A000023 -id:A000023 - cp's OEIS frontend

A277432 E.g.f.: sinh(sqrt(2)x)/(sqrt(2)(1-x)).

0, 1, 2, 8, 32, 164, 984, 6896, 55168, 496528, 4965280, 54618112, 655417344, 8520425536, 119285957504, 1789289362688, 28628629803008, 486686706651392, 8760360719725056, 166446853674776576, 3328937073495531520, 69907678543406162944, 1537968927954935584768

Offset: 0

Vladimir Reshetnikov, Oct 14 2016

nonn

Robert Israel, Table of n, a(n) for n = 0..449
Eric Weisstein's MathWorld, Incomplete Gamma Function

Cf. A000023, A263823, A277345, A277431.

I:=[1,2,8]; [0] cat [n le 3 select I[n] else n*Self(n-1) + 2*Self(n-2) - 2*(n-2)*Self(n-3): n in [1..30]]; // G. C. Greubel, Aug 19 2018

f:= gfun:-rectoproc({a(n) = n*a(n-1) + 2*a(n-2) - 2*(n-2)*a(n-3),a(0)=0,a(1)=1,a(2)=2},a(n),remember):
map(f, [$0..20]); # Robert Israel, Oct 30 2016

Round@Table[(Gamma[n + 1, Sqrt[2]] Exp[Sqrt[2]] - Gamma[n + 1, -Sqrt[2]]/Exp[Sqrt[2]])/(2 Sqrt[2]), {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster *)
Expand@Table[SeriesCoefficient[Sinh[Sqrt[2] x]/(Sqrt[2] (1 - x)), {x, 0, n}] n!, {n, 0, 20}]

x='x+O('x^30); concat([0], round(Vec(serlaplace(sinh(sqrt(2)*x)/( sqrt(2)*(1-x)))))) \\ G. C. Greubel, Aug 19 2018

a(n) = (Gamma(n+1, sqrt(2))*exp(sqrt(2)) - Gamma(n+1, -sqrt(2))/exp(sqrt(2))) / (2*sqrt(2)).
a(n) ~ sqrt(Pi)*sinh(sqrt(2))*n^(n+1/2)*exp(-n).
D-finite with recurrence: a(n) = n*a(n-1) + 2*a(n-2) - 2*(n-2)*a(n-3).
Gamma(n+1, sqrt(2))*exp(sqrt(2)) = A277431(n) + sqrt(2)*a(n).
Gamma(n+1, -sqrt(2))/exp(sqrt(2)) = A277431(n) - sqrt(2)*a(n).
For n > 0, a(2*n) = 2*n*a(2*n-1).

A335111 a(n) = n! * Sum_{k=0..n-1} (-2)^k / k!.

Original entry on oeis.org

0, 1, -2, 6, -8, 40, 48, 784, 5248, 49536, 490240, 5403904, 64822272, 842742784, 11798284288, 176974510080, 2831591636992, 48137058942976, 866467058614272, 16462874118651904, 329257482362552320, 6914407129635618816, 152116956851937476608, 3498690007594658430976

Offset: 0

Ilya Gutkovskiy, May 23 2020

sign

Inverse binomial transform of A000240.

Cf. A000023, A000240, A008290, A066534, A087981, A122803.

Table[n! Sum[(-2)^k/k!, {k, 0, n - 1}], {n, 0, 23}]
nmax = 23; CoefficientList[Series[Sum[k! x^k/(1 + 2 x)^(k + 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 23; CoefficientList[Series[x Exp[-2 x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!

a(n) = n! * sum(k=0, n-1, (-2)^k / k!); \\ Michel Marcus, May 23 2020

G.f.: Sum_{k>=1} k! * x^k / (1 + 2*x)^(k + 1).
E.g.f.: x*exp(-2*x) / (1 - x).
a(n) = A000023(n) - A122803(n).
a(n) ~ exp(-2) * n!. - Vaclav Kotesovec, Jun 08 2022
a(n) = Sum_{k=0..n} (-1)^k * k * A008290(n,k). - Alois P. Heinz, May 20 2023

A344901 Triangle read by rows: T(n,k) is the number of permutations of length n that have k same elements at the same positions with its inverse permutation for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 2, 0, 0, 4, 6, 8, 0, 0, 10, 24, 30, 40, 0, 0, 26, 160, 144, 180, 160, 0, 0, 76, 1140, 1120, 1008, 840, 700, 0, 0, 232, 8988, 9120, 8960, 5376, 4200, 2912, 0, 0, 764, 80864, 80892, 82080, 53760, 30240, 19656, 12768, 0, 0, 2620, 809856, 808640, 808920, 547200, 336000, 157248, 95760, 55680, 0, 0, 9496

