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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. 7
 1, 3, 1, 21, 10, 1, 231, 131, 21, 1, 3465, 2196, 446, 36, 1, 65835, 45189, 10670, 1130, 55, 1, 1514205, 1105182, 290599, 36660, 2395, 78, 1, 40883535, 31354119, 8951355, 1280419, 101325, 4501, 105, 1, 1267389585, 1012861224, 308846124, 48644344, 4421494, 240856, 7756, 136, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The Stirling-Frobenius cycle numbers are defined in A225470. 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 LINKS Peter Luschny, Generalized Eulerian polynomials. Peter Luschny, The Stirling-Frobenius numbers. FORMULA For a recurrence see the Sage program. T(n, 0) ~ A008545; T(n, n) ~ A000012; T(n, n-1) = A014105. Row sums ~ A047053; alternating row sums ~ A001813. From Wolfdieter Lang, May 29 2017: (Start) 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)}. E.g.f. of row polynomials in the variable x (i.e., of the triangle): (1 - 4*z)^{-(3+x)/4}. E.g.f. of column k: (1-4*x)^(-3/4)*(-(1/4)*log(1-4*x))^k/k!, k >= 0. 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. 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.) 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). 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) 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 EXAMPLE [n\k][    0,       1,      2,     3,    4,  5,  6 ]        1,        3,       1,       21,      10,      1,      231,     131,     21,     1,     3465,    2196,    446,    36,    1,    65835,   45189,  10670,  1130,   55,  1,  1514205, 1105182, 290599, 36660, 2395, 78,  1. ... From Wolfdieter Lang, Aug 11 2017: (Start) Recurrence: T(4, 2) = T(3, 1) + (4*4 - 1)*T(3, 2) = 131 +15*21 = 446. 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. (End) MATHEMATICA 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}]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 30 2018, after Wolfdieter Lang *) PROG (Sage) @CachedFunction def SF_C(n, k, m):     if k > n or k < 0 : return 0     if n == 0 and k == 0: return 1     return SF_C(n-1, k-1, m) + (m*n-1)*SF_C(n-1, k, m) for n in (0..8): [SF_C(n, k, 4) for k in (0..n)] CROSSREFS Cf. A132393 (m=1), A028338 (m=2), A225470 (m=3). Sequence in context: A107717 A143173 A000369 * A136236 A113090 A223549 Adjacent sequences:  A225468 A225469 A225470 * A225472 A225473 A225474 KEYWORD nonn,easy,tabl AUTHOR Peter Luschny, May 17 2013 STATUS approved

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Last modified July 2 02:41 EDT 2020. Contains 335389 sequences. (Running on oeis4.)