A028338 -id:A028338 - cp's OEIS frontend

A024197 a(n) = 3rd elementary symmetric function of the first n+2 odd positive integers.

15, 176, 950, 3480, 10045, 24640, 53676, 106800, 197835, 345840, 576290, 922376, 1426425, 2141440, 3132760, 4479840, 6278151, 8641200, 11702670, 15618680, 20570165, 26765376, 34442500, 43872400, 55361475, 69254640, 85938426, 105844200, 129451505

Offset: 1

Clark Kimberling

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.

Contribution from Johannes W. Meijer, Jun 08 2009: (Start)
Equals fourth right hand column of A028338 triangle.
Equals fourth left hand column of A109692 triangle.
Equals fourth right hand column of A161198 triangle divided by 2^m.
(End)

Vec(-x*(x^3+33*x^2+71*x+15)/(x-1)^7 + O(x^100)) \\ Colin Barker, Aug 15 2014

a(n) = n*(n+1)*(n+2)^2*(n^2+3*n+1)/6.
G.f.: -x*(x^3+33*x^2+71*x+15) / (x-1)^7. - Colin Barker, Aug 15 2014
a(n) = A004320(n)*A028387(n). - R. J. Mathar, Oct 01 2016
a(n) = A028338(n+2, n-1), n >= 1, (fourth diagonal). See a crossref. below. - Wolfdieter Lang, Jul 21 2017

A290319 Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4x)^(-1/4), (-1/4)log(1 - 4*x)). A generalized Stirling1 triangle.

Original entry on oeis.org

1, 1, 1, 5, 6, 1, 45, 59, 15, 1, 585, 812, 254, 28, 1, 9945, 14389, 5130, 730, 45, 1, 208845, 312114, 122119, 20460, 1675, 66, 1, 5221125, 8011695, 3365089, 633619, 62335, 3325, 91, 1, 151412625, 237560280, 105599276, 21740040, 2441334, 158760, 5964, 120, 1, 4996616625, 7990901865, 3722336388, 823020596, 102304062, 7680414, 355572, 9924, 153, 1, 184874815125, 300659985630, 145717348221, 34174098440, 4608270890, 386479380, 20836578, 722760, 15585, 190, 1

Offset: 0

Wolfdieter Lang, Aug 08 2017

This generalization of the unsigned Stirling1 triangle A132393 is called here |S1hat[4,1]|.
The signed matrix S1hat[4,1] with elements (-1)^(n-k)*|S1hat[4,1]|(n, k) is the inverse of the generalized Stirling2 Sheffer matrix S2hat[4,1] with elements S2[4,1](n, k)/d^k, where S2[4,1] is Sheffer (exp(x), exp(4*x) - 1), given in A285061. See also the P. Bala link below for the scaled and signed version s_{(4,0,1)}.
For the general |S1hat[d,a]| case see a comment in A286718.

			The triangle T(n, k) begins:
  n\k         0         1         2        3       4      5    6   7  8 ...
  0:          1
  1:          1         1
  2:          5         6         1
  3:         45        59        15        1
  4:        585       812       254       28       1
  5:       9945     14389      5130      730      45      1
  6:     208845    312114    122119    20460    1675     66    1
  7:    5221125   8011695   3365089   633619   62335   3325   91   1
  8:  151412625 237560280 105599276 21740040 2441334 158760 5964 120  1
  ...
From _Wolfdieter Lang_, Aug 11 2017: (Start)
Recurrence: T(4, 2) = T(3, 1) + (16 - 3)*T(3, 2) = 59 + 13*15 = 254.
Boas-Buck recurrence for column k=2 and n=4:
T(4, 2) = (4!/2)*(4*(1 + 8*(5/12))*T(2, 2)/2! + 1*(1 + 8*(1/2))*T(3,2)/3!) = (4!/2)*(2*13/3 + 5*15/3!) = 254. (End)

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).
Peter Bala, A 3 parameter family of generalized Stirling numbers.
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli Numbers, arXiv:math/1707.04451 [math.NT], July 2017.

