A016138 -id:A016138 - cp's OEIS frontend

A225469 Triangle read by rows, S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.

1, 3, 1, 9, 10, 1, 27, 79, 21, 1, 81, 580, 310, 36, 1, 243, 4141, 3990, 850, 55, 1, 729, 29230, 48031, 16740, 1895, 78, 1, 2187, 205339, 557571, 299131, 52745, 3689, 105, 1, 6561, 1439560, 6338620, 5044536, 1301286, 137592, 6524, 136, 1

Offset: 0

Peter Luschny, May 16 2013

The definition of the Stirling-Frobenius subset numbers: S_m(n, k) = (sum_{j=0..n} binomial(j, n-k)*A_m(n, j)) / (m^k*k!) where A_m(n, j) are the generalized Eulerian numbers (see the links for details).

This is the Sheffer triangle (exp(3*x),(1/4)*(exp(4*x -1))). See the P. Bala link where this is called exponential Riordan array S_{(4,0,3)}. - Wolfdieter Lang, Apr 13 2017

			[n\k][ 0,     1,     2,     3,    4,   5,  6]
[0]    1,
[1]    3,     1,
[2]    9,    10,     1,
[3]   27,    79,    21,     1,
[4]   81,   580,   310,    36,    1,
[5]  243,  4141,  3990,   850,   55,  1,
[6]  729, 29230, 48031, 16740, 1895, 78,  1.

Vincenzo Librandi, Rows n = 0..50, flattened
P. Bala, A 3 parameter family of generalized Stirling numbers.
Paweł Hitczenko, A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality, arXiv:2403.03422 [math.CO], 2024. See pp. 8-9.
Peter Luschny, Generalized Eulerian polynomials.
Peter Luschny, The Stirling-Frobenius numbers.
Shi-Mei Ma, Toufik Mansour, and Matthias Schork, Normal ordering problem and the extensions of the Stirling grammar, Russian Journal of Mathematical Physics, 2014, 21(2), arXiv 1308.0169 p. 12.

Cf. A048993 (m=1), A039755 (m=2), A225468 (m=3).

Cf. Columns: A000244, A016138, A018054.

SF_S := 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_S(n-1, k-1, m) + (m*(k+1)-1)*SF_S(n-1, k, m) end:
seq(print(seq(SF_S(n, k, 4), k=0..n)), n = 0..5);

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]]); SFS[n_, k_, m_] := Sum[ EulerianNumber[n, j, m]*Binomial[j, n-k], {j, 0, n}]/(k!*m^k); Table[ SFS[n, k, 4], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 29 2013, translated from 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_S(n, k, m):
    return add(EulerianNumber(n, j, m)*binomial(j, n - k) for j in (0..n))/(factorial(k)*m^k)
for n in (0..6): [SF_S(n, k, 4) for k in (0..n)]

T(n, k) = (sum_{j=0..n} binomial(j, n-k)*A_4(n, j)) / (4^k*k!) where A_4(n,j) = A225118.
For a recurrence see the Maple program.
T(n, 0) ~ A000244; T(n, 1) ~ A016138; T(n, 2) ~ A018054.
T(n, n) ~ A000012; T(n, n-1) ~ A014105.
From Wolfdieter Lang, Apr 13 2017: (Start)
E.g.f.: exp(3*z)*exp((x/4)*(exp(4*z -1))). Sheffer triangle (see a comment above).
E.g.f. column k: exp(3*x)*(exp(4*x) -1)^k/(4^k*k!), k >= 0 (Sheffer property).
O.g.f. column k: x^m/Product_{j=0..k} (1 - (3+4*j)*x), k >= 0.
(End)

A225467 Triangle read by rows, T(n, k) = 4^k*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.

