cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A296943 Number of bisymmetric and quasitrivial operations on an arbitrary n-element set.

Original entry on oeis.org

0, 1, 4, 14, 58, 292, 1754, 12280, 98242, 884180, 8841802, 97259824, 1167117890, 15172532572, 212415456010, 3186231840152, 50979709442434, 866655060521380, 15599791089384842, 296396030698312000, 5927920613966240002, 124486332893291040044
Offset: 0

Views

Author

J. Devillet, Dec 22 2017

Keywords

Links

Crossrefs

Cf. A002627, A296944.

Programs

  • Mathematica
    Join[{0}, Rest[ Range[0, 22]! CoefficientList[ Series[(2 Exp[x] -3)/(1 -x), {x, 0, 22}], x]]] (* Robert G. Wilson v, Dec 22 2017 *)
nxt[{n_,a_}]:={n+1,a(n+1)+2}; Join[{0},NestList[nxt,{1,1},20][[All,2]]] (* Harvey P. Dale, Jun 09 2021 *)

Formula

E.g.f.: (2*exp(x)-3)/(1-x).
a(n+1) = (n+1)*a(n)+2, a(0)=0, a(1)=1.
a(n) ~ (2*exp(1) - 3) * n!. - Vaclav Kotesovec, Jun 05 2019

A349788 Number of permutations of [n] having exactly one increasing cycle.

Original entry on oeis.org

0, 1, 1, 1, 5, 36, 234, 1597, 12459, 111451, 1116277, 12298958, 147655760, 1919465237, 26870436345, 403044639709, 6448695657957, 109628096021612, 1973308547820586, 37492874766408001, 749857477972731979, 15747006284752049759, 346434131946498886045
Offset: 0

Views

Author

Alois P. Heinz, Nov 30 2021

Keywords

Comments

Cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1) < b(2) < ... .
Exponential convolution of A000587 with A002627.

Examples

			a(4) = 5: (1)(243), (143)(2), (142)(3), (132)(4), (1234).

Links

Crossrefs

Column k=1 of A186754.
Cf. A000587, A002627, A131719, A186755, A186758.

Programs

  • Maple
    b:= proc(n) option remember; series(`if`(n=0, 1, add((x+
     (j-1)!-1)*binomial(n-1, j-1)*b(n-j), j=1..n)), x, 2)
    end:
a:= n-> coeff(b(n), x, 1):
seq(a(n), n=0..23);
  • Mathematica
    b[n_] := b[n] = Series[If[n == 0, 1, Sum[(x+
     (j-1)!-1)*Binomial[n-1, j-1]*b[n-j], {j, 1, n}]], {x, 0, 2}];
a[n_] := Coefficient[b[n], x, 1];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Apr 15 2022, after Alois P. Heinz *)

Formula

E.g.f.: exp(1-exp(x))*(exp(x)-1)/(1-x).
a(n) = A186758(n) - A186755(n).
a(n) = Sum_{j=0..n} binomial(n,j)*A000587(j)*A002627(n-j).
a(n) mod 2 = A131719(n).
a(n) ~ (exp(1) - 1) * exp(1 - exp(1)) * n!. - Vaclav Kotesovec, Dec 05 2021

A046803 Triangle of numbers related to Eulerian numbers.

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 1, 14, 22, 4, 1, 30, 105, 65, 5, 1, 62, 416, 581, 171, 6, 1, 126, 1491, 3920, 2695, 420, 7, 1, 254, 5034, 22506, 29310, 11180, 988, 8, 1, 510, 16365, 116667, 256317, 188361, 43041, 2259, 9, 1, 1022, 51892, 564667, 1945297, 2419897, 1090135
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			Triangle begins
  1;
  1,   2;
  1,   6,    3;
  1,  14,   22,    4;
  1,  30,  105,   65,    5;
  1,  62,  416,  581,  171,   6;
  1, 126, 1491, 3920, 2695, 420, 7;
  ...

References

  • D. Singh, The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers, Math. Student, 20 (1952), 66-70.

Links

Crossrefs

Row sums give A002627.
Cf. A008292 (Eulerian numbers), A046802.