Offset: 0

Mikhail Kurkov, Jun 01 2021

			Triangle T(n,k) begins:
     1;
     0,    1;
     0,    0,    2;
     2,    0,    0,    4;
     6,    8,    0,    0,   10;
    24,   30,   40,    0,    0,   26;
   160,  144,  180,  160,    0,    0, 76;
  1140, 1120, 1008,  840,  700,    0,  0, 232;
  8988, 9120, 8960, 5376, 4200, 2912,  0,   0, 764;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-1 give: A038205, A221145.
Row sums give A000142.
Main diagonal gives A000085.
Cf. A000023, A007318, A052571, A052849, A098558, A137482.

b:= proc(n, t) option remember; `if`(n=0, 1, add(b(n-j, t)*
      binomial(n-1, j-1)*(j-1)!, j=`if`(t=1, 1..min(2, n), 3..n)))
    end:
T:= (n, k)-> binomial(n, k)*b(k, 1)*b(n-k, 0):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Oct 28 2024

b[n_, t_] := b[n, t] = If[n == 0, 1, Sum[b[n-j, t]* Binomial[n-1, j-1]*(j-1)!, {j, If[t == 1, Range @ Min[2, n], Range[3, n]]}]];
T[n_, k_] := Binomial[n, k]*b[k, 1]*b[n-k, 0];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 24 2025, after Alois P. Heinz *)

T(n,k) = binomial(n,k)*A000085(k)*A038205(n-k).
From Alois P. Heinz, Oct 28 2024: (Start)
Sum_{k=0..n} k * T(n,k) = A052849(n) = A098558(n) for n>=2.
Sum_{k=0..n} (n-k) * T(n,k) = A052571(n).
Sum_{k=0..n} (-1)^k * T(n,k) = A000023(n).
T(n,0) + T(n,1) = A137482(n). (End)

A354941 a(n) = Sum_{k=0..n} binomial(n,k)^3 * k! * (-2)^(n-k).

Original entry on oeis.org

1, -1, -10, -2, 488, 4088, -9968, -730480, -9751936, -11540096, 2480655104, 62522038016, 680469314560, -8292439149568, -606011029669888, -15765339965278208, -183530875864317952, 4164677242501038080, 318357069130977181696, 10359690304436314505216, 176911847384965046337536

Offset: 0

Ilya Gutkovskiy, Jun 12 2022

sign

Cf. A000023, A274246, A295382, A343840, A354942.

Table[Sum[Binomial[n, k]^3 k! (-2)^(n - k), {k, 0, n}], {n, 0, 20}]
nmax = 20; CoefficientList[Series[BesselI[0, 2 Sqrt[x]] Sum[(-2)^k x^k/k!^3, {k, 0, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^3

a(n) = sum(k=0, n, binomial(n,k)^3 * k! * (-2)^(n-k)); \\ Michel Marcus, Jun 12 2022

Sum_{n>=0} a(n) * x^n / n!^3 = BesselI(0,2*sqrt(x)) * Sum_{n>=0} (-2)^n * x^n / n!^3.

A381082 Triangle T(n,k) read by rows, where the columns are the coefficients of the standard expansion of the function f(x) = (-log(1-x))^(k)exp(-mx)/k! for the case m=2.

Original entry on oeis.org

1, -2, 1, 4, -3, 1, -8, 8, -3, 1, 16, -18, 11, -2, 1, -32, 44, -20, 15, 0, 1, 64, -80, 94, 5, 25, 3, 1, -128, 272, 56, 294, 105, 49, 7, 1, 256, 112, 1868, 1596, 1169, 392, 98, 12, 1, -512, 5280, 12216, 16148, 10290, 4305, 1092, 186, 18, 1

Offset: 0

Igor Victorovich Statsenko, Feb 13 2025

			Triangle starts:
  [0]     1;
  [1]    -2,      1;
  [2]     4,     -3,       1;
  [3]    -8,      8,      -3,       1;
  [4]    16,    -18,      11,      -2,       1;
  [5]   -32,     44,     -20,      15,       0,        1;
  [6]    64,    -80,      94,       5,      25,        3,     1;
  [7]  -128,    272,      56,     294,     105,       49,     7,     1;
  [8]   256,    112,    1868,    1596,    1169,      392,    98,    12,    1;
  [9]  -512,   5280,   12216,   16148,   10290,     4305,  1092,   186,   18,     1;
  ...

Cf. A000023 (row sums).
Columns 0,1: A122803, A346397.
Triangles: for m = -3 is A327997; for m = -2 is A137346 (unsigned); for m = -1 is A094816; for m = 0 is A132393; for m = 1 is A269953.

