A001048 -id:A001048 - cp's OEIS frontend

A232187 Number T(n,k) of parity alternating permutations of [n] with exactly k descents from odd to even numbers; triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-1)/2)), read by rows.

1, 1, 2, 1, 1, 5, 3, 2, 8, 2, 20, 44, 8, 6, 66, 66, 6, 114, 594, 414, 30, 24, 624, 1584, 624, 24, 864, 8784, 14544, 4464, 144, 120, 6840, 36240, 36240, 6840, 120, 8280, 147720, 471120, 353520, 55320, 840, 720, 86400, 857520, 1739520, 857520, 86400, 720, 96480

Offset: 0

Alois P. Heinz, Nov 20 2013

T(2n+1,k) = T(2n+1,n-k).

T(2n+2,n) = T(2n+1,n) + T(2n+3,n+1).

			T(5,0) = 2: 12345, 34125.
T(5,1) = 8: 12543, 14325, 14523, 32145, 34521, 52143, 52341, 54123.
T(5,2) = 2: 32541, 54321.
T(6,2) = 8: 163254, 165432, 321654, 325416, 541632, 543216, 632541, 654321.
T(7,0) = 6: 1234567, 1256347, 3412567, 3456127, 5612347, 5634127.
T(7,1) = 66: 1234765, 1236547, 1236745, ..., 7456123, 7612345, 7634125.
T(7,2) = 66: 1254763, 1276543, 1432765, ..., 7652143, 7652341, 7654123.
T(7,3) = 6: 3254761, 3276541, 5432761, 5476321, 7632541, 7654321.
Triangle T(n,k) begins:
:  0 :   1;
:  1 :   1;
:  2 :   2;
:  3 :   1,    1;
:  4 :   5,    3;
:  5 :   2,    8,     2;
:  6 :  20,   44,     8;
:  7 :   6,   66,    66,     6;
:  8 : 114,  594,   414,    30;
:  9 :  24,  624,  1584,   624,   24;
: 10 : 864, 8784, 14544,  4464,  144;
: 11 : 120, 6840, 36240, 36240, 6840, 120;

Alois P. Heinz, Rows n = 0..29, flattened

Column k=0 gives: A199660.
Row sums give: A092186 (for n>0).
T(2n+1,n) = A000142(n).
T(2n+2,n) = A001048(n+1).

T(2n+1,k) = n! * A173018(n+1,k) = A000142(n) * A173018(n+1,k).

A301523 Integers which can be partitioned into two distinct factorials. 0! and 1! are not considered distinct.

Original entry on oeis.org

3, 7, 8, 25, 26, 30, 121, 122, 126, 144, 721, 722, 726, 744, 840, 5041, 5042, 5046, 5064, 5160, 5760, 40321, 40322, 40326, 40344, 40440, 41040, 45360, 362881, 362882, 362886, 362904, 363000, 363600, 367920, 403200, 3628801, 3628802, 3628806, 3628824, 3628920, 3629520, 3633840, 3669120, 3991680

Offset: 1

Seiichi Manyama, Mar 23 2018

nonn

Numbers of the form i! + j! where i > j > 0. - Altug Alkan, Mar 23 2018

Primes in this sequence are A088332(n) for n > 1.

			    + |   1    2    6   24
  ----+--------------------
    1 |
    2 |   3;
    6 |   7,   8;
   24 |  25,  26,  30;
  120 | 121, 122, 126, 144;

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000142, A001048, A038507, A059590, A088332, A301593.

Union[Total/@Subsets[Range[10]!,{2}]] (* Harvey P. Dale, Aug 25 2020 *)

A349280 Irregular triangle read by rows: T(n,k) is the number of arrangements of n labeled children with exactly k rounds; n >= 2, 1 <= k <= floor(n/2).

Original entry on oeis.org

2, 3, 8, 12, 30, 60, 144, 330, 120, 840, 2100, 1260, 5760, 15344, 11760, 1680, 45360, 127008, 113400, 30240, 403200, 1176120, 1169280, 428400, 30240, 3991680, 12054240, 13000680, 5821200, 831600, 43545600, 135508032, 155923680, 80415720, 16632000, 665280

Offset: 2

Steven Finch, Nov 13 2021

A round means the same as a directed ring or circle.

			Triangle starts:
[2]     2;
[3]     3;
[4]     8,     12;
[5]    30,     60;
[6]   144,    330,    120;
[7]   840,   2100,   1260;
[8]  5760,  15344,  11760,  1680;
[9] 45360, 127008, 113400, 30240;
...
For n = 4, there are 8 ways to make one round and 12 ways to make two rounds.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999 (Sec. 5.2)

Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.

