A001818 -id:A001818 - cp's OEIS frontend

A297010 Expansion of e.g.f. arcsinh(x*exp(x)).

0, 1, 2, 2, -8, -76, -264, 1672, 36800, 261648, -1443680, -66164704, -792152448, 2482671424, 289529373056, 5294082629760, 1648955815936, -2474170098704128, -65494141255724544, -303927676523118080, 35926135133071923200, 1341060635191667045376

Offset: 0

Ilya Gutkovskiy, Dec 23 2017

sign

			arcsinh(x*exp(x)) = x^1/1! + 2*x^2/2! + 2*x^3/3! - 8*x^4/4! - 76*x^5/5! - 264*x^6/6! + ...

Seiichi Manyama, Table of n, a(n) for n = 0..100

Cf. A001818, A009017, A009121, A009448, A009565, A009635, A009768, A191719, A216401, A297009.

a:=series(arcsinh(x*exp(x)),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 26 2019

nmax = 21; CoefficientList[Series[ArcSinh[x Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Log[x Exp[x] + Sqrt[1 + x^2 Exp[2 x]]], {x, 0, nmax}], x] Range[0, nmax]!

first(n) = my(x='x+O('x^n)); Vec(serlaplace(asinh(exp(x)*x)), -n) \\ Iain Fox, Dec 23 2017

A353913 Product_{n>=1} 1 / (1 - a(n)*x^n/n!) = 1 + arcsin(x).

Original entry on oeis.org

1, -2, 1, -28, 29, -194, 1583, -61328, 144153, -1697262, 20127867, -191762088, 3978820221, -66586416948, 1057400360235, -58260102945024, 370244721585681, -7992573879248406, 162968423791332339, -3399970067764816824, 88052648301403014789, -2360852841450177138924

Offset: 1

Ilya Gutkovskiy, May 10 2022

sign

Cf. A001818, A353818, A353873, A353914, A353915.

nn = 22; f[x_] := Product[1/(1 - a[n] x^n/n!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - ArcSin[x], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten

A353914 Product_{n>=1} 1 / (1 - a(n)*x^n/n!) = 1 + arcsinh(x).

Original entry on oeis.org

1, -2, -1, -20, -11, 46, -547, -29840, -27351, 232818, -3258663, -29911848, -390445563, 4450393260, -84140635815, -12153983817984, -18431412645519, 286688710444842, -6436900596281679, -169286474970429624, -2208721087854287811, 41892263643715799796, -1149793471388581053219

Offset: 1

Ilya Gutkovskiy, May 10 2022

sign

Cf. A001818, A353819, A353910, A353913, A353915.

nn = 23; f[x_] := Product[1/(1 - a[n] x^n/n!), {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - 1 - ArcSinh[x], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten

A087137 a(n) is the number of permutations in the symmetric group S_n that contain an odd cycle.

Original entry on oeis.org

0, 1, 1, 6, 15, 120, 495, 5040, 29295, 362880, 2735775, 39916800, 370945575, 6227020800, 68916822975, 1307674368000, 16813959537375, 355687428096000, 5214921734397375, 121645100408832000, 2004231846526284375, 51090942171709440000, 934957186489800849375

Offset: 0

Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 18 2003

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A059838, A001818.

CoefficientList[Series[1/(1-x)-1/Sqrt[1-x^2], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Sep 21 2014 *)

x='x+O('x^33); concat(0, Vec(serlaplace(1/(1-x)-1/sqrt(1-x^2)))) \\ Michel Marcus, Sep 21 2014

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

If n is odd then a(n) = n! else a(n) = n!-((n-1)!!)^2.

Formulae and more terms from Vladeta Jovovic, Oct 31 2003

Two more terms from Michel Marcus, Sep 21 2014

A161119 Triangle read by rows: T(n,k) is the number of fixed-point-free involutions of {1,2,...,2n} having k cycles with entries of opposite parities (0 <= k <= n).

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 0, 9, 0, 6, 9, 0, 72, 0, 24, 0, 225, 0, 600, 0, 120, 225, 0, 4050, 0, 5400, 0, 720, 0, 11025, 0, 66150, 0, 52920, 0, 5040, 11025, 0, 352800, 0, 1058400, 0, 564480, 0, 40320, 0, 893025, 0, 9525600, 0, 17146080, 0, 6531840, 0, 362880, 893025, 0, 44651250, 0, 238140000, 0, 285768000, 0, 81648000, 0, 3628800

Offset: 0

Emeric Deutsch, Jun 02 2009

T(n,k) is the number of basis elements in the order-n Brauer algebra that have propagation number k. - John M. Campbell, Dec 08 2021

			T(3,1)=9 because we have (12)(35)(46), (14)(26)(35), (16)(24)(35), (23)(15)(46), (25)(13)(46), (34)(15)(26), (36)(15)(24), (45)(13)(26), (56)(13)(24).
Triangle starts:
  1;
  0,  1;
  1,  0,  2;
  0,  9,  0,  6;
  9,  0, 72,  0, 24;

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Cf. A000142, A001147, A001818, A161120, A161121, A161122.

