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A225476 Triangle read by rows, k!*2^k*S_2(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.

%I A225476 #22 Aug 07 2015 03:48:59
%S A225476 1,1,1,1,4,2,1,13,18,6,1,40,116,96,24,1,121,660,1020,600,120,1,364,
%T A225476 3542,9120,9480,4320,720,1,1093,18438,74466,121800,94920,35280,5040,1,
%U A225476 3280,94376,576576,1394064,1653120,1028160,322560,40320,1,9841,478440,4319160
%N A225476 Triangle read by rows, k!*2^k*S_2(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.
%C A225476 The Stirling-Frobenius subset numbers are defined in A225468 (see also the Sage program).
%H A225476 Vincenzo Librandi, <a href="/A225476/b225476.txt">Rows n = 0..50, flattened</a>
%H A225476 Peter Luschny, <a href="http://www.luschny.de/math/euler/GeneralizedEulerianPolynomials.html">Generalized Eulerian polynomials.</a>
%H A225476 Peter Luschny, <a href="http://www.luschny.de/math/euler/StirlingFrobeniusNumbers.html">The Stirling-Frobenius numbers.</a>
%H A225476 Shi-Mei Ma, Toufik Mansour, Matthias Schork, <a href="http://arxiv.org/abs/1308.0169">Normal ordering problem and the extensions of the Stirling grammar</a>, Russian Journal of Mathematical Physics, 2014, 21(2), arXiv 1308.0169 p. 12.
%F A225476 T(n, k) = sum_{j=0..n} A_2(n, j)*binomial(j, n-k), where A_2(n, j) are the generalized Eulerian numbers of order m=2.
%F A225476 For a recurrence see the Maple program.
%e A225476 [n\k][0,   1,   2,    3,   4,   5 ]
%e A225476 [0]   1,
%e A225476 [1]   1,   1,
%e A225476 [2]   1,   4,   2,
%e A225476 [3]   1,  13,  18,    6,
%e A225476 [4]   1,  40, 116,   96,  24,
%e A225476 [5]   1, 121, 660, 1020, 600, 120.
%p A225476 SF_SSO := proc(n, k, m) option remember;
%p A225476 if n = 0 and k = 0 then return(1) fi;
%p A225476 if k > n or k < 0 then return(0) fi;
%p A225476 k*SF_SSO(n-1, k-1, m) + (m*(k+1)-1)*SF_SSO(n-1, k, m) end:
%p A225476 seq(print(seq(SF_SSO(n, k, 2), k=0..n)), n = 0..5);
%t A225476 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]]); SFSSO[n_, k_, m_] := Sum[ EulerianNumber[n, j, m]*Binomial[j, n - k], {j, 0, n}]/m^k; Table[ SFSSO[n, k, 2], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 29 2013, translated from Sage *)
%o A225476 (Sage)
%o A225476 @CachedFunction
%o A225476 def EulerianNumber(n, k, m) :
%o A225476     if n == 0: return 1 if k == 0 else 0
%o A225476     return (m*(n-k)+m-1)*EulerianNumber(n-1, k-1, m)+(m*k+1)*EulerianNumber(n-1, k, m)
%o A225476 def SF_SSO(n, k, m):
%o A225476     return add(EulerianNumber(n, j, m)*binomial(j, n - k) for j in (0..n))/m^k
%o A225476 for n in (0..6): [SF_SSO(n, k, 2) for k in (0..n)]
%Y A225476 T(n, 0) ~ A000012; T(n, 1) ~ A003462; T(n, 2) ~ A007798.
%Y A225476 T(n, n) ~ A000142; T(n, n-1) ~ A001563.
%Y A225476 Alternating row sum ~ A000364 (Euler secant numbers).
%Y A225476 Cf. A225468, A131689 (m=1).
%K A225476 nonn,tabl
%O A225476 0,5
%A A225476 _Peter Luschny_, May 19 2013