Row sums give A066166 (Stanley's children's game).
Column 1 gives A001048.
Right border element of row n is A001813(n/2) = |A067994(n)| for even n.
Cf. A008279, A265609, A331327.

ser := series((1 - x)^(-x*t), x, 20): xcoeff := n -> coeff(ser, x, n):
T := (n, k) -> n!*coeff(xcoeff(n), t, k):
seq(seq(T(n, k), k = 1..iquo(n,2)), n = 2..12); # Peter Luschny, Nov 13 2021
# second Maple program:
A349280 := (n,k) -> binomial(n,k)*k!*abs(Stirling1(n-k,k)):
seq(print(seq(A349280(n,k), k=1..iquo(n,2))), n=2..12); # Mélika Tebni, May 03 2023

f[k_, n_] := n! SeriesCoefficient[(1 - x)^(-x t), {x, 0, n}, {t, 0, k}]
Table[f[k, n], {n, 2, 12}, {k, 1, Floor[n/2]}]

G.f.: (1 - x)^(-x*t).
T(n, k) = binomial(n, k)*k!*|Stirling1(n-k, k)|. - Mélika Tebni, May 03 2023
The above formula can also be written as T(n, k) = A008279(n, k)*A331327(n, k) or as T(n, k) = A265609(n + 1, k)*A331327(n, k). - Peter Luschny, May 03 2023

A180013 Triangular array read by rows: T(n,k) = number of fixed points in the permutations of {1,2,...,n} that have exactly k cycles; n>=1, 1<=k<=n.

Original entry on oeis.org

1, 0, 2, 0, 3, 3, 0, 8, 12, 4, 0, 30, 55, 30, 5, 0, 144, 300, 210, 60, 6, 0, 840, 1918, 1575, 595, 105, 7, 0, 5760, 14112, 12992, 5880, 1400, 168, 8, 0, 45360, 117612, 118188, 60921, 17640, 2898, 252, 9, 0, 403200, 1095840, 1181240, 672840, 224490, 45360, 5460, 360, 10

Offset: 1

Geoffrey Critzer, Jan 13 2011

Row sums = n! which is the number of fixed points in all the permutations of {1,2,...,n}.
It appears that column k = 2 is A001048 (with different offset).
From Olivier Gérard, Oct 23 2012: (Start)
This is a multiple of the triangle of Stirling numbers of the first kind, A180013(n,k) = (n)*A132393(n-1,k).
Another interpretation is : T(n,n-k) is the total number of ways to insert the symbol n among the cycles of permutations of [n-1] with (n+1-k) cycles to form a canonical cycle representation of a permutation of [n]. For each cycle of length c, there are c places to insert a symbol, and for each permutation there is the possibility to create a new cycle (a fixed point).
(End)

			T(4,3)= 12 because there are 12 fixed points in the permutations of 4 that have 3 cycles: (1)(2)(4,3); (1)(3,2)(4); (1)(4,2)(3); (2,1)(3)(4); (3,1)(2)(4); (4,1)(2)(3) where the permutations are represented in their cycle notation.
1
0   2
0   3    3
0   8   12    4
0  30   55   30   5
0 144  300  210  60    6
0 840 1918 1575 595  105   7

G. C. Greubel, Table of n, a(n) for n = 1..5050

Cf. A000142, A001048. Diagonal, lower diagonal give: A000027, A027480(n+1).

egf:= k-> x * (log(1/(1-x)))^(k-1) / (k-1)!:
T:= (n,k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(seq(T(n, k), k=1..n), n=1..10); # Alois P. Heinz, Jan 16 2011
# As coefficients of polynomials:
with(PolynomialTools): with(ListTools): A180013_row := proc(n)
`if`(n=0, 1,(n+1)!*hypergeom([-n,1-x],[1],1)); CoefficientList(simplify(%),x) end: FlattenOnce([seq(A180013_row(n), n=0..9)]); # Peter Luschny, Jan 28 2016

Flatten[Table[Table[(n + 1) Abs[StirlingS1[n, k]], {k, 0, n}], {n, 0, 9}],1] (* Olivier Gérard, Oct 23 2012 *)

E.g.f.: for column k: x*(log(1/(1-x)))^(k-1)/(k-1)!.

T(n, k) = [x^k] (n+1)!*hypergeom([-n,1-x],[1],1) for n>0. - Peter Luschny, Jan 28 2016

More terms from Alois P. Heinz, Jan 16 2011

A195326 Numerators of fractions leading to e - 1/e (A174548).