T := proc (n, k) if `mod`(n-k, 2) = 1 then 0 else binomial(n, k)^2*factorial(k)*(product(2*j-1, j = 1 .. (1/2)*n-(1/2)*k))^2 end if end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form

dfo(n) = if (n<0, (-1)^n/dfo(-n), (2*n)! / n! / 2^n); \\ A001147
T(n,k) = if ((n-k)%2, 0, k!*binomial(n,k)^2*dfo((n-k)/2)^2);
row(n) = vector(n+1, k, T(n,k-1)) \\ Michel Marcus, Dec 09 2021

T(n,k) = k!*binomial(n,k)^2*(n-k-1)!!^2 if n-k is even; T(n,k) = 0 if n-k is odd.
Sum of row n = (2n-1)!! = A001147(n).
T(n,n) = n! = A000142(n).
T(2n,0) = A001818(n).
Sum_{k>=0} k*T(n,k) = n^2*(2n-3)!! = A161120(n).

A161121 Triangle read by rows: T(n,k) is the number of fixed-point-free involutions of {1,2,...,2n} having k cycles with entries of the same parity (0 <= k <= 2*floor(n/2)).

Original entry on oeis.org

1, 1, 2, 0, 1, 6, 0, 9, 24, 0, 72, 0, 9, 120, 0, 600, 0, 225, 720, 0, 5400, 0, 4050, 0, 225, 5040, 0, 52920, 0, 66150, 0, 11025, 40320, 0, 564480, 0, 1058400, 0, 352800, 0, 11025, 362880, 0, 6531840, 0, 17146080, 0, 9525600, 0, 893025, 3628800, 0, 81648000, 0

Offset: 0

Emeric Deutsch, Jun 02 2009

Row n contains 1 + 2*floor(n/2) terms.
Sum of row n = (2n-1)!! (A001147).
a(n,0) = n! (A000142).
a(2n,2n) = A001818(n).
Sum_{k>=0} k*T(n,k) = n*(n-1)*(2n-3)!! = A161122(n).

			T(3,2)=9 because we have (12)(35)(46), (14)(26)(35), (16)(24)(35), (23)(15)(46), (25)(13)(46), (34)(15)(26), (36)(15)(24), (45)(13)(26), (56)(13)(24).
Triangle starts:
    1;
    1;
    2,   0,   1;
    6,   0,   9;
   24,   0,  72,   0,   9;
  120,   0, 600,   0, 225;

Robert Israel, Table of n, a(n) for n = 0..10081 (rows 0 to 141, flattened)

Cf. A000142, A001147, A001818, A161119, A161120, A161122.

T := proc (n, k) if n < k then 0 elif `mod`(k, 2) = 0 then binomial(n, k)^2*factorial(n-k)*(product(2*j-1, j = 1 .. (1/2)*k))^2 else 0 end if end proc: for n from 0 to 10 do seq(T(n, k), k = 0 .. 2*floor((1/2)*n)) end do; # yields sequence in triangular form

T[n_, k_] := If[EvenQ[k], (n-k)! Binomial[n, k]^2 ((k-1)!!)^2, 0];
Table[T[n, k], {n, 0, 10}, {k, 0, 2 Quotient[n, 2]}] // Flatten (* Jean-François Alcover, Feb 01 2023 *)

T(n,k) = (n-k)!*binomial(n,k)^2*((k-1)!!)^2 if k is even; T(n,k) = 0 if k is odd.

A197037 Decimal expansion of the modified Struve L-function of order 0 at 1.

Original entry on oeis.org

7, 1, 0, 2, 4, 3, 1, 8, 5, 9, 3, 7, 8, 9, 0, 8, 8, 8, 7, 3, 8, 5, 2, 6, 6, 7, 7, 8, 1, 1, 6, 5, 0, 7, 4, 4, 9, 4, 4, 4, 0, 9, 9, 5, 8, 3, 2, 9, 7, 0, 3, 1, 9, 8, 1, 2, 3, 0, 1, 5, 1, 6, 5, 5, 7, 3, 9, 6, 9, 1, 1, 5, 6, 8, 8, 2, 3, 8, 1, 8, 0, 9, 7, 7, 3, 5, 4, 0, 3, 9, 1, 5, 7, 9, 1, 1, 7, 2, 5, 8, 3, 3, 6, 3, 6, 1, 0

Offset: 0

R. J. Mathar, Oct 08 2011

			0.710243185937890888738526...

