A130534 -id:A130534 - cp's OEIS frontend

A349782 Triangle read by rows, T(n, k) = Sum_{j=0..k} |Stirling1(n, j)|.

1, 0, 1, 0, 1, 2, 0, 2, 5, 6, 0, 6, 17, 23, 24, 0, 24, 74, 109, 119, 120, 0, 120, 394, 619, 704, 719, 720, 0, 720, 2484, 4108, 4843, 5018, 5039, 5040, 0, 5040, 18108, 31240, 38009, 39969, 40291, 40319, 40320, 0, 40320, 149904, 268028, 335312, 357761, 362297, 362843, 362879, 362880

Offset: 0

Peter Luschny, Dec 02 2021

T(n, k) is the number of permutations of n objects that contain at most k cycles.

			Triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 1,    2;
[3] 0, 2,    5,     6;
[4] 0, 6,    17,    23,    24;
[5] 0, 24,   74,    109,   119,   120;
[6] 0, 120,  394,   619,   704,   719,   720;
[7] 0, 720,  2484,  4108,  4843,  5018,  5039,  5040;
[8] 0, 5040, 18108, 31240, 38009, 39969, 40291, 40319, 40320;

Row sums: A121586, central terms: A349783.

Cf. A132393, A048994, A008275, A000142, A033312, A000774, A081046, A130534.

T := (n, k) -> add(abs(Stirling1(n,j)), j = 0..k):
seq(seq(T(n, k), k = 0..n), n = 0..9);

T[n_, k_] := Sum[Abs[StirlingS1[n, j]], {j, 0, k}]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Dec 09 2021 *)

T(n, k) = sum(j=0, k, abs(stirling(n, j, 1))); \\ Michel Marcus, Dec 09 2021

T(n,k) = Sum_{j=0..k} A132393(n,j). - Alois P. Heinz, Dec 10 2021

A355266 Triangle read by rows, T(n, k) = (-1)^(n-k)Bell(k)Stirling1(n+1, k+1), for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 6, 11, 12, 5, 24, 50, 70, 50, 15, 120, 274, 450, 425, 225, 52, 720, 1764, 3248, 3675, 2625, 1092, 203, 5040, 13068, 26264, 33845, 29400, 16744, 5684, 877, 40320, 109584, 236248, 336420, 336735, 235872, 110838, 31572, 4140

Offset: 0

Peter Luschny and Mélika Tebni, Jul 05 2022

			Triangle T(n, k) begins:
[0]    1;
[1]    1,      1;
[2]    2,      3,     2;
[3]    6,     11,    12,      5;
[4]   24,     50,    70,     50,    15;
[5]  120,    274,   450,    425,   225,     52;
[6]  720,   1764,  3248,   3675,  2625,   1092,  203;
[7] 5040,  13068, 26264,  33845, 29400,  16744, 5684, 877;

Cf. A002720 (row sums), A000166 (alternating row sums), A000110 (main diagonal), A000142 (column 0).

Cf. A105479, A000254, A130534, A053557, A053556.

T := (n, k) -> (-1)^(n-k)*combinat:-bell(k)*Stirling1(n+1, k+1):
seq(seq(T(n, k), k = 0..n), n = 0..8);

from functools import cache
@cache
def b(n: int, k=0):
    return int(n < 1) or k * b(n - 1, k) + b(n - 1, k + 1)
@cache
def s(n: int) -> list[int]:
    if n == 0: return [1]
    row = [0] + s(n - 1)
    for k in range(1, n): row[k] = row[k] + (n - 1) * row[k + 1]
    return row
def A355266_row(n):
    return [s * b(k - 1) for k, s in enumerate(s(n + 1))][1:]
for n in range(9): print(A355266_row(n))

T(n, k) = A000110(k) * A130534(n, k).
Sum_{k=0..n} T(n, k) = n!*Laguerre(n, -1) = A002720(n).
Sum_{k=0..n} (-1)^k*T(n, k) = !n = n!*A053557(n)/A053556(n) = A000166(n).

A371741 Triangle read by rows: g.f. (1 - t)^(-x) * (1 + t)^(3-x).

