1      1      1      1       1       1       1 ...
1  |     3      5      7      9      11      13      15 ...
2  |    13     29     51     79     113     153     199 ...
3  |    71    195    409    737    1203    1831    2645 ...
4  |   461   1493   3623   7427   13601   22961   36443 ...
5  |  3447  12823  35285  81009  164371  304667  526833 ...
6  | 29093 122125 375591 954419 2124937 4289433 8025755 ...
  ...

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+1)*v[k+1] + sum(j=1, k, v[j])); r[1+i] = v[1..n+1]); Mat(r)}
{ A(6) }

Crossrefs

Row 2 appears to be essentially A144391. - Joerg Arndt, Feb 17 2024

Cf. A003319.

Sequence in context: A105064 A073496 A176122 * A091623 A215474 A348114

Adjacent sequences: A370377 A370378 A370379 * A370381 A370382 A370383

Keyword

nonn,tabl,changed

Author

Mikhail Kurkov, Feb 17 2024

Status

approved