A380179 Triangle T(n,k) read by rows: T(n,k) = -binomial(n+1,k) + Sum_{i=0..k} Sum_{j=0..i+1} (i+1)^(n-i+j)*(-1)^(k-i)/(j!*(k-i)!) for 0 <= k <= n.

1, 1, 1, 1, 5, 1, 1, 14, 14, 1, 1, 33, 68, 30, 1, 1, 72, 257, 218, 55, 1, 1, 151, 873, 1189, 553, 91, 1, 1, 310, 2812, 5734, 4094, 1204, 140, 1, 1, 629, 8802, 25916, 26484, 11598, 2352, 204, 1, 1, 1268, 27107, 112718, 158840, 96702, 28566, 4236, 285, 1

Offset: 0

Author

Mikhail Kurkov, Jan 14 2025

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1;
  1,    1;
  1,    5,     1;
  1,   14,    14,      1;
  1,   33,    68,     30,      1;
  1,   72,   257,    218,     55,     1;
  1,  151,   873,   1189,    553,    91,     1;
  1,  310,  2812,   5734,   4094,  1204,   140,    1;
  1,  629,  8802,  25916,  26484, 11598,  2352,  204,   1;
  1, 1268, 27107, 112718, 158840, 96702, 28566, 4236, 285, 1;
		

Crossrefs

Cf. A347420.

Programs

PARI

T(n,k) = if(k >= 0 && n >= k, -binomial(n+1, k) + sum(i=0, k, sum(j=0, i+1, (i+1)^(n-i+j)*(-1)^(k-i)/(j!*(k-i)!))))

Formula

Conjecture: A347420(n) = 2^n + Sum_{k=1..n-1} T(n-1, k) for n >= 0.