Original entry on oeis.org

1, 3, 3, 1, 1, 3, 0, 7, 1, 0, 5, 3, 0, 11, 12, 1, 0, 9, 12, 3, 0, 30, 47, 18, 1, 0, 26, 45, 22, 3, 0, 114, 215, 125, 25, 1, 0, 102, 205, 135, 35, 3, 0, 552, 1174, 855, 265, 33, 1, 0, 504, 1122, 885, 315, 51, 3, 0, 3240, 7518, 6349, 2520, 490, 42, 1, 0, 3000, 7210, 6447, 2800, 630, 70, 3

Offset: 0

Peter Bala, Apr 05 2024

			Triangle begins
  n\k |  0     1     2     3     4      5
  - - - - - - - - - - - - - - - - - - - -
   0  |  1
   1  |  3
   2  |  3     1
   3  |  1     3
   4  |  0     7     1
   5  |  0     5     3
   6  |  0    11    12     1
   7  |  0     9    12     3
   8  |  0    30    47    18     1
   9  |  0    26    45    22     3
  10  |  0   114   215   125    25     1
  11  |  0   102   205   135    35     3
  ...

Cf. A130534, A132393, A371740.

with(combinat):
T := proc (n, k); if irem(n, 2) = 0 then abs(Stirling1((1/2)*n, k)) + (3*n/2)*abs(Stirling1((n-2)/2, k)) else 3*abs(Stirling1((n-1)/2, k)) + ((n-1)/2)*abs(Stirling1((n-3)/2, k)) end if; end proc:
seq(print(seq(T(n, k), k = 0..floor(n/2))), n = 0..12);

G.f.: (1 - t)^(-x)*(1 + t)^(3-x) = Sum_{n >= 0} R(n, x)*t^n/floor(n/2)! = 1 + 3*t + (3 + x)^t^2/1! + (1 + 3*x)*t^3/1! + x*(7 + x)*t^4/2! + x*(5 + 3*x)*t^5/2! + x*(1 + x)*(11 + x)*t^6/3! + x*(1 + x)*(9 + 3*x)*t^7/3! + x*(1 + x)*(2 + x)*(15 + x)*t^8/4! + x*(1 + x)*(2 + x)*(13 + 3*x)*t^9/4! + ....
Row polynomials: R(2*n, x) =  (4*n - 1 + x) * Product_{i = 0..n-2} (x + i) for n >= 1.
R(2*n+1, x) = (4*n - 3 + 3*x) * Product_{i = 0..n-2} (x + i) for n >= 1.
T(2*n, k) = |Stirling1(n, k)| + 3*n*|Stirling1(n-1, k)| = A132393(n, k) + 3*n*A132393(n-1, k).
T(2*n+1, k) = 3*|Stirling1(n, k)| + n*|Stirling1(n-1, k)| = 3*A132393(n, k) + n*A132393(n-1, k).
T(2*n, k) = (4*n - 1)*A132393(n-1, k) + A132393(n-1, k-1).
T(2*n+1, k) = (4*n - 3)*A132393(n-1, k) + 3*A132393(n-1, k-1).
n-th row sums equals 4*floor(n/2)! for n >= 2.

A376863 Triangle of generalized Stirling numbers of the lower level of the hierarchy (section m=1).

Original entry on oeis.org

1, 3, 1, 13, 7, 1, 73, 50, 12, 1, 501, 400, 125, 18, 1, 4051, 3609, 1335, 255, 25, 1, 37633, 36463, 15214, 3485, 460, 33, 1, 394353, 408694, 186949, 48769, 7805, 763, 42, 1, 4596553, 5036792, 2479602, 714364, 131299, 15708, 1190, 52, 1, 58941091, 67714809, 35419350, 11045558, 2256933, 312375, 29190, 1770, 63, 1, 824073141, 986271823, 543025851, 180766890, 40194965, 6221397, 676893, 50970, 2535, 75, 1

Offset: 0

Igor Victorovich Statsenko, Oct 07 2024

			Triangle starts:
[0]        1;
[1]        3,        1;
[2]       13,        7,        1;
[3]       73,       50,       12,       1;
[4]      501,      400,      125,      18,       1;
[5]     4051,     3609,     1335,     255,      25,       1;
[6]    37633,    36463,    15214,    3485,     460,      33,      1;
[7]   394353,   408694,   186949,   48769,    7805,     763,     42,    1;
[8]  4596553,  5036792,  2479602,  714364,  131299,   15708,   1190,   52,     1;

Igor Victorovich Statsenko, Relationships of “P”-generalized Stirling numbers of the first kind with other generalized Stirling numbers, Innovation science No 10-1, State Ufa, Aeterna Publishing House, 2024, pp. 19-12. In Russian.