Original entry on oeis.org

1, 3, 4, 9, 40, 16, 27, 316, 336, 64, 81, 2320, 4960, 2304, 256, 243, 16564, 63840, 54400, 14080, 1024, 729, 116920, 768496, 1071360, 485120, 79872, 4096, 2187, 821356, 8921136, 19144384, 13502720, 3777536, 430080, 16384, 6561, 5758240, 101417920, 322850304

Offset: 0

Peter Luschny, May 08 2013

The definition of the Stirling-Frobenius subset numbers of order m is in A225468.

This is the Sheffer triangle (exp(3*x), exp(4*x) - 1). See also the P. Bala link under A225469, the Sheffer triangle (exp(3*x),(1/4)*(exp(4*x) - 1)), which is named there exponential Riordan array S_{(4,0,3)}. - Wolfdieter Lang, Apr 13 2017

			[n\k][  0,      1,       2,        3,        4,       5,      6,     7]
[0]     1,
[1]     3,      4,
[2]     9,     40,      16,
[3]    27,    316,     336,       64,
[4]    81,   2320,    4960,     2304,      256,
[5]   243,  16564,   63840,    54400,    14080,    1024,
[6]   729, 116920,  768496,  1071360,   485120,   79872,   4096,
[7]  2187, 821356, 8921136, 19144384, 13502720, 3777536, 430080, 16384.
...
From _Wolfdieter Lang_, Aug 11 2017: (Start)
Recurrence (see the Maple program): T(4, 2) = 4*T(3, 1) + (4*2+3)*T(3, 2) = 4*316 + 11*336 = 4960.
Boas-Buck recurrence for column k = 2, and n = 4: T(4, 2) = (1/2)*(2*(6 + 4*2)*T(3, 2) + 2*6*(-4)^2*Bernoulli(2)*T(2, 2)) = (1/2)*(28*336 + 12*16*(1/6)*16) = 4960. (End)

Vincenzo Librandi, Rows n = 0..50, flattened
Paweł Hitczenko, A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality, arXiv:2403.03422 [math.CO], 2024. See p. 9.
Peter Luschny, Eulerian polynomials.
Peter Luschny, The Stirling-Frobenius numbers.

Cf. A048993 (m=1), A154537 (m=2), A225466 (m=3). A225469 (scaled).

Cf. Columns: A000244, 4*A016138, 16*A018054. A225118.

SF_SS := 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*SF_SS(n-1, k-1, m) + (m*(k+1)-1)*SF_SS(n-1, k, m) end:
seq(print(seq(SF_SS(n, k, 4), k=0..n)), n=0..5);

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]]); SFSS[n_, k_, m_] := Sum[ EulerianNumber[n, j, m]*Binomial[j, n-k], {j, 0, n}]/k!; Table[ SFSS[n, k, 4], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 29 2013, translated from Sage *)

T(n, k) = sum(m=0, k, binomial(k, m)*(-1)^(m - k)*((3 + 4*m)^n)/k!);
for(n = 0, 10, for(k=0, n, print1(T(n, k),", ");); print();) \\ Indranil Ghosh, Apr 13 2017

from sympy import binomial, factorial
def T(n, k): return sum(binomial(k, m)*(-1)**(m - k)*((3 + 4*m)**n)//factorial(k) for m in range(k + 1))
for n in range(11): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Apr 13 2017

@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_SS(n, k, m):
    return add(EulerianNumber(n,j,m)*binomial(j,n-k) for j in (0..n))/factorial(k)
def A225467(n): return SF_SS(n, k, 4)

