A225472 - OEIS

A225472

Triangle read by rows, k!*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.

1, 2, 3, 4, 21, 18, 8, 117, 270, 162, 16, 609, 2862, 4212, 1944, 32, 3093, 26550, 72090, 77760, 29160, 64, 15561, 230958, 1031940, 1953720, 1662120, 524880, 128, 77997, 1941030, 13429962, 39735360, 57561840, 40415760, 11022480, 256, 390369, 15996222, 165198852

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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

P. Bala, A 3 parameter family of generalized Stirling numbers.

Peter Luschny, Generalized Eulerian polynomials.

Peter Luschny, The Stirling-Frobenius numbers.

FORMULA

For a recurrence see the Maple program.

T(n, 0) ~ A000079; T(n, 1) ~ A005057; T(n, n) ~ A032031.

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

Cf. A131689 (m=1), A145901 (m=2), A225473 (m=4).

Cf. A225466, A225468, columns: A000079, 3*A016127, 3^2*2!*A016297, 3^3*3!*A025999.