Programs

  • Mathematica
    egf = Exp[x*y]*(Exp[x]-1)*((y-1)/(y*Exp[x] - Exp[x*y])); row[n_] := Last[ CoefficientList[ Series[egf, {x, 0, n}, {y, 0, n}], {x, y}]]*n!; Flatten[ Table[ row[n], {n, 1, 10}]] (* Jean-François Alcover, Dec 20 2012, after Vladeta Jovovic *)
  • PARI
    T(n)={my(A=O(x*x^n)); [Vecrev(p) | p<-Vec(serlaplace(exp(x*y + A)*(exp(x + A)-1)*(y-1)/(y*exp(x + A)-exp(x*y + A))))]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Mar 07 2020
  • PARI
    \\ here U(n,k) is A008292.
U(n, k)={sum(j=0, k, (-1)^j * (k-j)^n * binomial( n+1, j))};
T(n, k)={sum(i=1, n, binomial(n,i)*U(n-i, k-1))} \\ Andrew Howroyd, Mar 07 2020

Formula

T(n, k) = Sum_{i=1..n} binomial(n,i) * A008292(n-i, k-1).
E.g.f.: exp(x*y)*(exp(x)-1)*(y-1)/(y*exp(x)-exp(x*y)). - Vladeta Jovovic, Sep 20 2003

Extensions

More terms from Vladeta Jovovic, Sep 20 2003

A173516 a(1)=1, a(n) = 3*n*a(n-1) + 1, n > 1.

Original entry on oeis.org

1, 7, 64, 769, 11536, 207649, 4360630, 104655121, 2825688268, 84770648041, 2797431385354, 100707529872745, 3927593665037056, 164958933931556353, 7423152026920035886, 356311297292161722529, 18171876161900247848980
Offset: 1

Views

Author

Gary Detlefs, Feb 20 2010

Keywords

Comments

If s(n) is a sequence defined as s(0) = x, s(n) = 3n*s(n-1)+k then s(n) = 3^n*n!*x + a(n)*k, n > 0.

Crossrefs

Cf. A002627, A056541.

Formula

E.g.f.: (exp(x)-1) / (1-3*x). - Kevin Ryde, Apr 16 2024

A213937 Row sums a(n) of triangle A213936: number of representative necklaces with n beads (C_N symmetry) corresponding to all color signatures given by the partitions [1^n], [2,1^(n-2)], ..., [n-1,1], [n].

Original entry on oeis.org

1, 2, 4, 11, 42, 207, 1238, 8661, 69282, 623531, 6235302, 68588313, 823059746, 10699776687, 149796873606, 2246953104077, 35951249665218, 611171244308691, 11001082397556422, 209020565553572001
Offset: 1

Views

Author

Wolfdieter Lang, Jul 10 2012

Keywords

Comments

See A213936 and A212359 for more details, references and links.

Examples

			n=4: the representative necklaces (of a color class) correspond to the color signatures c[.] c[.] c[.] c[.], c[.]^2 c[.] c[.], c[.]^3 c[.]^1 and c[.]^4 (the reverse partition order compared to Abramowitz-Stegun without 2^2). The corresponding necklaces are (we use j for color c[j]): cyclic(1234), coming in all-together 6  permutations of the present colors, cyclic(1123) coming in 3 permutions, cyclic(1112) and cyclic(1111), adding up to the 11 = a(4) necklaces. Not all 4 colors are present, except for the first signature (partition).

Crossrefs

Cf. A002627, A231936, A248669.

Formula

a(n) = A002627(n-1) + 1, n>=1.
a(n) = Sum_{k=1..n} A213936(n,k), n>=1.
a(n) = 1 + Sum_{k=1..n-1} (n-1)!/k! = 1 + A002627(n-1), n>=1.
a(n) = 1 + Sum_{k=1..n} A248669(n-1,k), n>=1. - Greg Dresden, Mar 31 2022

A296964 Expansion of e.g.f. (exp(x)-x)*x/(1-x).

Original entry on oeis.org

0, 1, 2, 9, 40, 205, 1236, 8659, 69280, 623529, 6235300, 68588311, 823059744, 10699776685, 149796873604, 2246953104075, 35951249665216, 611171244308689, 11001082397556420, 209020565553571999, 4180411311071440000, 87788637532500240021, 1931350025715005280484, 44421050591445121451155
Offset: 0

Views

Author

J. Devillet, Dec 22 2017

Keywords

Comments

Number of quasilinear weak orderings R on {1,...,n} and for which {1,...,n} has exactly one maximal element for the quasilinear weak ordering R.
Essentially the same as A038156. - R. J. Mathar, Jan 02 2018

Links

Crossrefs

Cf. A002627, A038156.