T(n, k) = (1/k!)*sum_{j=0..n} binomial(j, n-k)*A_4(n, j) where A_m(n, j) are the generalized Eulerian numbers A225118.
For a recurrence see the Maple program.
T(n, 0) ~ A000244; T(n, 1) ~ A190541.
T(n, n) ~ A000302; T(n, n-1) ~ A002700.
From Wolfdieter Lang, Apr 13 2017: (Start)
T(n, k) = Sum_{m=0..k} binomial(k,m)*(-1)^(m-k)*((3+4*m)^n)/k!, 0 <= k <= n.
In terms of Stirling2 = A048993: T(n, m) = Sum_{k=0..n} binomial(n, k)* 3^(n-k)*4^k*Stirling2(k, m), 0 <= m <= n.
E.g.f. exp(3*z)*exp(x*(exp(4*z) - 1)) (Sheffer property).
E.g.f. column k: exp(3*x)*((exp(4*x) - 1)^k)/k!, k >= 0.
O.g.f. column k: (4*x)^k/Product_{j=0..k} (1 - (3 + 4*j)*x), k >= 0.
(End)
Boas-Buck recurrence for column sequence k: T(n, k) = (1/(n - k))*((n/2)*(6 + 4*k)*T(n-1, k) + k*Sum_{p=k..n-2} binomial(n, p)*(-4)^(n-p)*Bernoulli(n-p)*T(p, k)), for n > k >= 0, with input T(k, k) = 4^k. See a comment and references in A282629. An example is given below. - Wolfdieter Lang, Aug 11 2017

A190541 a(n) = 7^n - 3^n.

Original entry on oeis.org

0, 4, 40, 316, 2320, 16564, 116920, 821356, 5758240, 40333924, 282416200, 1977149596, 13840755760, 96887416084, 678218289880, 4747547161036, 33232887522880, 232630384847044, 1628413210489960, 11398894023111676, 79792262810827600, 558545853622930804, 3909821017201928440

Offset: 0

Vincenzo Librandi, Jun 02 2011

Length-n words from letters {1,2,...,7} with at least one letter greater than 3. - Joerg Arndt, Jun 02 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (10,-21).

Cf. A000244, A000420.

Similar sequences: A121213, A016169.

Magma
```
[7^n - 3^n: n in [0..30]];
```

A190541:=n->7^n-3^n: seq(A190541(n), n=0..25); # Wesley Ivan Hurt, Oct 04 2014

Table[7^n - 3^n, {n, 0, 25}] (* or *) CoefficientList[Series[4 x /((1 - 3 x) (1 - 7 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 04 2014 *)
LinearRecurrence[{10,-21},{0,4},20] (* Harvey P. Dale, Mar 30 2015 *)

a(n)=7^n-3^n \\ Charles R Greathouse IV, Jun 02 2011

a(n) = 10*a(n-1) - 21*a(n-2).
G.f.: 4*x/((1-3*x)*(1-7*x)). - Vincenzo Librandi, Oct 04 2014
a(n) = A000420(n) - A000244(n). - Wesley Ivan Hurt, Oct 04 2014
E.g.f.: 2*exp(5*x)*sinh(2*x). - Elmo R. Oliveira, Mar 31 2025
a(n) = 4*A016138(n-1). - R. J. Mathar, Jun 07 2025

A165147 a(n) = (3*7^n-3^n)/2.

Original entry on oeis.org

1, 9, 69, 501, 3561, 25089, 176109, 1234221, 8643921, 60520569, 423683349, 2965901541, 20761665081, 145332718449, 1017332217789, 7121335090461, 49849374331041, 348945706410729, 2442620203155429, 17098342196928981

Offset: 0

Klaus Brockhaus, Sep 15 2009

Partial sums are in A016138.

Binomial transform of A016129. Inverse binomial transform of A165148.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (10,-21).

Cf. A016138, A016129, A165148.

Magma
```
[ (3*7^n-3^n)/2: n in [0..19] ];
```

LinearRecurrence[{10, -21}, {1, 9}, 25] (* Paolo Xausa, Apr 22 2024 *)
Table[(3*7^n-3^n)/2,{n,0,20}] (* Harvey P. Dale, Aug 05 2025 *)

a(n) = 10*a(n-1)-21*a(n-2) for n > 1; a(0) = 1, a(1) = 9.

G.f.: (1-x)/((1-3*x)*(1-7*x)).

cp's OEIS Frontend

A225469 Triangle read by rows, S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A225467 Triangle read by rows, T(n, k) = 4^k*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Sage

Formula

A190541 a(n) = 7^n - 3^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A165147 a(n) = (3*7^n-3^n)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula