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A225470 Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

%I A225470 #41 Apr 21 2025 03:46:19
%S A225470 1,2,1,10,7,1,80,66,15,1,880,806,231,26,1,12320,12164,4040,595,40,1,
%T A225470 209440,219108,80844,14155,1275,57,1,4188800,4591600,1835988,363944,
%U A225470 39655,2415,77,1,96342400,109795600,46819324,10206700,1276009,95200,4186,100,1
%N A225470 Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.
%C A225470 The Stirling-Frobenius subset numbers S_{m}(n,k), for m >= 1 fixed, regarded as an infinite lower triangular matrix, can be inverted by Sum_{k} S_{m}(n,k)*s_{m}(k,j)*(-1)^(n-k) = [j = n]. The inverse numbers s_{m}(k,j), which are unsigned, are the Stirling-Frobenius cycle numbers. For m = 1 this gives the classical Stirling cycle numbers A132393. The Stirling-Frobenius subset numbers are defined in A225468.
%C A225470 Triangle T(n,k), read by rows, given by (2, 3, 5, 6, 8, 9, 11, 12, 14, 15, ... (A007494)) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, May 14 2015
%H A225470 Peter Bala, <a href="/A143395/a143395.pdf">A 3 parameter family of generalized Stirling numbers</a>
%H A225470 Peter Luschny, <a href="http://www.luschny.de/math/euler/GeneralizedEulerianPolynomials.html">Generalized Eulerian polynomials</a>
%H A225470 Peter Luschny, <a href="http://www.luschny.de/math/euler/StirlingFrobeniusNumbers.html">The Stirling-Frobenius numbers</a>
%F A225470 For a recurrence see the Maple program.
%F A225470 From _Wolfdieter Lang_, May 18 2017: (Start)
%F A225470 This is the Sheffer triangle (1/(1 - 3*x)^{-2/3}, -(1/3)*log(1-3*x)). See the P. Bala link where this is called exponential Riordan array, and the signed version is  denoted by s_{(3,0,2)}.
%F A225470 E.g.f. of row polynomials in the variable x (i.e., of the triangle): (1 - 3*z)^{-(2+x)/3}.
%F A225470 E.g.f. of column k: (1-3*x)^(-2/3)*(-(1/3)*log(1-3*x))^k/k!, k >= 0.
%F A225470 Recurrence for row polynomials R(n, x) = Sum_{k=0..n} T(n, k)*x^k: R(n, x) = (x+2)*R(n-1,x+3), with R(0, x) = 1.
%F A225470 R(n, x) = risefac(3,2;x,n) := Product_{j=0..(n-1)} (x + (2 + 3*j)). (See the P. Bala link, eq. (16) for the signed s_{3,0,2} row polynomials.)
%F A225470 T(n, k) = Sum_{j=0..(n-m)} binomial(n-j, k)* S1p(n, n-j)*2^(n-k-j)*3^j, with S1p(n, m) = A132393(n, m). (End)
%F A225470 Boas-Buck type recurrence for column sequence k: T(n, k) = (n!/(n - k)) * Sum_{p=k..n-1} 3^(n-1-p)*(2 + 3*k*beta(n-1-p))*T(p, k)/p!, for n > k >= 0, with input T(k, k) = 1, and beta(k) = A002208(k+1)/A002209(k+1), beginning {1/2, 5/12, 3/8, 251/720, ...}. See a comment and references in A286718. - _Wolfdieter Lang_, Aug 11 2017
%e A225470 Triangle starts:
%e A225470   [n\k][    0,      1,     2,     3,    4,  5,  6]
%e A225470   [0]       1,
%e A225470   [1]       2,      1,
%e A225470   [2]      10,      7,     1,
%e A225470   [3]      80,     66,    15,     1,
%e A225470   [4]     880,    806,   231,    26,    1,
%e A225470   [5]   12320,  12164,  4040,   595,   40,  1,
%e A225470   [6]  209440, 219108, 80844, 14155, 1275, 57,  1.
%e A225470   ...
%e A225470 From _Wolfdieter Lang_, Aug 11 2017: (Start)
%e A225470 Recurrence (see Maple program): T(4, 2) = T(3, 1) + (3*4 - 1)*T(3, 2) = 66 + 11*15 = 231.
%e A225470 Boas-Buck type recurrence for column k = 2 and n = 4: T(4, 2) = (4!/2)*(3*(2 + 6*(5/12))*T(2, 2)/2! + 1*(2 + 6*(1/2))*T(3,2)/3!) = (4!/2)*(3*9/4 + 5*15/3!) = 231. (End)
%p A225470 SF_C := proc(n, k, m) option remember;
%p A225470 if n = 0 and k = 0 then return(1) fi;
%p A225470 if k > n or  k < 0 then return(0) fi;
%p A225470 SF_C(n-1, k-1, m) + (m*n-1)*SF_C(n-1, k, m) end:
%p A225470 seq(print(seq(SF_C(n, k, 3), k = 0..n)), n = 0..8);
%t A225470 SFC[0, 0, _] = 1; SFC[n_, k_, _] /; (k > n || k < 0) = 0; SFC[n_, k_, m_] := SFC[n, k, m] = SFC[n-1, k-1, m] + (m*n-1)*SFC[n-1, k, m]; Table[SFC[n, k, 3], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 26 2013, after Maple *)
%Y A225470 Cf. A225468; A132393 (m=1), A028338 (m=2), A225471(m=4).
%Y A225470 Column k=0..4 give A008544, A024395(n-1), A286722(n-2), A383221, A383222.
%Y A225470 T(n, n) ~ A000012; T(n, n-1) ~ A005449; T(n, n-2) ~ A024391; T(n, n-3) ~ A024392.
%Y A225470 row sums ~ A032031; alternating row sums ~ A007559.
%Y A225470 Cf. A132393.
%K A225470 nonn,easy,tabl
%O A225470 0,2
%A A225470 _Peter Luschny_, May 16 2013