Original entry on oeis.org

0, 2, 2, 7, 7, 47, 47, 5923, 5923, 426457, 426457, 15636757, 15636757, 7318002277, 7318002277, 1536780478171, 1536780478171, 603180793741, 603180793741, 142957467201379447, 142957467201379447

Offset: 0

Paul Curtz, Oct 12 2011

The sequence of approximations of exp(1) obtained by truncating the Taylor series of exp(x) after n terms is A061354(n)/A061355(n) = 1, 2, 5/2, 8/3, 65/24, ...
A Taylor series of exp(-1) is 1, 0, 1/2, 1/3, 3/8, ... and (apart from the first 2 terms) given by A000255(n)/A001048(n). Subtracting both sequences term by term we obtain a series for exp(1) - exp(-1) = 0, 2, 2, 7/3, 7/3, 47/20, 47/20, 5923/2520, 5923/2520, 426457/181440, 426457/181440, ... which defines the numerators here.
Each second of the denominators (that is 3, 2520, 19958400, ...) is found in A085990 (where each third term, that is 60, 19958400, ...) is to be omitted.
This numerator sequence here is basically obtained by doubling entries of A051397, A009628, A087208, or A186763, caused by the standard associations between cosh(x), sinh(x) and exp(x).

			a(0) =  1  -  1;
a(1) =  2  -  0;
a(2) = 5/2 - 1/2.

Cf. A001113, A068985, A197222, A197223.

taylExp1 := proc(n)
        add(1/j!,j=0..n) ;
end proc:
A000255 := proc(n)
        if n <=1 then
                1;
        else
                n*procname(n-1)+(n-1)*procname(n-2) ;
        end if;
end proc:
A001048 := proc(n)
        n!+(n-1)! ;
end proc:
A195326 := proc(n)
        if n = 0 then
                0;
        elif n =1 then
                2;
        else
                taylExp1(n) -A000255(n-2)/A001048(n-1);
        end if;
          numer(%);
end proc:
seq(A195326(n),n=0..20) ; # R. J. Mathar, Oct 14 2011

Material meant to be placed in other sequences removed by R. J. Mathar, Oct 14 2011

A349426 Irregular triangle read by rows: T(n,k) is the number of arrangements of n labeled children with exactly k nontrivial rounds; n >= 3, 1 <= k <= floor(n/3).

Original entry on oeis.org

3, 8, 30, 144, 90, 840, 840, 5760, 7280, 45360, 66528, 7560, 403200, 657720, 151200, 3991680, 7064640, 2356200, 43545600, 82285632, 34890240, 1247400, 518918400, 1035365760, 521080560, 43243200, 6706022400, 14013679680, 8034586560, 1059458400

Offset: 3

Steven Finch, Nov 17 2021

A nontrivial round means the same as a ring or circle consisting of more than one child.

			Triangle starts:
[3]           3;
[4]           8;
[5]          30;
[6]         144,          90;
[7]         840,         840;
[8]        5760,        7280;
[9]       45360,       66528,       7560;
[10]     403200,      657720,     151200;
[11]    3991680,     7064640,    2356200;
[12]   43545600,    82285632,   34890240,    1247400;
[13]  518918400,  1035365760,  521080560,   43243200;
[14] 6706022400, 14013679680, 8034586560, 1059458400;
...
For n = 6, there are 144 ways to make one round and 90 ways to make two rounds.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999 (Sec. 5.2)

Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.

Row sums give A066165 (variant of Stanley's children's game).
Column 1 gives A001048.
Right border element of row n is A166334(n/3) for each n divisible by 3.
Cf. A066166, A349280 (correspond to Stanley's original game).

f[k_, n_] := n! SeriesCoefficient[(1 - x)^(-x t) Exp[-x^2 t], {x, 0, n}, {t, 0, k}]
Table[f[k, n], {n, 2, 14}, {k, 1, Floor[n/3]}]

E.g.f.: (1 - x)^(-x*t) * exp(-x^2*t).

A373967 Triangle read by rows: T(n,k) = (-1)^n * n! + (-1)^(k+1) * k! for n >= 2 and 1 <= k <= n-1.

Original entry on oeis.org

3, -5, -8, 25, 22, 30, -119, -122, -114, -144, 721, 718, 726, 696, 840, -5039, -5042, -5034, -5064, -4920, -5760, 40321, 40318, 40326, 40296, 40440, 39600, 45360, -362879, -362882, -362874, -362904, -362760, -363600, -357840, -403200, 3628801, 3628798, 3628806, 3628776, 3628920, 3628080, 3633840, 3588480, 3991680

Offset: 2

Mohammed Yaseen, Jun 24 2024

			Triangle begins:
      3;
     -5,    -8;
     25,    22,    30;
   -119,  -122,  -114,  -144;
    721,   718,   726,   696,   840;
  -5039, -5042, -5034, -5064, -4920, -5760;
  ...

Cf. A000142, A153805, A373966.

Unsigned diagonals: A001048, A213167.