Programs

  • Mathematica
    Join[{0,1},Drop[With[{nn=30},CoefficientList[Series[(Exp[x]-x)*x/(1-x),{x,0,nn}],x] Range[0,nn]!],2]] (* Harvey P. Dale, Apr 02 2018 *)
  • Sage
    x = QQ[['x']].gen()
f = (exp(x) - x) * x / (1 - x)
f.egf_to_ogf()  # F. Chapoton, Jul 21 2025

Formula

a(n) = A002627(n)-1, a(0)=0, a(1)=1.

A319013 a(n) is the sum over each permutation of S_n of the least element of the descent set.

Original entry on oeis.org

0, 1, 7, 37, 201, 1231, 8653, 69273, 623521, 6235291, 68588301, 823059733, 10699776673, 149796873591, 2246953104061, 35951249665201, 611171244308673, 11001082397556403, 209020565553571981, 4180411311071439981, 87788637532500240001, 1931350025715005280463
Offset: 1

Views

Author

Peter Kagey, Sep 07 2018

Keywords

Comments

a(1) = 0 since the descent set of the identity permutation is empty.
Lim_{n->infinity} a(n)/n! = e - 1.

Examples

			For n = 3, the least element of the descent set for each permutation in S_3 is given by the table:
+-------------+-------------+----------------------+
| permutation | descent set | least element (or 0) |
+-------------+-------------+----------------------+
| 123         | {}          | 0                    |
| 132         | {2}         | 2                    |
| 213         | {1}         | 1                    |
| 231         | {2}         | 2                    |
| 312         | {1}         | 1                    |
| 321         | {1,2}       | 1                    |
+-------------+-------------+----------------------+
Thus a(3) = 0 + 2 + 1 + 2 + 1 + 1 = 7.

References

  • R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 2011; see Section 1.4, pp. 38.

Links

Crossrefs

Cf. A002627.

Programs

  • Mathematica
    Table[Sum[k^2*Binomial[n, k + 1]*(n - k - 1)!, {k, 1, n - 1}], {n, 1, 15}]
  • PARI
    a(n) = sum(k=1, n-1, k^2*binomial(n, k+1)*(n-k-1)!); \\ Michel Marcus, Nov 28 2019

Formula

a(n) = Sum_{k=1..n-1} k^2*binomial(n, k+1)*(n - k - 1)!.
a(n+1) = (n+1)*a(n) + n^2, with a(1) = 0.
a(n) = A002627(n) - n.

A341101 T(n, k) = Sum_{j=0..k} binomial(n, k - j)*Stirling1(n - k + j, j)*(-1)^(n-k). Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 2, 0, 1, 4, 0, 2, 6, 8, 0, 6, 19, 24, 16, 0, 24, 80, 110, 80, 32, 0, 120, 418, 615, 500, 240, 64, 0, 720, 2604, 4046, 3570, 1960, 672, 128, 0, 5040, 18828, 30604, 28777, 17360, 6944, 1792, 256, 0, 40320, 154944, 261656, 259056, 167874, 74592, 22848, 4608, 512
Offset: 0

Views

Author

Peter Luschny, Feb 09 2021

Keywords

Examples

			Triangle starts:
n\k  0     1     2     3     4     5     6     7     8 ...
0:   1
1:   0     2
2:   0     1     4
3:   0     2     6     8
4:   0     6    19    24    16
5:   0    24    80   110    80    32
6:   0   120   418   615   500   240    64
7:   0   720  2604  4046  3570  1960   672   128
8:   0  5040 18828 30604 28777 17360  6944  1792   256

Links

  • Özmen, N., Erkuş-Duman, E. (2019). On the Generalized Sylvester Polynomials. In: Lindahl, K., Lindström, T., Rodino, L., Toft, J., Wahlberg, P. (eds) Analysis, Probability, Applications, and Computation. Trends in Mathematics. Birkhäuser, Cham. See page 48.

Crossrefs

Alternating row sums: (-1)^n*(n+1) = A181983(n+1).
Cf. A000522 (row sums), A097204 (row sums - 2^n), A002627 (row sums - n!).
Cf. A000166, A340264.

