A024392 -id:A024392 - cp's OEIS frontend

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.

1, 2, 1, 10, 7, 1, 80, 66, 15, 1, 880, 806, 231, 26, 1, 12320, 12164, 4040, 595, 40, 1, 209440, 219108, 80844, 14155, 1275, 57, 1, 4188800, 4591600, 1835988, 363944, 39655, 2415, 77, 1, 96342400, 109795600, 46819324, 10206700, 1276009, 95200, 4186, 100, 1

Offset: 0

Peter Luschny, May 16 2013

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.

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

			Triangle starts:
  [n\k][    0,      1,     2,     3,    4,  5,  6]
  [0]       1,
  [1]       2,      1,
  [2]      10,      7,     1,
  [3]      80,     66,    15,     1,
  [4]     880,    806,   231,    26,    1,
  [5]   12320,  12164,  4040,   595,   40,  1,
  [6]  209440, 219108, 80844, 14155, 1275, 57,  1.
  ...
From _Wolfdieter Lang_, Aug 11 2017: (Start)
Recurrence (see Maple program): T(4, 2) = T(3, 1) + (3*4 - 1)*T(3, 2) = 66 + 11*15 = 231.
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)

Peter Bala, A 3 parameter family of generalized Stirling numbers
Peter Luschny, Generalized Eulerian polynomials
Peter Luschny, The Stirling-Frobenius numbers

Cf. A225468; A132393 (m=1), A028338 (m=2), A225471(m=4).
Column k=0..4 give A008544, A024395(n-1), A286722(n-2), A383221, A383222.
T(n, n) ~ A000012; T(n, n-1) ~ A005449; T(n, n-2) ~ A024391; T(n, n-3) ~ A024392.
row sums ~ A032031; alternating row sums ~ A007559.
Cf. A132393.

SF_C := 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;
SF_C(n-1, k-1, m) + (m*n-1)*SF_C(n-1, k, m) end:
seq(print(seq(SF_C(n, k, 3), k = 0..n)), n = 0..8);

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 *)

For a recurrence see the Maple program.
From Wolfdieter Lang, May 18 2017: (Start)
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)}.
E.g.f. of row polynomials in the variable x (i.e., of the triangle): (1 - 3*z)^{-(2+x)/3}.
E.g.f. of column k: (1-3*x)^(-2/3)*(-(1/3)*log(1-3*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+2)*R(n-1,x+3), with R(0, x) = 1.
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.)
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)
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

A024395 a(n) = n-th elementary symmetric function of the first n+1 positive integers congruent to 2 mod 3.

Original entry on oeis.org

1, 7, 66, 806, 12164, 219108, 4591600, 109795600, 2951028000, 88084714400, 2891353030400, 103521905491200, 4015191638617600, 167714507921497600, 7506196028811110400, 358368551285791692800, 18180562447078051328000

Offset: 0

Clark Kimberling

Comment by R. J. Mathar, Oct 01 2016 (Start):
The k-th elementary symmetric functions of the integers 2+j*3, j=0..n-1, form a triangle T(n,k), 0<=k<=n, n>=0:
1
1 2
1 7 10
1 15 66 80
1 26 231 806 880
1 40 595 4040 12164 12320
1 57 1275 14155 80844 219108 209440
1 77 2415 39655 363944 1835988 4591600 4188800
1 100 4186 95200 1276009 10206700 46819324 109795600 96342400
This here is the first subdiagonal. The diagonal seems to be A008544. The first columns are A000012, A005449, A024391, A024392. (End)

			From _Gheorghe Coserea_, Dec 24 2015: (Start)
For n=1 we have a(1) = 2*5*(1/2 + 1/5) = 7.
For n=2 we have a(2) = 2*5*8*(1/2 + 1/5 + 1/8) = 66.
For n=3 we have a(3) = 2*5*8*11*(1/2 + 1/5 + 1/8 + 1/11) = 806.
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A024216, A225470 (second column).

