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 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. 5
 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 (list; table; graph; refs; listen; history; text; internal format)
 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. 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. Sequence in context: A246391 A303973 A225466 * A176234 A058780 A183458 Adjacent sequences:  A225469 A225470 A225471 * A225473 A225474 A225475 KEYWORD nonn,easy,tabl AUTHOR Peter Luschny, May 17 2013 STATUS approved

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Last modified June 17 16:29 EDT 2019. Contains 324195 sequences. (Running on oeis4.)