A254881 Triangle read by rows, T(n,k) = sum(j=0..k-1, S(n+1,j+1)*S(n,k-j)) where S denotes the Stirling cycle numbers A132393, T(0,0)=1, n>=0, 0<=k<=2n.

1, 0, 1, 1, 0, 2, 5, 4, 1, 0, 12, 40, 51, 31, 9, 1, 0, 144, 564, 904, 769, 376, 106, 16, 1, 0, 2880, 12576, 23300, 24080, 15345, 6273, 1650, 270, 25, 1, 0, 86400, 408960, 840216, 991276, 748530, 381065, 133848, 32523, 5370, 575, 36, 1, 0, 3628800, 18299520

Offset

0,6

Comments

These are also the coefficients of the polynomials interpolating the sequence k -> n!*((n+k)!/k!)*binomial(n+k-1,k-1) (for fixed n>=0). Divided by n! these polynomials generate the rows of Lah numbers L(n+k, k) = ((n+k)!/k!)* binomial(n+k-1,k-1).

Links

Table of n, a(n) for n=0..51.

Formula

T(n, n) = A187235(n) for n>=1 (after the explicit formula of Vaclav Kotesovec).

Example

[1]
[0, 1, 1]
[0, 2, 5, 4, 1]
[0, 12, 40, 51, 31, 9, 1]
[0, 144, 564, 904, 769, 376, 106, 16, 1]
[0, 2880, 12576, 23300, 24080, 15345, 6273, 1650, 270, 25, 1]
For example in the case n=3 the polynomial (k^6+9*k^5+31*k^4+51*k^3+40*k^2+12*k)/3! generates the Lah numbers 0, 24, 240, 1200, 4200, 11760, 28224, ... (A253285).

Maple

# This is a special case of the recurrence given in A246117.
t := proc(n, k) option remember; if n=0 and k=0 then 1 elif
k <= 0 or k>n then 0 else iquo(n, 2)*t(n-1, k)+t(n-1, k-1) fi end:
A254881 := (n, k) -> t(2*n, k):
seq(print(seq(A254881(n, k), k=0..2*n)), n=0..5);
# Illustrating the comment:
restart: with(PolynomialTools): with(CurveFitting): for N from 0 to 5 do
CoefficientList(PolynomialInterpolation([seq([k, N!*((N+k)!/k!)*binomial(N+k-1, k-1)], k=0..2*N)], n), n) od;

Mathematica

Flatten[{1, Table[Table[Sum[Abs[StirlingS1[n+1, j+1]] * Abs[StirlingS1[n, k-j]], {j, 0, k-1}], {k, 0, 2*n}], {n, 1, 10}]}] (* Vaclav Kotesovec, Feb 10 2015 *)

Prog

(Sage)
def T(n, k):
    if n == 0: return 1
    return sum(stirling_number1(n+1, j+1)*stirling_number1(n, k-j) for j in range(k))
for n in range (6): [T(n, k) for k in (0..2*n)]

Crossrefs

Cf. A246117, A254882, A187235.

The sequences A000012, A002378, A083374, A253285 are the Lah number rows generated by the polynomials divided by n! for n=0, 1, 2, 3 respectivly.

Sequence in context: A106315 A285295 A217563 * A100946 A200019 A282824

Adjacent sequences: A254878 A254879 A254880 * A254882 A254883 A254884

Keyword

nonn,tabf

Author

Peter Luschny, Feb 10 2015

Status

approved