A371740 Triangle read by rows: g.f. (1 - t)^(-x) * (1 + t)^(2-x).

1, 2, 1, 1, 0, 4, 0, 3, 1, 0, 6, 6, 0, 5, 6, 1, 0, 16, 24, 8, 0, 14, 23, 10, 1, 0, 60, 110, 60, 10, 0, 54, 105, 65, 15, 1, 0, 288, 600, 420, 120, 12, 0, 264, 574, 435, 145, 21, 1, 0, 1680, 3836, 3150, 1190, 210, 14, 0, 1560, 3682, 3199, 1330, 280, 28, 1, 0, 11520, 28224, 25984, 11760, 2800, 336, 16

Offset

0,2

Links

Table of n, a(n) for n=0..71.

Formula

G.f.: (1 - t)^(-x)*(1 + t)^(2-x) = Sum_{n >= 0} R(n, x)*t^n/floor((n+1)/2)! = 1 + 2*t/1! + (1 + x)*t^2/1! + 4*x*t^3/2! + x*(3 + x)*t^4/2! + 6*x*(1 + x)*t^5/3! + x*(1 + x)*(5 + x)*t^6/3! + 8*x*(1 + x)*(2 + x)*t^7/3! + x*(1 + x)*(2 + x)*(7 + x)*t^8/4! + 10*x*(1 + x)*(2 + x)*(3 + x)*t^9/5! + ....

Row polynomials: R(2*n, x) = (2*n - 1 + x) * Product_{i = 0..n-2} (x + i) for n >= 1.

R(2*n+1, x) = (2*n + 2) * Product_{i = 0..n-1} (x + i) for n >= 0.

T(2*n, k) = |Stirling1(n, k)| + n*|Stirling1(n-1, k)| = A132393(n, k) + n* A132393(n-1, k);

T(2*n+1, k) = (2*n + 2)*|Stirling1(n, k)| = (2*n + 2)*A132393(n, k).

n-th row sum equals 2 * floor((n+1)/2)! for n >= 1.

Example

Triangle begins
 n\k |  0    1     2    3    4   5
 - - - - - - - - - - - - - - - - -
  0  |  1
  1  |  2
  2  |  1    1
  3  |  0    4
  4  |  0    3     1
  5  |  0    6     6
  6  |  0    5     6    1
  7  |  0   16    24    8
  8  |  0   14    23   10    1
  9  |  0   60   110   60   10
 10  |  0   54   105   65   15   1
 ...

Maple

with(combinat):
T := proc (n, k); if irem(n, 2) = 0 then abs(Stirling1((1/2)*n, k)) + (n/2)*abs(Stirling1((n-2)/2, k)) else (n+1)*abs(Stirling1((n-1)/2, k)) end if; end proc:
seq(print(seq(T(n, k), k = 0..floor(n/2))), n = 0..12);

Crossrefs

Cf. A052849, A132393, A371741.

Sequence in context: A351322 A305736 A324642 * A326757 A147787 A247288

Adjacent sequences: A371737 A371738 A371739 * A371741 A371742 A371743

Keyword

nonn,tabf,easy

Author

Peter Bala, Apr 09 2024

Status

approved