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A202189 Triangle T(n,k) = coefficient of x^n in expansion of x^k*(1+x+x^2)^(k*x) = sum(n>=k, T(n,k) x^n*k!/n!). 0
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 (list; table; graph; refs; listen; history; text; internal format)
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).

EXAMPLE

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);

(Sage)

# The function bell_transform is defined in 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

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Last modified February 18 05:30 EST 2019. Contains 320245 sequences. (Running on oeis4.)