S2[d,a] for [d,a] = [1,0], [2,1], [3,1], [3,2], [4,1] and [4,3] is A048993, A154537, A282629, A225466, A285061 and A225467, respectively.
|S1hat[d,a]| for [d,a] = [1,0], [2,1], [3,1], [3,2] and [4,3] is A132393, A028338, A286718, A225470 and A225471, respectively.
Columns k=0..3 give A007696, A024382(n-1), A383700, A383701.
Row sums: A001813. Alternating row sums: A000007.

FoldList[Join[Table[If[i == 1, 0, #[[i-1]]] + (4*#2 - 3)*#[[i]], {i, Length[#]}], {1}] &, {1}, Range[10]] (* Paolo Xausa, Aug 18 2025 *)

Recurrence: T(n, k) = T(n-1, k-1) + (4*n - 3)*T(n-1, k), for n >= 1, k = 0..n, and T(n, -1) = 0, T(0, 0) = 1 and T(n, k) = 0 for n < k.
E.g.f. of row polynomials R(n, x) = Sum_{k=0..n} T(n, k)*x^k (i.e., e.g.f. of the triangle): (1 - 4*z)^{-(x + 1)/4}.
E.g.f. of column k is (1 - 4*x)^(-1/4)*((-1/4)*log(1 - 4*x))^k/k!.
Recurrence for row polynomials is R(n, x) = (x+1)*R(n-1, x+4), with R(0, x) = 1. Row polynomial R(n, x) = risefac(4,1;x,n) with the rising factorial risefac(d,a;x,n) :=Product_{j=0..n-1} (x + (a + j*d)). (For the signed case see the Bala link, eq. (16)).
T(n, k) = sigma^{(n)}{n-k}(a_0, a_1, ..., a{n-1}) with the elementary symmetric functions with indeterminates a_j = 1 + 4*j.
T(n, k) = Sum_{j=0..n-k} binomial(n-j, k)*|S1|(n, n-j)*4^j, with the unsigned Stirling1 triangle |S1| = A132393.
Boas-Buck type recurrence for column sequence k: T(n, k) = (n!/(n - k)) * Sum_{p=k..n-1} 4^(n-1-p)*(1 + 4*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 with {1/2, 5/12, 3/8, 251/720, ...}. See a comment and references in A286718. - Wolfdieter Lang, Aug 11 2017

A225475 Triangle read by rows, k!*s_2(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 1, 3, 4, 2, 15, 23, 18, 6, 105, 176, 172, 96, 24, 945, 1689, 1900, 1380, 600, 120, 10395, 19524, 24278, 20880, 12120, 4320, 720, 135135, 264207, 354662, 344274, 241080, 116760, 35280, 5040, 2027025, 4098240, 5848344, 6228096, 4993296, 2956800, 1229760

Offset: 0

Peter Luschny, May 19 2013

The Stirling-Frobenius cycle numbers are defined in A225470.

			[n\k][ 0,    1,    2,    3,   4,   5]
[0]    1,
[1]    1,    1,
[2]    3,    4,    2,
[3]   15,   23,   18,    6,
[4]  105,  176,  172,   96,  24,
[5]  945, 1689, 1900, 1380, 600, 120.

Vincenzo Librandi, Rows n = 0..50, flattened
Peter Luschny, Generalized Eulerian polynomials.
Peter Luschny, The Stirling-Frobenius numbers.

Cf. A028338, A225479 (m=1), A048594.