A000262 (column 0), A052852 (row sums).

Triangle for m=0: A130534.

T:=(m,n,k)->add(add(Stirling1(n-j,k)*binomial(n+m,i)*binomial(n,j)*binomial(j,i)*i!*m^(j-i), j=i..n),i=0..n):m:=1:seq(seq(T(m,n,k),k=0..n),n=0..10);

T(m, n, k) = Sum_{i=0..n} Sum_{j=i..n} Stirling1(n-j, k) * binomial(n+m, i) * binomial(n, j)* binomial(j, i) * i! * m^(j - i), for m = 1.

A221916 Array of products of the list entries of the combinations of n, taken in reverse standard order.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 6, 6, 3, 2, 3, 2, 1, 1, 24, 24, 12, 8, 6, 12, 8, 6, 4, 3, 2, 4, 3, 2, 1, 1, 120, 120, 60, 40, 30, 24, 60, 40, 30, 24, 20, 15, 12, 10, 8, 6, 20, 15, 12, 10, 8, 6, 5, 4, 3, 2, 5, 4, 3, 2, 1, 1, 720, 720, 360, 240, 180, 144, 120, 360, 240, 180, 144, 120, 120, 90, 72, 60, 60, 48, 40, 36, 30, 24, 120, 90, 72, 60, 60, 48, 40, 36, 30, 24, 30, 24

Offset: 0

Wolfdieter Lang, Feb 08 2013

The sequence of the row lengths is 2^n = A000079(n), n >= 0.
As a preliminary some notation. The list choose(n) consists of the 2^n combinations of n, written as lists. The first entry of choose(n) is the empty combination []. The other entries are ordered lexicographically within entries of the same length, called m, and m increases from 1 to n (m=0 is reserved for the empty combination).  E.g., choose(3) = [[], [1], [2], [3], [1, 2], [1, 3], [2, 3], [1, 2, 3]].
The array entry a(n,k=0) := n!, n >= 0, and for k = 1, 2, ... , 2^n-1 one takes a(n,k) =  n!/product(comb(n,k,l), l = 1..m(n,k)), with m(n,k) the length of the (k+1)-th entry comb(n,k) of choose(n), and comb(n,k,l) is the l-th entry of comb(n,k). E.g., comb(3,5) = [1, 3], comb(3,5,1) = 1 and comb(3,5,2) = 3, hence a(3,5) = 3!/(1*3) = 2.
This array  a(n,k) is the row reversed array A(n,k) = A221914(n,k) if one adds there A(n,0) = 1 for n >= 0.
If all entries of the present array which belong to the same m= m(n,k) value are summed one obtains the unsigned Stirling1(n+1,m+1) triangle A130534(n,m) because this is sigma_{n-m}(1,2,...,n) with the (n-m)-th elementary symmetric function of 1,2, ..., n.
For row No. n the sum of entries is (n+1)! = A000142(n+1), like for the triangle A130534.