Table[ (-1)^(n+1)*Sum[(-3)^(n - k) k (-1)^(n - k) StirlingS1[n+1, k + 1], {k, 0, n}], {n, 1, 30}]
Join[{1},Table[Module[{c=NestList[3+#&,2,n+1]},Times@@c*Total[1/c]],{n,0,20}]] (* Harvey P. Dale, Jul 09 2019 *)

n = 16; a = vector(n); a[1] = 7; a[2] = 66;
for (k=2, n-1, a[k+1] = (6*k+7) * a[k] - (3*k+2)^2 * a[k-1]);
print(concat(1,a))  \\ Gheorghe Coserea, Aug 30 2015

E.g.f. (for offset 1): -(1/3)*log(1-3*x)/(1-3*x)^(2/3). - Vladeta Jovovic, Sep 26 2003
For n >= 1, a(n-1) = 3^(n-1)*n!*sum(binomial(k-1/3,k)/(n-k), k = 0..n-1). - Milan Janjic, Dec 14 2008, corrected by Peter Bala, Oct 08 2013
a(n) ~ (n+1)! * 3^n * (log(n) + gamma - Pi*sqrt(3)/6 + 3*log(3)/2) / (n^(1/3)*GAMMA(2/3)), where "GAMMA" is the Gamma function and "gamma" is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Oct 07 2013
a(n+1) = (6*n+7) * a(n) - (3*n+2)^2 * a(n-1). - Gheorghe Coserea, Aug 30 2015
a(n) = A225470(n+1, 1), n >= 0. - Wolfdieter Lang, May 29 2017

Formula (see Mathematica line), correction and more terms from Victor Adamchik (adamchik(AT)cs.cmu.edu), Jul 21 2001

A024403 [ (4th elementary symmetric function of S(n))/(3rd elementary symmetric function of S(n)) ], where S(n) = {first n+3 positive integers congruent to 2 mod 3}.

Original entry on oeis.org

1, 3, 5, 9, 13, 18, 24, 30, 37, 45, 54, 64, 74, 85, 97, 109, 122, 136, 151, 167, 183, 200, 218, 236, 255, 275, 296, 318, 340, 363, 387, 411, 436, 462, 489, 517, 545, 574, 604, 634, 665, 697, 730, 764, 798, 833, 869, 905, 942, 980, 1019, 1059, 1099, 1140, 1182, 1224

Offset: 1

Clark Kimberling

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A024392, A024393.

[(n*(405*n^4+4590*n^3+18495*n^2+30534*n+16376))  div (40*(3*n+10)*(9*(n+1)^2+33*n+55)): n in [1..60]]; // Vincenzo Librandi, Jul 07 2019

Table[Floor[(n (405 n^4 + 4590 n^3 + 18495 n^2 + 30534 n + 16376)) / (40 (3 n + 10) (9 (n+1)^2 + 33 n + 55))], {n, 1, 100}] (* Vincenzo Librandi, Jul 07 2019 *)

Empirical g.f.: x*(x^2-x+1)*(x^8-x^7-x^5+x^4-x^3-x-1) / ((x-1)^3*(x^2+1)*(x^4+1)). - Colin Barker, Aug 16 2014

a(n) = floor(A024393(n) / A024392(n + 1)). - Sean A. Irvine, Jul 07 2019

A024399 a(n) = [ (3rd elementary symmetric function of S(n))/(first elementary symmetric function of S(n)) ], where S(n) = {first n+2 positive integers congruent to 2 mod 3}.

Original entry on oeis.org

5, 31, 101, 248, 515, 952, 1619, 2586, 3930, 5738, 8107, 11141, 14954, 19670, 25420, 32345, 40596, 50331, 61718, 74935, 90167, 107609, 127466, 149950, 175283, 203697, 235431, 270734, 309865, 353090, 400685, 452936, 510136, 572588, 640605, 714507

Offset: 1

Clark Kimberling

nonn

Cf. A024392, A005449.

Conjecture: a(n)= +4*a(n-1) -6*a(n-2) +5*a(n-3) -5*a(n-4) +6*a(n-5) -4*a(n-6) +a(n-7). G.f. x*(-5-11*x-7*x^2-5*x^3+x^4) / ( (1+x+x^2)*(x-1)^5 ). - R. J. Mathar, Oct 08 2011

a(n) = floor(A024392(n) / A005449(n + 2)). - Sean A. Irvine, Jul 06 2019

cp's OEIS Frontend

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.

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A024395 a(n) = n-th elementary symmetric function of the first n+1 positive integers congruent to 2 mod 3.

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A024403 [ (4th elementary symmetric function of S(n))/(3rd elementary symmetric function of S(n)) ], where S(n) = {first n+3 positive integers congruent to 2 mod 3}.

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A024399 a(n) = [ (3rd elementary symmetric function of S(n))/(first elementary symmetric function of S(n)) ], where S(n) = {first n+2 positive integers congruent to 2 mod 3}.

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A024401 a(n) = [ (3rd elementary symmetric function of S(n))/(2nd elementary symmetric function of S(n)) ], where S(n) = {first n+2 positive integers congruent to 2 mod 3}.

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