SFCO[n_, k_, m_] := SFCO[n, k, m] = If[ k > n || k < 0, Return[0], If[ n == 0 && k == 0, Return[1], Return[ k*SFCO[n - 1, k - 1, m] + (m*n - 1)*SFCO[n - 1, k, m]]]]; Table[ SFCO[n, k, 2], {n, 0, 8}, {k, 0, n}] // Flatten  (* Jean-François Alcover, Jul 02 2013, translated from Sage *)

@CachedFunction
def SF_CO(n, k, m):
    if k > n or k < 0 : return 0
    if n == 0 and k == 0: return 1
    return k*SF_CO(n-1, k-1, m) + (m*n-1)*SF_CO(n-1, k, m)
for n in (0..8): [SF_CO(n, k, 2) for k in (0..n)]

For a recurrence see the Sage program.
T(n, 0) ~ A001147; T(n, 1) ~ A004041.
T(n, n) ~ A000142; T(n, n-1) ~ A001563.
T(n,k) = A028338(n,k)*A000142(k). - Philippe Deléham, Jun 24 2015

A225477 Triangle read by rows, 3^k*s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

Original entry on oeis.org

1, 2, 3, 10, 21, 9, 80, 198, 135, 27, 880, 2418, 2079, 702, 81, 12320, 36492, 36360, 16065, 3240, 243, 209440, 657324, 727596, 382185, 103275, 13851, 729, 4188800, 13774800, 16523892, 9826488, 3212055, 586845, 56133, 2187, 96342400, 329386800, 421373916, 275580900, 103356729, 23133600, 3051594, 218700, 6561

Offset: 0

Peter Luschny, May 17 2013

Triangle T(n,k), read by rows, given by (2, 3, 5, 6, 8, 9, 11, 12, 14, ... (A007494)) DELTA (3, 0, 3, 0, 3, 0, 3, 0, 3, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, May 15 2015

			[n\k][   0,      1,      2,      3,      4,     5,   6 ]
[0]      1,
[1]      2,      3,
[2]     10,     21,      9,
[3]     80,    198,    135,     27,
[4]    880,   2418,   2079,    702,     81,
[5]  12320,  36492,  36360,  16065,   3240,   243,
[6] 209440, 657324, 727596, 382185, 103275, 13851, 729.

Peter Luschny, Generalized Eulerian polynomials.
Peter Luschny, The Stirling-Frobenius numbers.

T(n, 0) ~ A008544; T(n, n) ~ A000244.
row sums ~ A034000; alternating row sums ~ A008544.
Cf. A225470, A132393 (m=1), A028338 (m=2), A225478 (m=4).

s[][0, 0] = 1; s[m][n_, k_] /; (k > n || k < 0) = 0; s[m_][n_, k_] := s[m][n, k] = s[m][n - 1, k - 1] + (m*n - 1)*s[m][n - 1, k];
T[n_, k_] := 3^k*s[3][n, k];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 02 2019 *)

@CachedFunction
def SF_CS(n, k, m):
    if k > n or k < 0 : return 0
    if n == 0 and k == 0: return 1
    return m*SF_CS(n-1, k-1, m) + (m*n-1)*SF_CS(n-1, k, m)
for n in (0..8): [SF_CS(n, k, 3) for k in (0..n)]

For a recurrence see the Sage program.

T(n,k) = 3^k * A225470(n,k). - Philippe Deléham, May 14 2015.

A225478 Triangle read by rows, 4^k*s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

Original entry on oeis.org

1, 3, 4, 21, 40, 16, 231, 524, 336, 64, 3465, 8784, 7136, 2304, 256, 65835, 180756, 170720, 72320, 14080, 1024, 1514205, 4420728, 4649584, 2346240, 613120, 79872, 4096, 40883535, 125416476, 143221680, 81946816, 25939200, 4609024, 430080, 16384, 1267389585, 4051444896, 4941537984, 3113238016, 1131902464, 246636544, 31768576, 2228224, 65536

Offset: 0

Peter Luschny, May 17 2013

Triangle T(n,k), read by rows, given by (3, 4, 7, 8, 11, 12, 15, 16, ... (A014601)) DELTA (4, 0, 4, 0, 4, 0, 4, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, May 14 2015.

			[n\k][    0,       1,       2,       3,      4,     5,    6 ]
[0]       1,
[1]       3,       4,
[2]      21,      40,      16,
[3]     231,     524,     336,      64,
[4]    3465,    8784,    7136,    2304,    256,
[5]   65835,  180756,  170720,   72320,  14080,  1024,
[6] 1514205, 4420728, 4649584, 2346240, 613120, 79872, 4096.