			The array a(n,k) begins:
n\k  0   1   2  3  4   5  6  7  8  9  10  11  12  13  14  15
0:   1
1:   1   1
2:   2   2   1  1
3:   6   6   3  2  3   2  1  1
4:  24  24  12  8  6  12  8  6  4  3   2   4   3   2   1   1
...
Row n=5: 120, 120, 60, 40, 30, 24, 60, 40, 30, 24, 20, 15, 12, 10, 8, 6, 20, 15, 12, 10, 8, 6, 5, 4, 3, 2, 5, 4, 3, 2, 1, 1.
Row n=6: 720, 720, 360, 240, 180, 144, 120, 360, 240, 180, 144, 120, 120, 90, 72, 60, 60, 48, 40, 36, 30, 24, 120, 90, 72, 60, 60, 48, 40, 36, 30, 24, 30, 24, 20, 18, 15, 12, 12, 10, 8, 6, 30, 24, 20, 18, 15, 12, 12, 10, 8, 6, 6, 5, 4, 3, 2, 6, 5, 4, 3, 2, 1, 1.
The combinations for row n are choose(4) = [[], [1], [2], [3], [4], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4], [1, 2, 3], [1, 2, 4], [1, 3, 4], [2, 3, 4], [1, 2, 3, 4]]. For k=0 one takes 4! = 24. For k >= 1 one obtains 4!/1, 4!/2, 4!/3, 4!/4; 4!/(1*2), 4!/(1*3), 4!/(1*4), 4!/(2*3), 4!/(2*4), 4!/(3*4); 4!/(1*2*3),  4!/(1*2*4), 4!/(1*3*4), 4!/(2*3*4);  4!/(1*2*3*4) giving row n=4. The semicolons separate the binomial(4,m) entries with m values from 1 to 4. The example in the comment above was k=13 leading to 4!/(1*3*4) = 2 = a(4,13).

Cf. A221914, A130534, |A008275|.

a(n,k) := n! for k=0, and for k =1,2, ..., 2^n-1 it is n!/product(comb(n,k,l),l=1..|comb(n,k)|) with |comb(n,k)| the number of entries of comb(n,k) which is the (k+1)-th entry of the list of combinations choose(n) (starting with the empty combination for k=0), and comb(n,k,l) is the l-th entry of the list comb(n,k). See a comment above how |comb(n,k)| = m(n,k) is determined.

A323630 Expansion of e.g.f. exp(log(1 - x)^2/2)/(1 - x). This is also the transform of the involution numbers given by the signless Stirling cycle numbers.

Original entry on oeis.org

1, 1, 3, 12, 62, 390, 2884, 24472, 234086, 2490030, 29139306, 371878056, 5138306700, 76398336924, 1215973642584, 20624305367520, 371309259462972, 7071037633297116, 141997246553420052, 2998654325698019280, 66426777891686458728, 1540117294435707244488, 37296711627004301923056

Offset: 0

Ilya Gutkovskiy, Jan 21 2019

nonn

Cf. A000085, A004211, A008275, A130534, A319360.

seq(n!*coeff(series(exp(log(1-x)^2/2)/(1-x),x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 28 2019

nmax = 22; CoefficientList[Series[Exp[Log[1 - x]^2/2]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] HypergeometricU[-k/2, 1/2, -1/2]/(-1/2)^(k/2), {k, 0, n}], {n, 0, 22}]

my(x='x + O('x^25)); Vec(serlaplace(exp(log(1 - x)^2/2)/(1 - x))) \\ Michel Marcus, Jan 24 2019

a(n) = Sum_{k=0..n} |Stirling1(n,k)|*A000085(k).
From Emanuele Munarini, Jul 09 2022: (Start)
a(n) = Sum_{k=0..n/2} |Stirling1(n+1,2*k+1)|*binomial(2*k,k)*k!/2^k.
a(n+1) = (n+1)*a(n) - Sum_{k=1..n} binomial(n,k)*(k-1)!*a(n-k). (End)

A372973 Triangle read by rows: the exponential almost-Riordan array ( 1/(1-x) | 1/(1-x), log(1/(1-x)) ).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 6, 2, 3, 1, 24, 6, 11, 6, 1, 120, 24, 50, 35, 10, 1, 720, 120, 274, 225, 85, 15, 1, 5040, 720, 1764, 1624, 735, 175, 21, 1, 40320, 5040, 13068, 13132, 6769, 1960, 322, 28, 1, 362880, 40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1

Offset: 0

Stefano Spezia, May 26 2024

			The triangle begins:
    1;
    1,   1;
    2,   1,   1;
    6,   2,   3,   1;
   24,   6,  11,   6,  1;
  120,  24,  50,  35, 10,  1;
  720, 120, 274, 225, 85, 15, 1;
  ...