T[n_,k_]:= (-1)^n*n! + (-1)^(k+1)*k!; Table[T[n,k],{n,2,10},{k,n-1}]// Flatten (* Stefano Spezia, Jun 24 2024 *)

Integral_{1..e} (log(x)^k - log(x)^n) dx = A373966(n,k)*e + T(n,k).

A380338 Expansion of e.g.f. log(1 - x * log(1 - x)).

Original entry on oeis.org

0, 0, 2, 3, -4, -30, 54, 1260, 3856, -36288, -279000, 2970000, 56725008, 109343520, -5495740992, -26086263840, 1293641890560, 21771049466880, -45508965806592, -4589738336217600, 10493846174810880, 2423866077943511040, 34328754265480012800, -358930542362135546880

Offset: 0

Seiichi Manyama, Jan 21 2025

sign

Cf. A052804, A052858.
Cf. A089064, A380339.
Cf. A001048, A375684.

my(N=30, x='x+O('x^N)); concat([0, 0], Vec(serlaplace(log(1-x*log(1-x)))))

a(n) = n!*sum(k=1, n\2, (-1)^(k-1)*(k-1)!*abs(stirling(n-k, k, 1))/(n-k)!);

a(n) = n! * Sum_{k=1..floor(n/2)} (-1)^(k-1) * (k-1)! * |Stirling1(n-k,k)|/(n-k)!.

a(0) = a(1) = 0; a(n) = n * (n-2)! - Sum_{k=2..n-1} k * (k-2)! * binomial(n-1,k) * a(n-k).

A052225 (n+1)!*(n+3)-3.

Original entry on oeis.org

5, 27, 141, 837, 5757, 45357, 403197, 3991677, 43545597, 518918397, 6706022397, 93405311997, 1394852659197, 22230464255997, 376610217983997, 6758061133823997, 128047474114559997, 2554547108585471997

Offset: 1

Andreas Ulvaer (aulvaer(AT)yahoo.com), Feb 20 2000

			a(2)=27 because 27=(2+1)!*(2+3)-3 or 3*2*1*5-3.

a(n) = (n+1)!*(n+3) - 3 \\ Michel Marcus, Jun 19 2013

a(n) = A001048(n+2) - 3. - Michel Marcus, Jun 19 2013

More terms from James Sellers, Feb 22 2000

A130494 Row sums of triangle A130478.

Original entry on oeis.org

1, 4, 11, 37, 163, 907, 6067, 47107, 415027, 4084147, 44363827, 526994227, 6793931827, 94451224627, 1408352613427, 22418320792627, 379413423256627, 6802709918872627, 128803497755800627, 2568107879638168627, 53780695151756440627, 1180214324937540760627

Offset: 1

Gary W. Adamson, May 31 2007

nonn

			a(5) = 163 sum of row 5 terms of triangle A130478: (120 + 30 + 8 + 3 + 2); where (30, 8, 3, 2) = the first 4 reversed terms of A001048.
a(5) = 163 = 5! + A130495(4) = 120 + 43.
a(5) = 163 = 5! + (4! + 3!) + (3! + 2!) + (2! + 1!) + (1! + 1).

Cf. A130495, A001048, A130478.

Row sums of triangle A130478. a(1) = 1; a(n), n>1 = n! + A130495(n-1). a(n) = n! + ((n-1)! + (n-2)!) + ((n-2)! + (n-3)!) + ... + (1! + 1). a(n) = n! + sum of first (n-1) terms of A001048 in reverse, where A001048 = (2, 3, 8, 30, 144, ...).

More terms from Alois P. Heinz, Dec 02 2018

cp's OEIS Frontend

A232187 Number T(n,k) of parity alternating permutations of [n] with exactly k descents from odd to even numbers; triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-1)/2)), read by rows.

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A301523 Integers which can be partitioned into two distinct factorials. 0! and 1! are not considered distinct.

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A349280 Irregular triangle read by rows: T(n,k) is the number of arrangements of n labeled children with exactly k rounds; n >= 2, 1 <= k <= floor(n/2).

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A180013 Triangular array read by rows: T(n,k) = number of fixed points in the permutations of {1,2,...,n} that have exactly k cycles; n>=1, 1<=k<=n.

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A195326 Numerators of fractions leading to e - 1/e (A174548).

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A349426 Irregular triangle read by rows: T(n,k) is the number of arrangements of n labeled children with exactly k nontrivial rounds; n >= 3, 1 <= k <= floor(n/3).

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A373967 Triangle read by rows: T(n,k) = (-1)^n * n! + (-1)^(k+1) * k! for n >= 2 and 1 <= k <= n-1.

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A380338 Expansion of e.g.f. log(1 - x * log(1 - x)).

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