Peter Luschny, Generalized Eulerian polynomials.
Peter Luschny, The Stirling-Frobenius numbers.

T(n, 0) ~ A008545; T(n, n) ~ A000302; T(n, n-1) ~ A002700.
row sums ~ A034176; alternating row sums ~ A008545.
Cf. A225471, A132393 (m=1), A028338 (m=2), A225477 (m=3).

s[][0, 0] = 1; s[m][n_, k_] /; (k > n || k < 0) = 0; s[m_][n_, k_] := s[m][n, k] = s[m][n - 1, k - 1] + (m*n - 1)*s[m][n - 1, k];
T[n_, k_] := 4^k*s[4][n, k];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 02 2019 *)

@CachedFunction
def SF_CS(n, k, m):
    if k > n or k < 0 : return 0
    if n == 0 and k == 0: return 1
    return m*SF_CS(n-1, k-1, m) + (m*n-1)*SF_CS(n-1, k, m)
for n in (0..8): [SF_CS(n, k, 4) for k in (0..n)]

For a recurrence see the Sage program.

T(n,k) = 4^k * A225471(n,k). - Philippe Deléham, May 14 2015.

A225474 Triangle read by rows, k!2^ks_2(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 2, 3, 8, 8, 15, 46, 72, 48, 105, 352, 688, 768, 384, 945, 3378, 7600, 11040, 9600, 3840, 10395, 39048, 97112, 167040, 193920, 138240, 46080, 135135, 528414, 1418648, 2754192, 3857280, 3736320, 2257920, 645120, 2027025, 8196480, 23393376, 49824768, 79892736

Offset: 0

Peter Luschny, May 19 2013

The Stirling-Frobenius cycle numbers are defined in A225470.

			[n\k][ 0,    1,    2,     3,    4,    5]
[0]    1,
[1]    1,    2,
[2]    3,    8,    8,
[3]   15,   46,   72,    48,
[4]  105,  352,  688,   768,  384,
[5]  945, 3378, 7600, 11040, 9600, 3840.

Peter Luschny, Generalized Eulerian polynomials.
Peter Luschny, The Stirling-Frobenius numbers.

Cf. A028338, A225479 (m=1), A048594, A161198, A225475.

Cf. A000079, A000142, A000165, A001147, A014479, A028338.

SFCSO[n_, k_, m_] := SFCSO[n, k, m] = If[k>n || k<0, 0, If[n == 0 && k == 0, 1, m*k*SFCSO[n-1, k-1, m] + (m*n-1)*SFCSO[n-1, k, m]]]; Table[SFCSO[n, k, 2], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 05 2014, translated from Sage *)

@CachedFunction
def SF_CSO(n, k, m):
    if k > n or k < 0 : return 0
    if n == 0 and k == 0: return 1
    return m*k*SF_CSO(n-1, k-1, m) + (m*n-1)*SF_CSO(n-1, k, m)
for n in (0..8): [SF_CSO(n, k, 2) for k in (0..n)]

For a recurrence see the Sage program.
T(n, 0) ~ A001147; T(n, n) ~ A000165; T(n, n-1) ~ A014479.
T(n,k) = A028338(n,k) * A000165(k) = A225475(n,k) * A000079(k) = A161198(n,k) * A000142(k). - Philippe Deléham, Jun 25 2015

cp's OEIS Frontend

A024197 a(n) = 3rd elementary symmetric function of the first n+2 odd positive integers.

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A290319 Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1 triangle.

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A225475 Triangle read by rows, k!*s_2(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

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A225477 Triangle read by rows, 3^k*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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A225478 Triangle read by rows, 4^k*s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

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A225474 Triangle read by rows, k!*2^k*s_2(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

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A290319 Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4x)^(-1/4), (-1/4)log(1 - 4*x)). A generalized Stirling1 triangle.

A225474 Triangle read by rows, k!2^ks_2(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.