A352650 Triangle read by rows: T(n,k) = n * T(n-1,k) + (-1)^(n-k) for 0 <= k <= n with initial values T(n,k) = 0 if n < 0 or k < 0 or k > n.

1, 0, 1, 1, 1, 1, 2, 4, 2, 1, 9, 15, 9, 3, 1, 44, 76, 44, 16, 4, 1, 265, 455, 265, 95, 25, 5, 1, 1854, 3186, 1854, 666, 174, 36, 6, 1, 14833, 25487, 14833, 5327, 1393, 287, 49, 7, 1, 133496, 229384, 133496, 47944, 12536, 2584, 440, 64, 8, 1, 1334961, 2293839, 1334961, 479439, 125361, 25839, 4401, 639, 81, 9, 1

Offset: 0

Author

Werner Schulte, Apr 04 2022

Keywords

• nonn
• easy
• tabl

Comments

Conjecture 1: T(n,k) = Sum_{i=0..n-k} (-1)^(n+k+i) * A326326(n-k,i) * n^i for 0 <= k <= n.

Conjecture 2: T(n,k) = T(n-k,0) + Sum_{i=1..n-k} T(n-k,i) * T(i+k,k) * k / (i + k - 1) for 0 < k <= n.

Examples

			The triangle T(n,k) for 0 <= k <= n starts:
n\k :       0       1       2      3      4     5    6   7  8  9
================================================================
  0 :       1
  1 :       0       1
  2 :       1       1       1
  3 :       2       4       2      1
  4 :       9      15       9      3      1
  5 :      44      76      44     16      4     1
  6 :     265     455     265     95     25     5    1
  7 :    1854    3186    1854    666    174    36    6   1
  8 :   14833   25487   14833   5327   1393   287   49   7  1
  9 :  133496  229384  133496  47944  12536  2584  440  64  8  1
  etc.
		

Crossrefs

Cf. A000166 (column 0 and 2), A002467 (column 1), A006347 (column 3), A006348 (column 4), A009179 (row sums, signed), A352988 (matrix inverse).

Cf. A094587, A097807, A326326.

Programs

Maple

T := proc(n,k) option remember;
if k > n then 0 else n * T(n-1,k) + (-1)^(n-k) fi end:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od; # Peter Luschny, Apr 11 2022

Formula

T(n,n) = 1 for n >= 0.

T(n,n-1) = n - 1 for n > 0.

T(n,n-2) = (n - 1)^2 for n > 1.

T(n,0) = A000166(n) for n >= 0.

T(n,1) = A002467(n) for n > 0.

T(n,2) = A000166(n) for n > 1.

T(n,k) + T(n,k+1) = (n!) / (k!) for 0 <= k <= n.

T(n,k) = (n - 1) * (T(n-1,k) + T(n-2,k)) for 0 <= k < n-1.

T(n,k) = (T(n,k-2) - (k - 2) * T(n,k-1)) / (k - 1) for 1 < k <= n.

The row polynomials p(n,x) = Sum_{k=0..n} T(n,k) * x^k satisfy recurrence equation p(n,x) = (n - 1) * (p(n-1,x) + p(n-2,x)) + x^n for n > 0 with initial value p(0,x) = 1.

Row sums are p(n,1) = abs(A009179(n)) for n >= 0.

Alternating row sums are p(n,-1) = (-1)^n for n >= 0.

T(n,k) * T(n+1,k+1) - T(n+1,k) * T(n,k+1) = (-1)^(n-k) * A094587(n,k) for 0 <= k <= n.

Define 3x3-matrices T(i,j) with n <= i <= n+2 and k <= j <= k+2. Then we have: det(T(i,j)) = 0^(n-k) for 0 <= k <= n.

E.g.f. of column k >= 0: Sum_{n>=k} T(n,k) * t^n / (n!) = (Sum_{n>=k} (-t)^n / (n!)) * (-1)^k / (1 - t).

E.g.f.: Sum_{n>=0, k=0..n} T(n,k) * x^k * t^n / (n!) = (x * exp(x * t) + exp(-t)) / ((1 + x) * (1 - t)).

p(n,x) = Sum_{k=0..n} ((n!)/(k!))*(x^(k+1) + (-1)^k)/(x + 1) for n >= 0.

T(n,k) = Sum_{i=0..n-k} (-1)^i * (n!) / ((k+i)!) for 0 <= k <= n.

T(n,k) equals matrix product of A094587 and A097807.