Programs

  • Maple
    T := (n, k) -> add(binomial(n, k - j)*Stirling1(n - k + j, j)*(-1)^(n-k), j=0..k):
seq(print(seq(T(n,k), k = 0..n)), n = 0..9);
# Alternative:
SP := (n, x) -> (x^n)*hypergeom([-n, x], [], -1/x):
row := n -> seq(coeff(simplify(SP(n, x)), x, k), k = 0..n):
for n from 0 to 8 do row(n) od; # Peter Luschny, Nov 23 2022
  • Mathematica
    T[ n_, k_] := If[ n<0, 0, n! * Coefficient[ SeriesCoefficient[ E^(x * z) / (1 - z)^x, {z, 0, n}], x, k]]; (* Michael Somos, Nov 23 2022 *)
  • PARI
    T(n, k) = sum(j=0, k, binomial(n, k-j)*stirling(n-k+j, j, 1)*(-1)^(n-k)); \\ Michel Marcus, Feb 11 2021
  • PARI
    {T(n, k) = if( n<0, 0, n! * polcoeff( polcoeff( exp(x*y) / (1 - x + x * O(x^n))^y, n), k))}; /* Michael Somos, Nov 23 2022 */
  • Python
    from math import factorial
from sympy import Symbol, Poly
x = Symbol("x")
def Coeffs(p) -> list[int]:
    return list(reversed(Poly(p, x).all_coeffs()))
def L(n, m, x):
    if n == 0:
        return 1
    if n == 1:
        return 1 - m - 2*x
    return ((2 * (n  - x) - m - 1) * L(n - 1, m, x) / n
          - (n  - x - m - 1) * L(n - 2, m, x) / n)
def Sylvester(n):
    return (-1)**n * factorial(n) * L(n, n, x)
for n in range(7):
    print(Coeffs(Sylvester(n))) # Peter Luschny, Dec 13 2022

Formula

Sum_{k=0..n-1} T(n, k) = Sum_{k=0..n} binomial(n, k)*(k! - 1) = A097204(n).
E.g.f. for row polynomials: P(x, z) := Sum_{k>=0} T(n, k) * x^n * z^k/k! = e^(x*z) / (1 - z)^x = 1 + (2*x) * z + (x + 4*x^2) * z^2/2! + ... - Michael Somos, Nov 23 2022
From Peter Luschny, Nov 24 2022: (Start)
T(n, k) = [x^k] (x^n)*hypergeom([-n, x], [], -1/x).
T(n, k) = [x^k] (-1)^n * n! * L(n, -x - n, x), where L(n, a, x) is the n-th generalized Laguerre polynomial. (End)

A349088 a(n) = n! * Sum_{k=0..floor((n-1)/3)} 1 / (3*k+1)!.

Original entry on oeis.org

0, 1, 2, 6, 25, 125, 750, 5251, 42008, 378072, 3780721, 41587931, 499055172, 6487717237, 90828041318, 1362420619770, 21798729916321, 370578408577457, 6670411354394226, 126737815733490295, 2534756314669805900, 53229882608065923900, 1171057417377450325801
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 25 2022

Keywords

Crossrefs

Cf. A002627, A009628, A143820, A186763, A337725, A349089, A352659.

Programs

  • Mathematica
    Table[n! Sum[1/(3 k + 1)!, {k, 0, Floor[(n - 1)/3]}], {n, 0, 22}]
nmax = 22; CoefficientList[Series[(Exp[x] - 2 Exp[-x/2] Sin[(Pi - 3 Sqrt[3] x)/6])/(3 (1 - x)), {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: (exp(x) - 2 * exp(-x/2) * sin((Pi - 3*sqrt(3)*x)/6)) / (3*(1 - x)).
a(n) = floor(c * n!) for n > 0, where c = 1.041865355... = A143820.

A349089 a(n) = n! * Sum_{k=0..floor((n-1)/4)} 1 / (4*k+1)!.

Original entry on oeis.org

0, 1, 2, 6, 24, 121, 726, 5082, 40656, 365905, 3659050, 40249550, 482994600, 6278929801, 87905017214, 1318575258210, 21097204131360, 358652470233121, 6455744464196178, 122659144819727382, 2453182896394547640, 51516840824285500441, 1133370498134281009702
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 25 2022

Keywords

Crossrefs

Cf. A002627, A009628, A186763, A334363, A337728, A349088, A352660.

Programs

  • Mathematica
    Table[n! Sum[1/(4 k + 1)!, {k, 0, Floor[(n - 1)/4]}], {n, 0, 22}]
nmax = 22; CoefficientList[Series[(Sin[x] + Sinh[x])/(2 (1 - x)), {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: (sin(x) + sinh(x)) / (2*(1 - x)).
a(n) = floor(c * n!) for n > 0, where c = 1.008336089... = A334363.
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