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Chapter 12.2.
Eric Weisstein, Modified Struve Function, MathWorld

Maple
```
StruveL(0,1); evalf(%) ;
```

RealDigits[ StruveL[0, 1], 10, 107] // First (* Jean-François Alcover, Feb 20 2013 *)

L_0(1) = sum_{k>=0} 1/(2*4^k*Gamma(3/2+k)^2) = (2/Pi)*sum_{k>=1} 1/A001818(k).

A296728 Expansion of e.g.f. arcsin(x*cos(x)) (odd powers only).

Original entry on oeis.org

1, -2, -16, 8, 12672, 571264, -44351360, -12355211520, -452681248768, 478190483394560, 132554796040912896, -18854516962334277632, -27186884683859043123200, -5502410397289951851773952, 6273206188133923322747420672, 5389680791235134726930445369344

Offset: 0

Ilya Gutkovskiy, Dec 19 2017

sign

			arcsin(x*cos(x)) = x/1! - 2*x^3/3! - 16*x^5/5! + 8*x^7/7! + 12672*x^9/9! + ...

Cf. A001818, A009015, A009016, A009446, A009447, A009633, A009634, A012495, A012780, A101928, A191512, A296464, A296466, A296679, A296680, A296729, A296730, A296731, A296740.

nmax = 16; Table[(CoefficientList[Series[ArcSin[x Cos[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

first(n) = x='x+O('x^(2*n)); vecextract(Vec(serlaplace(asin(x*cos(x)))), (4^n - 1)/3) \\ Iain Fox, Dec 19 2017

a(n) = (2*n+1)! * [x^(2*n+1)] arcsin(x*cos(x)).

A296729 Expansion of e.g.f. arcsin(x*cosh(x)) (odd powers only).

Original entry on oeis.org

1, 4, 44, 1912, 156816, 21506816, 4420845376, 1271132964480, 487161448339712, 239980527068474368, 147742478026391141376, 111153314734461183924224, 100339775128577885016985600, 107037870347952811373977239552, 133204585741561810426003651444736

Offset: 0

Ilya Gutkovskiy, Dec 19 2017

nonn

			arcsin(x*cosh(x)) = x/1! + 4*x^3/3! + 44*x^5/5! + 1912*x^7/7! + 156816*x^9/9! + ...

Cf. A001818, A009015, A009016, A009446, A009447, A009633, A009634, A012495, A012780, A101928, A191512, A296464, A296466, A296679, A296680, A296728, A296730, A296731, A296740.

nmax = 15; Table[(CoefficientList[Series[ArcSin[x Cosh[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

first(n) = x='x+O('x^(2*n)); vecextract(Vec(serlaplace(asin(x*cosh(x)))), (4^n - 1)/3) \\ Iain Fox, Dec 19 2017

a(n) = (2*n+1)! * [x^(2*n+1)] arcsin(x*cosh(x)).

A302605 a(n) = n! * [x^n] exp(nx)arcsin(x).

Original entry on oeis.org

0, 1, 4, 28, 272, 3384, 51300, 917064, 18884672, 440168832, 11454902500, 329208395264, 10355322975120, 353851897861760, 13052503620917124, 516917167506777600, 21875427250996723968, 985164766018898243584, 47043119138733155306052, 2374168079889664129576960

Offset: 0

Ilya Gutkovskiy, Apr 10 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..384
N. J. A. Sloane, Transforms

Cf. A001818, A115416, A291482, A293191, A302583, A302584, A302585, A302586, A302587, A302606, A302608, A302609.

Table[n! SeriesCoefficient[Exp[n x] ArcSin[x], {x, 0, n}], {n, 0, 19}]

a(n) = Sum_{k=1..n} binomial(n,k)*(k-2)!!^2*n^(n-k)*(1-(-1)^k)/2. - Fabian Pereyra, Oct 05 2024

cp's OEIS Frontend

A297010 Expansion of e.g.f. arcsinh(x*exp(x)).

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A353913 Product_{n>=1} 1 / (1 - a(n)*x^n/n!) = 1 + arcsin(x).

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A353914 Product_{n>=1} 1 / (1 - a(n)*x^n/n!) = 1 + arcsinh(x).

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A087137 a(n) is the number of permutations in the symmetric group S_n that contain an odd cycle.

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A161119 Triangle read by rows: T(n,k) is the number of fixed-point-free involutions of {1,2,...,2n} having k cycles with entries of opposite parities (0 <= k <= n).

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A161121 Triangle read by rows: T(n,k) is the number of fixed-point-free involutions of {1,2,...,2n} having k cycles with entries of the same parity (0 <= k <= 2*floor(n/2)).

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A197037 Decimal expansion of the modified Struve L-function of order 0 at 1.

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Mathematica

Formula

A296728 Expansion of e.g.f. arcsin(x*cos(x)) (odd powers only).

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A302605 a(n) = n! * [x^n] exp(nx)arcsin(x).