A395345 Triangle read by rows: T(n, k) = k * T(n, k+1) + T(n-1, k-1) - (n-k-1) * T(n-1, k) with T(n, 0) = 0^n, T(n, k) = 0 for k < 0 or k > n.

1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 7, 11, 6, 1, 0, 34, 55, 35, 10, 1, 0, 213, 349, 240, 85, 15, 1, 0, 1630, 2695, 1939, 770, 175, 21, 1, 0, 14747, 24527, 18186, 7749, 2030, 322, 28, 1, 0, 153946, 257175, 194795, 87066, 24969, 4662, 546, 36, 1, 0, 1821473, 3053041, 2349660, 1087085, 331905, 68943, 9660, 870, 45, 1

Offset

0,8

Links

Table of n, a(n) for n=0..65.

Tomasz Kania, Closed form for certain integer coefficients based on A075856 using A291980, answer to question on MathOverflow, 2026.

Formula

Due to Tomasz Kania: (Start)

T(n,k) = Sum_{j=0..n-k} (-1)^(n+k+j) * binomial(n+j-1, k-1) * A075856(n-k,j).

T(n+1, k+1) = Sum_{j=k..n} T(n, j) * A291980(j, k).

E.g.f. for column k: A(x)^k / k! where A(x) = Sum_{n>=0} A074059(n) * x^n / n! such that x = 2 * A(x) - (1 + A(x)) * log(1 + A(x)). (End)

Example

Triangle begins:
  1;
  0,      1;
  0,      1,      1;
  0,      2,      3,      1;
  0,      7,     11,      6,     1;
  0,     34,     55,     35,    10,     1;
  0,    213,    349,    240,    85,    15,    1;
  0,   1630,   2695,   1939,   770,   175,   21,   1;
  0,  14747,  24527,  18186,  7749,  2030,  322,  28,  1;
  0, 153946, 257175, 194795, 87066, 24969, 4662, 546, 36, 1;

Maple

T := proc(n, k) option remember; if n = k then 1 elif k <= 0 or k > n then 0 else
k * T(n, k+1) + T(n-1, k-1) - (n-k-1) * T(n-1, k) fi end:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, Apr 20 2026

Prog

(PARI) rows(n) = my(v1 = vector(n+1, i, vector(i, j, 0))); for(i=0, n, v1[i+1][i+1] = 1; forstep(j=i-1, 1, -1, v1[i+1][j+1] = j*v1[i+1][j+2] + v1[i][j] - (i-j-1)*v1[i][j+1])); v1
(PARI) \\ using function inverse_bell_matrix_row from A354794
row(n) = if(n==0, [1], concat(0, inverse_bell_matrix_row(n, x->(!(x%2)<<1 - 1)*(x-1)!))) \\ Mikhail Kurkov, May 12 2026

Crossrefs

Cf. A074059, A075856, A291980.

Sequence in context: A048994 A132393 A344172 * A121434 A296455 A383656

Adjacent sequences: A395342 A395343 A395344 * A395346 A395347 A395348

Keyword

nonn,tabl,easy

Author

Mikhail Kurkov, Apr 20 2026

Status

approved