OFFSET
0,2
COMMENTS
The Stirling-Frobenius subset numbers are defined in A225468 (see also the Sage program).
LINKS
Vincenzo Librandi, Rows n = 0..50, flattened
Peter Luschny, Generalized Eulerian polynomials.
Peter Luschny, The Stirling-Frobenius numbers.
FORMULA
For a recurrence see the Maple program.
From Wolfdieter Lang, Apr 10 2017: (Start)
E.g.f. for sequence of column k: exp(2*x)*(exp(3*x) - 1)^k, k >= 0. From the Sheffer triangle S2[3,2] = A225466 with column k multiplied with k!.
O.g.f. for sequence of column k is 3^k*k!*x^k/Product_{j=0..k} (1 - (2+3*j)*x), k >= 0.
T(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(k, j)*(2+3*j)^n, 0 <= k <= n.
Three term recurrence (see the Maple program): T(n, k) = 0 if n < k , T(n, -1) = 0, T(0,0) = 1, T(n, k) = 3*k*T(n-1, k-1) + (2 + 3*k)*T(n-1, k) for n >= 1, k=0..n.
For the column scaled triangle (with diagonal 1s) see A225468, and the Bala link with (a,b,c) = (3,0,2), where Sheffer triangles are called exponential Riordan triangles.
(End)
The e.g.f. of the row polynomials R(n, x) = Sum_{k=0..n} T(n, k)*x^k is exp(2*z)/(1 - x*(exp(3*z) - 1)). - Wolfdieter Lang, Jul 12 2017
EXAMPLE
[n\k][0, 1, 2, 3, 4, 5, 6 ]
[0] 1,
[1] 2, 3,
[2] 4, 21, 18,
[3] 8, 117, 270, 162,
[4] 16, 609, 2862, 4212, 1944,
[5] 32, 3093, 26550, 72090, 77760, 29160,
[6] 64, 15561, 230958, 1031940, 1953720, 1662120, 524880.
MAPLE
SF_SO := proc(n, k, m) option remember;
if n = 0 and k = 0 then return(1) fi;
if k > n or k < 0 then return(0) fi;
m*k*SF_SO(n-1, k-1, m) + (m*(k+1)-1)*SF_SO(n-1, k, m) end:
seq(print(seq(SF_SO(n, k, 3), k=0..n)), n = 0..5);
MATHEMATICA
EulerianNumber[n_, k_, m_] := EulerianNumber[n, k, m] = (If[ n == 0, Return[If[k == 0, 1, 0]]]; Return[(m*(n-k)+m-1)*EulerianNumber[n-1, k-1, m] + (m*k+1)*EulerianNumber[n-1, k, m]]); SFSO[n_, k_, m_] := Sum[ EulerianNumber[n, j, m]*Binomial[j, n-k], {j, 0, n}]; Table[ SFSO[n, k, 3], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 29 2013, translated from Sage *)
PROG
(Sage)
@CachedFunction
def EulerianNumber(n, k, m) :
if n == 0: return 1 if k == 0 else 0
return (m*(n-k)+m-1)*EulerianNumber(n-1, k-1, m)+ (m*k+1)*EulerianNumber(n-1, k, m)
def SF_SO(n, k, m):
return add(EulerianNumber(n, j, m)*binomial(j, n - k) for j in (0..n))
for n in (0..6): [SF_SO(n, k, 3) for k in (0..n)]
CROSSREFS
KEYWORD
AUTHOR
Peter Luschny, May 17 2013
STATUS
approved