T:=(m,n,k)->add(Stirling1(n-i,k)*binomial(n,i)*m^(i)*(-1)^(n-k), i=0..n):
m:=2:seq(print(seq(T(m,n,k), k=0..n)), n=0..9);

T(n,k) = Sum_{i=0..n} Stirling1(n-i, k)*binomial(n, i)*m^(i)*(-1)^(n-k), where m = 2.

A383380 Expansion of e.g.f. exp(-2*x) / (1-x)^4.

Original entry on oeis.org

1, 2, 8, 40, 248, 1808, 15136, 142784, 1496960, 17254144, 216740864, 2945973248, 43065951232, 673626675200, 11224114860032, 198447384666112, 3710328985124864, 73136238041563136, 1515739708283944960, 32947698735175172096, 749499782353468522496, 17806903161183314378752

Offset: 0

Seiichi Manyama, Apr 24 2025

Cf. A000261, A383344, A383378.
Cf. A000023, A052124, A087981, A383381.
Cf. A000255.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-2*x)/(1-x)^4))

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A000255.
a(n) = n! * Sum_{k=0..n} (-2)^(n-k) * binomial(k+3,3)/(n-k)!.
a(0) = 1, a(1) = 2; a(n) = (n+1)*a(n-1) + 2*(n-1)*a(n-2).
a(n) ~ sqrt(Pi) * n^(n + 7/2) / (3*sqrt(2)*exp(n+2)). - Vaclav Kotesovec, Apr 25 2025

A033167 Positions of the incrementally largest terms in the continued fraction expansion of zeta(3), offset 1 variant.

Original entry on oeis.org

1, 2, 4, 29, 63, 572, 1556, 2013, 2530, 2760, 3019, 4159, 4741, 6820, 10565, 11666, 32859, 139893, 392130, 707970, 1049722, 2081165, 14990404, 36112276, 39552835, 42710787, 199618806

Offset: 1

N. J. A. Sloane

Positions in this sequence correspond to the n-th term of A013631 at index n-1.

See A229055 for another version.

Eric Weisstein's World of Mathematics, Apery's Constant Continued Fraction

Cf. A229055 (= a(n) - 1), A013631 (continued fraction of zeta(3)), A033165, A000023, A033166.

More terms from Eric W. Weisstein, Aug 23 2000
More terms from Robert Gerbicz, Aug 22 2006
Edited (with more terms taken from A229055) by N. J. A. Sloane, Jun 16 2021
Edited for offset change in A013631. - Andrew Howroyd, Jul 10 2024

A069146 Numbers m such that m = sigma(abs(k)) - 3k, where k = sigma(m) - 3m.

Original entry on oeis.org

1248, 1596, 28272, 30240, 32760, 463296, 2178540, 12865770, 23569920, 30998250, 45532800, 142990848, 1379454720, 1912369152, 2623977450, 43861478400, 66433720320, 153003540480, 403031236608, 489622536192, 704575228896

Offset: 1

Jason Earls, Apr 08 2002

1.5*10^12 < a(22) <= 7834005405696. If 2^k-1 > 3 is a prime (A000023), then 2^(k-1)*3*19*(2^k-1) is a term. - Giovanni Resta, Dec 11 2019

			Let n = 1248. The sum of the divisors of n is 3528, so k = 3528 - 3*1248 = -216. The sum of the divisors of 216 is 600 and 600 - 3*(-216) = 1248, so 1248 is in the sequence.

Cf. A000203, A069085, A001065.

Select[Range[5*10^5], DivisorSigma[1, Abs[(k = DivisorSigma[1, #] - 3#)]] -3k == # &] (* Amiram Eldar, Dec 11 2019 *)

More terms from David Wasserman, Apr 15 2003
a(12)-a(15) from Amiram Eldar, Dec 11 2019
a(16)-a(21) from Giovanni Resta, Dec 11 2019

A080252 a(n) = na(n-1)+4a(n-2)-4(n-2)a(n-3).

Original entry on oeis.org

0, 1, 2, 10, 40, 216, 1296, 9136, 73088, 658048, 6580480, 72386304, 868635648, 11292267520, 158091745280, 2371376195584, 37942019129344, 645014325264384, 11610257854758912, 220594899240681472, 4411897984813629440

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Feb 10 2003

Cf. A000023, A010842.

