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A225471 Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

%I A225471 #37 May 06 2025 10:28:43
%S A225471 1,3,1,21,10,1,231,131,21,1,3465,2196,446,36,1,65835,45189,10670,1130,
%T A225471 55,1,1514205,1105182,290599,36660,2395,78,1,40883535,31354119,
%U A225471 8951355,1280419,101325,4501,105,1,1267389585,1012861224,308846124,48644344,4421494,240856,7756,136,1
%N A225471 Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.
%C A225471 The Stirling-Frobenius cycle numbers are defined in A225470.
%C A225471 Triangle T(n,k), read by rows, given by (3, 4, 7, 8, 11, 12, 15, 16, 19, 20, ... (A014601)) 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 A225471 P. Bala, <a href="/A143395/a143395.pdf">A 3 parameter family of generalized Stirling numbers</a>.
%H A225471 Peter Luschny, <a href="http://www.luschny.de/math/euler/GeneralizedEulerianPolynomials.html">Generalized Eulerian polynomials.</a>
%H A225471 Peter Luschny, <a href="http://www.luschny.de/math/euler/StirlingFrobeniusNumbers.html">The Stirling-Frobenius numbers.</a>
%F A225471 For a recurrence see the Sage program.
%F A225471 T(n, 0) ~ A008545; T(n, n) ~ A000012; T(n, n-1) = A014105.
%F A225471 Row sums ~ A047053; alternating row sums ~ A001813.
%F A225471 From _Wolfdieter Lang_, May 29 2017: (Start)
%F A225471 This is the Sheffer triangle (1/(1 - 4*x)^{-3/4}, -(1/4)*log(1-4*x)). See the P. Bala link where this is called exponential Riordan array, and the signed version is denoted by s_{(4,0,3)}.
%F A225471 E.g.f. of row polynomials in the variable x (i.e., of the triangle): (1 - 4*z)^{-(3+x)/4}.
%F A225471 E.g.f. of column k: (1-4*x)^(-3/4)*(-(1/4)*log(1-4*x))^k/k!, k >= 0.
%F A225471 Recurrence for row polynomials R(n, x) = Sum_{k=0..n} T(n, k)*x^k: R(n, x) = (x+3)*R(n-1,x+4), with R(0, x) = 1.
%F A225471 R(n, x) = risefac(4,3;x,n) := Product_{j=0..(n-1)} (x + (3 + 4*j)). (See the P. Bala link, eq. (16) for the signed s_{4,0,3} row polynomials.)
%F A225471 T(n, k) = Sum_{j=0..(n-m)} binomial(n-j, k)* S1p(n, n-j)*3^(n-k-j)*4^j, with S1p(n, m) = A132393(n, m).
%F A225471 T(n, k) = sigma[4,3]^{(n)}_{n-k}, with the elementary symmetric functions sigma[4,3]^{(n)}_m of degree m in the n numbers 3, 7, 11, ..., 3+4*(n-1), with sigma[4,3]^{(n)}_0 := 1. (End)
%F A225471 Boas-Buck type recurrence for column sequence k: T(n, k) = (n!/(n - k)) * Sum_{p=k..n-1} 4^(n-1-p)*(3 + 8*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 with {1/2, 5/12, 3/8, 251/720, ...}. See a comment and references in A286718. - _Wolfdieter Lang_, Aug 11 2017
%e A225471 [n\k][    0,       1,      2,     3,    4,  5,  6 ]
%e A225471 [0]       1,
%e A225471 [1]       3,       1,
%e A225471 [2]      21,      10,      1,
%e A225471 [3]     231,     131,     21,     1,
%e A225471 [4]    3465,    2196,    446,    36,    1,
%e A225471 [5]   65835,   45189,  10670,  1130,   55,  1,
%e A225471 [6] 1514205, 1105182, 290599, 36660, 2395, 78,  1.
%e A225471 ...
%e A225471 From _Wolfdieter Lang_, Aug 11 2017: (Start)
%e A225471 Recurrence: T(4, 2) = T(3, 1) + (4*4 - 1)*T(3, 2) = 131 +15*21 = 446.
%e A225471 Boas-Buck recurrence for column k=2 and n=4: T(4, 2) = (4!/2)*(4*(3+8*(5/12)) *T(2, 2)/2! + 1*(3 + 8*(1/2))*T(3,2)/3!) = (4!/2)*(4*(19/3)/2  + 7*21/3!) =  446.
%e A225471 (End)
%t A225471 T[0, 0] = 1; T[n_, k_] := Sum[Binomial[n - j, k]*Abs[StirlingS1[n, n - j]]* 3^(n - k - j)*4^j, {j, 0, n - k}];
%t A225471 Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 30 2018, after _Wolfdieter Lang_ *)
%o A225471 (Sage)
%o A225471 @CachedFunction
%o A225471 def SF_C(n, k, m):
%o A225471     if k > n or k < 0 : return 0
%o A225471     if n == 0 and k == 0: return 1
%o A225471     return SF_C(n-1, k-1, m) + (m*n-1)*SF_C(n-1, k, m)
%o A225471 for n in (0..8): [SF_C(n, k, 4) for k in (0..n)]
%Y A225471 Columns k=0..3 give A008545, A286723(n-1), A383702, A383703.
%Y A225471 Cf. A132393 (m=1), A028338 (m=2), A225470 (m=3).
%K A225471 nonn,easy,tabl
%O A225471 0,2
%A A225471 _Peter Luschny_, May 17 2013