Y. Alp and E. G. Kocer, Exponential Almost-Riordan Arrays, Results Math 79, 173 (2024). See page 6.

Cf. A000012 (right diagonal), A000254, A000399 (k=3), A000454 (k=4), A000482 (k=5), A001233 (k=6), A001234 (k=7), A098558 (row sums), A179865 (subdiagonal), A243569 (k=8), A243570 (k=9).

Triangle A130534 with 1st column A000142.

T[n_,0]:=n!; T[n_,k_]:=(n-1)!/(k-1)!SeriesCoefficient[1/(1-x)Log[1/(1-x)]^(k-1),{x,0,n-1}]; Table[T[n,k],{n,0,9},{k,0,n}]//Flatten

T(n,0) = n!; T(n,k) = (n-1)!/(k-1)! * [x^(n-1)] log(1/(1-x))^(k-1)/(1-x).
T(n,1) = (n-1)! for n > 0.
T(n,2) = A000254(n-1) for n > 1.

A377058 Triangle of generalized Stirling numbers of the lower level of the hierarchy (case m=2).

Original entry on oeis.org

1, 5, 1, 32, 11, 1, 248, 113, 18, 1, 2248, 1230, 263, 26, 1, 23272, 14534, 3765, 505, 35, 1, 270400, 186992, 55654, 9115, 865, 45, 1, 3479744, 2612000, 865186, 163779, 19110, 1372, 56, 1, 49079936, 39434448, 14235388, 3013164, 408569, 36288, 2058, 68, 1

Offset: 0

Igor Victorovich Statsenko, Oct 14 2024

nonn

These numbers are a subset of the generalized Stirling numbers introduced in A370518. Therefore, we assume them to be numbers of the lower level of hierarchy with respect to A370518.

			[0]           1;
[1]           5,          1;
[2]          32,         11,         1;
[3]         248,        113,        18,        1;
[4]        2248,       1230,       263,       26,       1;
[5]       23272,      14534,      3765,      505,      35,      1;
[6]      270400,     186992,     55654,     9115,     865,     45,     1;
[7]     3479744,    2612000,    865186,   163779,   19110,   1372,    56,    1;
[8]    49079936,   39434448,  14235388,  3013164,  408569,  36288,  2058,   68,    1;

Igor Victorovich Statsenko, Relationships of "P"-generalized Stirling numbers of the first kind with other generalized Stirling numbers, Innovation science No 10-1, State Ufa, Aeterna Publishing House, 2024, pp. 19-22. In Russian.

A361649 (row sums).
Triangle for m=0: A130534.
Triangle for m=1: A376863.

T := (m,n,k) -> add(add(Stirling1(n-j,k)*binomial(n+m,i)*binomial(n,j)*binomial(j,i)*i!*m^(j-i), j=i..n), i=0..n): m:=2: seq(seq(T(m,n,k), k=0..n), n=0..10);

T(m, n, k) = Sum_{i=0..n} Sum_{j=i..n} Stirling1(n-j, k)*binomial(n+m, i)*binomial(n, j)* binomial(j, i)*i!*m^(j-i), for m = 2.

cp's OEIS Frontend

A349782 Triangle read by rows, T(n, k) = Sum_{j=0..k} |Stirling1(n, j)|.

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A355266 Triangle read by rows, T(n, k) = (-1)^(n-k)*Bell(k)*Stirling1(n+1, k+1), for 0 <= k <= n.

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A371741 Triangle read by rows: g.f. (1 - t)^(-x) * (1 + t)^(3-x).

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A376863 Triangle of generalized Stirling numbers of the lower level of the hierarchy (section m=1).

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A221916 Array of products of the list entries of the combinations of n, taken in reverse standard order.

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A323630 Expansion of e.g.f. exp(log(1 - x)^2/2)/(1 - x). This is also the transform of the involution numbers given by the signless Stirling cycle numbers.

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A372973 Triangle read by rows: the exponential almost-Riordan array ( 1/(1-x) | 1/(1-x), log(1/(1-x)) ).

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A377058 Triangle of generalized Stirling numbers of the lower level of the hierarchy (case m=2).

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A355266 Triangle read by rows, T(n, k) = (-1)^(n-k)Bell(k)Stirling1(n+1, k+1), for 0 <= k <= n.