A080252 := n -> (exp(2)*GAMMA(1+n,2) - exp(-2)*GAMMA(1+n,-2))/4:
seq(simplify(A080252(n)), n=0..20); # Peter Luschny, Dec 18 2017

c = CoefficientList[Series[(Sinh[z]*Cosh[z])/(1 - z), {z, 0, 25}], z]; For[n = 0, n < 25, n++; Print[c[[n]]*(n - 1)! ]]
RecurrenceTable[{a[0]==0,a[1]==1,a[2]==2,a[n]==n*a[n-1]+4a[n-2]-4(n-2)* a[n-3]}, a,{n,20}] (* Harvey P. Dale, Nov 17 2013 *)

x='x+O('x^99); concat([0], Vec(serlaplace(sinh(x)*cosh(x)/(1-x)))) \\ Altug Alkan, Dec 18 2017

E.g.f.: sinh(z)*cosh(z)/(1-z).
a(n) ~ n!*(e^2-1/e^2)/4. - Vaclav Kotesovec, Oct 13 2012
a(n) = (exp(2)*Gamma(1+n,2) - exp(-2)*Gamma(1+n,-2))/4 = (A010842(n) - A000023(n))/4. - Peter Luschny, Dec 18 2017

A188143 Binomial transform of A187848.

Original entry on oeis.org

1, 5, 29, 193, 1453, 12209, 113237, 1149241, 12675661, 151095569, 1937411429, 26614052617, 390244490749, 6087782363009, 100728768290645, 1762767028074937, 32542231109506285, 632202858036492593, 12895661952702667205

Offset: 0

Groux Roland, Mar 24 2011

nonn

a(n) is also the INVERTi transform of A010842(n+1) starting at n=2.
a(n) is also the moment of order n for the measure of density: exp(x-2) / ((Ei(x-2))^2+Pi^2) over the interval 2..infinity with Ei the exponential integral.
More generally, for every integer k, the sequence a(n,k)=int(x^n*exp(x-k) / ((Ei(x-k))^2+Pi^2), x=k..infinity) is the INVERTi transform of the sequence b(n+1,k), starting at n=2, with b(n,k)=int(x^n*exp(x-k), x=k..infinity) whose e.g.f. is exp(k*x)/(1-x).

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A000023.

with(LinearAlgebra):
c:= proc(n) option remember; add(n!/k!, k=0..n) end:
b:= n-> (-1)^(n+1) * Determinant(Matrix(n+2,
        (i, j)-> `if`(0<=i+1-j, c(i+1-j), 0))):
a:= proc(n) add(b(k) *binomial(n,k), k=0..n) end:
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 24 2011

c[n_] := c[n] = Sum[n!/k!, {k, 0, n}]; b[n_] := (-1)^(n+1)*Det[Table[If[0 <= i+1-j, c[i+1-j], 0], {i, 1, n+2}, {j, 1, n+2}]]; a[n_] := Sum[b[k] * Binomial[n, k], {k, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 28 2017, after Alois P. Heinz *)

a(n) = Integral_{x=2..oo} x^n*exp(x-2)/((Ei(x-2))^2 + Pi^2) dx.
G.f.: 1/x^2 - 3/x - Q(0)/x^2, where Q(k) = 1 - 2*x - x*(k+1)/(1-x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 19 2013
a(n) ~ exp(2) * n^2 * n!. - Vaclav Kotesovec, Nov 02 2023

cp's OEIS Frontend

A277432 E.g.f.: sinh(sqrt(2)*x)/(sqrt(2)*(1-x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A335111 a(n) = n! * Sum_{k=0..n-1} (-2)^k / k!.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

A344901 Triangle read by rows: T(n,k) is the number of permutations of length n that have k same elements at the same positions with its inverse permutation for 0 <= k <= n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A354941 a(n) = Sum_{k=0..n} binomial(n,k)^3 * k! * (-2)^(n-k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A381082 Triangle T(n,k) read by rows, where the columns are the coefficients of the standard expansion of the function f(x) = (-log(1-x))^(k)*exp(-m*x)/k! for the case m=2.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Formula

A383380 Expansion of e.g.f. exp(-2*x) / (1-x)^4.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A033167 Positions of the incrementally largest terms in the continued fraction expansion of zeta(3), offset 1 variant.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A069146 Numbers m such that m = sigma(abs(k)) - 3k, where k = sigma(m) - 3m.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A080252 a(n) = n*a(n-1)+4*a(n-2)-4*(n-2)*a(n-3).

Views

A277432 E.g.f.: sinh(sqrt(2)x)/(sqrt(2)(1-x)).

A381082 Triangle T(n,k) read by rows, where the columns are the coefficients of the standard expansion of the function f(x) = (-log(1-x))^(k)exp(-mx)/k! for the case m=2.

A080252 a(n) = na(n-1)+4a(n-2)-4(n-2)a(n-3).