A202189 Triangle read by rows: T(n,k) = (n!/k!) * [x^n] x^k*(1+x+x^2)^(k*x).

1, 0, 1, 6, 0, 1, 12, 24, 0, 1, -20, 60, 60, 0, 1, 540, 240, 180, 120, 0, 1, -882, 6300, 2100, 420, 210, 0, 1, -6720, -8736, 35280, 8960, 840, 336, 0, 1, 189936, 181440, 13608, 136080, 27720, 1512, 504, 0, 1

Offset

1,4

Comments

Also the Bell transform of (n+1)!*Sum_{k=0..n} Sum_{i=k..n-k} (-1)^(i-k)*S1(i,k)*binomial(i,n-k-i)/i! where S1 are the Stirling cycle numbers A132393. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 15 2016

Links

Table of n, a(n) for n=1..45.

Formula

T(n,m) = (n!/m!)*Sum_{k=0..n-m} (Sum_{i=k..n-m-k} Stirling1(i,k)*binomial(i,n-m-k-i)/i!)*m^k.

E.g.f. for column k: (1/k!)*x^k*(1+x+x^2)^(k*x). - Jason Yuen, Feb 17 2025

Example

Triangle begins:
  1
  0, 1,
  6, 0, 1,
  12, 24, 0, 1,
  -20, 60, 60, 0, 1,
  540, 240, 180, 120, 0, 1,
  -882, 6300, 2100, 420, 210, 0, 1

Mathematica

Table[n!/m! Sum[Sum[(StirlingS1[i, k] Binomial[i, n - m - k - i])/i!, {i, k, n - m - k}] m^k, {k, 0, n - m}], {n, 9}, {m, n}] // Flatten (* Michael De Vlieger, Jan 15 2016 *)

Prog

(Maxima)
T(n, m):=n!/m!*sum((sum((stirling1(i, k)*binomial(i, n-m-k-i))/i!, i, k, n-m-k))*m^k, k, 0, n-m);
(SageMath) # uses[bell_transform from A264428]
# Adds a column 1, 0, 0, 0, ... at the left side of the triangle.
def A202189_row(n):
    f = lambda n: factorial(n+1)*sum(sum((-1)^(i-k)*stirling_number1(i, k)* binomial(i, n-k-i)/factorial(i) for i in (k..n-k)) for k in (0..n))
    return bell_transform(n, [f(k) for k in (0..n)])
[A202189_row(n) for n in (0..9)] # Peter Luschny, Jan 15 2016

Crossrefs

Sequence in context: A215080 A317446 A137943 * A202183 A227612 A221273

Adjacent sequences: A202186 A202187 A202188 * A202190 A202191 A202192

Keyword

sign,tabl

Author

Vladimir Kruchinin, Dec 13 2011

Status

approved