A371741 Triangle read by rows: g.f. (1 - t)^(-x) * (1 + t)^(3-x).

1, 3, 3, 1, 1, 3, 0, 7, 1, 0, 5, 3, 0, 11, 12, 1, 0, 9, 12, 3, 0, 30, 47, 18, 1, 0, 26, 45, 22, 3, 0, 114, 215, 125, 25, 1, 0, 102, 205, 135, 35, 3, 0, 552, 1174, 855, 265, 33, 1, 0, 504, 1122, 885, 315, 51, 3, 0, 3240, 7518, 6349, 2520, 490, 42, 1, 0, 3000, 7210, 6447, 2800, 630, 70, 3

Offset

0,2

Links

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

Formula

G.f.: (1 - t)^(-x)*(1 + t)^(3-x) = Sum_{n >= 0} R(n, x)*t^n/floor(n/2)! = 1 + 3*t + (3 + x)^t^2/1! + (1 + 3*x)*t^3/1! + x*(7 + x)*t^4/2! + x*(5 + 3*x)*t^5/2! + x*(1 + x)*(11 + x)*t^6/3! + x*(1 + x)*(9 + 3*x)*t^7/3! + x*(1 + x)*(2 + x)*(15 + x)*t^8/4! + x*(1 + x)*(2 + x)*(13 + 3*x)*t^9/4! + ....

Row polynomials: R(2*n, x) = (4*n - 1 + x) * Product_{i = 0..n-2} (x + i) for n >= 1.

R(2*n+1, x) = (4*n - 3 + 3*x) * Product_{i = 0..n-2} (x + i) for n >= 1.

T(2*n, k) = |Stirling1(n, k)| + 3*n*|Stirling1(n-1, k)| = A132393(n, k) + 3*n*A132393(n-1, k).

T(2*n+1, k) = 3*|Stirling1(n, k)| + n*|Stirling1(n-1, k)| = 3*A132393(n, k) + n*A132393(n-1, k).

T(2*n, k) = (4*n - 1)*A132393(n-1, k) + A132393(n-1, k-1).

T(2*n+1, k) = (4*n - 3)*A132393(n-1, k) + 3*A132393(n-1, k-1).

n-th row sums equals 4*floor(n/2)! for n >= 2.

Example

Triangle begins
  n\k |  0     1     2     3     4      5
  - - - - - - - - - - - - - - - - - - - -
   0  |  1
   1  |  3
   2  |  3     1
   3  |  1     3
   4  |  0     7     1
   5  |  0     5     3
   6  |  0    11    12     1
   7  |  0     9    12     3
   8  |  0    30    47    18     1
   9  |  0    26    45    22     3
  10  |  0   114   215   125    25     1
  11  |  0   102   205   135    35     3
  ...

Maple

with(combinat):
T := proc (n, k); if irem(n, 2) = 0 then abs(Stirling1((1/2)*n, k)) + (3*n/2)*abs(Stirling1((n-2)/2, k)) else 3*abs(Stirling1((n-1)/2, k)) + ((n-1)/2)*abs(Stirling1((n-3)/2, k)) end if; end proc:
seq(print(seq(T(n, k), k = 0..floor(n/2))), n = 0..12);

Crossrefs

Cf. A130534, A132393, A371740.

Sequence in context: A021306 A214281 A125300 * A303992 A365213 A126717

Adjacent sequences: A371738 A371739 A371740 * A371742 A371743 A371744

Keyword

nonn,tabf,easy

Author

Peter Bala, Apr 05 2024

Status

approved