A225472 - OEIS

%I #21 Jul 12 2017 05:02:54

%S 1,2,3,4,21,18,8,117,270,162,16,609,2862,4212,1944,32,3093,26550,

%T 72090,77760,29160,64,15561,230958,1031940,1953720,1662120,524880,128,

%U 77997,1941030,13429962,39735360,57561840,40415760,11022480,256,390369,15996222,165198852

%N 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.

%C The Stirling-Frobenius subset numbers are defined in A225468 (see also the Sage program).

%H Vincenzo Librandi, <a href="/A225472/b225472.txt">Rows n = 0..50, flattened</a>

%H P. Bala, <a href="/A143395/a143395.pdf">A 3 parameter family of generalized Stirling numbers</a>.

%H Peter Luschny, <a href="http://www.luschny.de/math/euler/GeneralizedEulerianPolynomials.html">Generalized Eulerian polynomials.</a>

%H Peter Luschny, <a href="http://www.luschny.de/math/euler/StirlingFrobeniusNumbers.html">The Stirling-Frobenius numbers.</a>

%F For a recurrence see the Maple program.

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

%F From _Wolfdieter Lang_, Apr 10 2017: (Start)

%F 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!.

%F 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.

%F T(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(k, j)*(2+3*j)^n, 0 <= k <= n.

%F 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.

%F 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.

%F (End)

%F 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

%e [n\k][0,     1,      2,       3,       4,       5,      6 ]

%e [0]   1,

%e [1]   2,     3,

%e [2]   4,    21,     18,

%e [3]   8,   117,    270,     162,

%e [4]  16,   609,   2862,    4212,    1944,

%e [5]  32,  3093,  26550,   72090,   77760,   29160,

%e [6]  64, 15561, 230958, 1031940, 1953720, 1662120, 524880.

%p SF_SO := proc(n, k, m) option remember;

%p if n = 0 and k = 0 then return(1) fi;

%p if k > n or k < 0 then return(0) fi;

%p m*k*SF_SO(n-1, k-1, m) + (m*(k+1)-1)*SF_SO(n-1, k, m) end:

%p seq(print(seq(SF_SO(n, k, 3), k=0..n)), n = 0..5);

%t 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 *)

%o (Sage)

%o @CachedFunction

%o def EulerianNumber(n, k, m) :

%o     if n == 0: return 1 if k == 0 else 0

%o     return (m*(n-k)+m-1)*EulerianNumber(n-1, k-1, m)+ (m*k+1)*EulerianNumber(n-1, k, m)

%o def SF_SO(n, k, m):

%o     return add(EulerianNumber(n, j, m)*binomial(j, n - k) for j in (0..n))

%o for n in (0..6): [SF_SO(n, k, 3) for k in (0..n)]

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

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

%K nonn,easy,tabl

%O 0,2

%A _Peter Luschny_, May 17 2013