A003727 -id:A003727 - cp's OEIS frontend

A009189 Expansion of e.g.f.: exp(cos(x)*x).

1, 1, 1, -2, -11, -24, 61, 624, 1737, -7424, -88679, -242560, 2086525, 23499776, 45950997, -1002251264, -9763133167, -2151563264, 705668046769, 5583112077312, -17356978593659, -666018502836224, -3823112141007763, 39230927775531008, 788728947108214489

Offset: 0

R. H. Hardin

Seiichi Manyama, Table of n, a(n) for n = 0..560

Cf. A003727, A352252.

With[{nn=30},CoefficientList[Series[Exp[Cos[x]*x],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 15 2018 *)

a(n):=(sum(binomial(n,k)*(-1)^((n-k)/2)*(1+(-1)^(n-k))/(2^(k))*sum(binomial(k,i)*(k-2*i)^(n-k),i,0,floor((k-1)/2)),k,1,n-1))+1; /* Vladimir Kruchinin, Apr 21 2011 */

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x*cos(x)))) \\ Seiichi Manyama, Mar 26 2022

a(n) = (sum(k=1..n-1, binomial(n,k)*(-1)^((n-k)/2)*(1+(-1)^(n-k))/(2^(k))*sum(i=0..floor((k-1)/2)), binomial(k,i)*(k-2*i)^(n-k)))+1. - Vladimir Kruchinin, Apr 21 2011

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-1)^k * binomial(n-1,2*k) * (2*k+1) * a(n-2*k-1). - Ilya Gutkovskiy, Mar 10 2022

Extended with signs by Olivier Gérard, Mar 15 1997

Definition clarified and prior Mathematica program replaced by Harvey P. Dale, Mar 15 2018

A274804 The exponential transform of sigma(n).

Original entry on oeis.org

1, 1, 4, 14, 69, 367, 2284, 15430, 115146, 924555, 7991892, 73547322, 718621516, 7410375897, 80405501540, 914492881330, 10873902417225, 134808633318271, 1738734267608613, 23282225008741565, 323082222240744379, 4638440974576329923, 68794595993688306903

Offset: 0

Johannes W. Meijer, Jul 27 2016

nonn

The exponential transform [EXP] transforms an input sequence b(n) into the output sequence a(n). The EXP transform is the inverse of the logarithmic transform [LOG], see the Weisstein link and the Sloane and Plouffe reference. This relation goes by the name of Riddell's formula. For information about the logarithmic transform see A274805. The EXP transform is related to the multinomial transform, see A274760 and the second formula.
The definition of the EXP transform, see the second formula, shows that n >= 1. To preserve the identity LOG[EXP[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the exponential transform, otherwise information about b(0) will be lost in transformation.
In the a(n) formulas, see the examples, the multinomial coefficients A178867 appear.
We observe that a(0) = 1 and provides no information about any value of b(n), this notwithstanding it is customary to start the a(n) sequence with a(0) = 1.
The Maple programs can be used to generate the exponential transform of a sequence. The first program uses a formula found by Alois P. Heinz, see A007446 and the first formula. The second program uses the definition of the exponential transform, see the Weisstein link and the second formula. The third program uses information about the inverse of the exponential transform, see A274805.
Some EXP transform pairs are, n >= 1: A000435(n) and A065440(n-1); 1/A000027(n) and A177208(n-1)/A177209(n-1); A000670(n) and A075729(n-1); A000670(n-1) and A014304(n-1); A000045(n) and A256180(n-1); A000290(n) and A033462(n-1); A006125(n) and A197505(n-1); A053549(n) and A198046(n-1); A000311(n) and A006351(n); A030019(n) and A134954(n-1); A038048(n) and A053529(n-1); A193356(n) and A003727(n-1).

			Some a(n) formulas, see A178867:
a(0) = 1
a(1) = x(1)
a(2) = x(1)^2 + x(2)
a(3) = x(1)^3 + 3*x(1)*x(2) + x(3)
a(4) = x(1)^4 + 6*x(1)^2*x(2) + 4*x(1)*x(3) + 3*x(2)^2 + x(4)
a(5) = x(1)^5 + 10*x(1)^3*x(2) + 10*x(1)^2*x(3) + 15*x(1)*x(2)^2 + 5*x(1)*x(4) + 10*x(2)*x(3) + x(5)

Frank Harary and Edgar M. Palmer, Graphical Enumeration, 1973.
Robert James Riddell, Contributions to the theory of condensation, Dissertation, University of Michigan, Ann Arbor, 1951.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

Alois P. Heinz, Table of n, a(n) for n = 0..531
M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers, Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric W. Weisstein MathWorld, Exponential Transform.

Cf. A274805, A274760, A178867, A000203, A007446.
Cf. A177208, A177209, A006351, A197505, A144180, A256180, A033462, A198046, A134954, A145460, A188489, A005432, A029725, A124213, A002801.
Cf. A036040, another version.

nmax:=21: with(numtheory): b := proc(n): sigma(n) end: a:= proc(n) option remember; if n=0 then 1 else add(binomial(n-1, j-1) * b(j) *a(n-j), j=1..n) fi: end: seq(a(n), n=0..nmax); # End first EXP program.
nmax:= 21: with(numtheory): b := proc(n): sigma(n) end: t1 := exp(add(b(n)*x^n/n!, n=1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n!*coeff(t2, x, n) end: seq(a(n), n=0..nmax); # End second EXP program.
nmax:=21: with(numtheory): b := proc(n): sigma(n) end: f := series(log(1+add(q(n)*x^n/n!, n=1..nmax+1)), x, nmax+1): d := proc(n): n!*coeff(f, x, n) end: a(0):=1: q(0):=1: a(1):=b(1): q(1):=b(1): for n from 2 to nmax+1 do q(n) := solve(d(n)-b(n), q(n)): a(n):=q(n): od: seq(a(n), n=0..nmax); # End third EXP program.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n-1, j-1]*DivisorSigma[1, j]*a[n-j], {j, 1, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 22 2017 *)
nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k]*x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 08 2021 *)

a(n) = Sum_{j=1..n} (binomial(n-1,j-1) * b(j) * a(n-j)), n >= 1 and a(0) = 1, with b(n) = A000203(n) = sigma(n).

E.g.f.: exp(Sum_{n >= 1} b(n)*x^n/n!) with b(n) = sigma(n) = A000203(n).

A351937 Expansion of e.g.f. exp( (sinh(x) + x*cosh(x)) / 2 ).

Original entry on oeis.org

1, 1, 1, 3, 9, 24, 99, 418, 1769, 9320, 49541, 278912, 1764825, 11319784, 77850287, 570610472, 4290387409, 34316005632, 285335249065, 2455224885440, 22165590003849, 206191758121856, 1989511661589435, 19903718061574144, 204795484665487865, 2179948112062667392

Offset: 0

Ilya Gutkovskiy, Feb 26 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..568

Cf. A000248, A003724, A003727, A349103, A349105.

nmax = 25; CoefficientList[Series[Exp[(Sinh[x] + x Cosh[x])/2], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, 2 k] (k + 1) a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 25}]

my(x='x+O('x^30)); Vec(serlaplace(exp((sinh(x) + x*cosh(x))/2))) \\ Michel Marcus, Feb 26 2022

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

A354518 Expansion of e.g.f. cosh(x)^exp(x).

Original entry on oeis.org

1, 0, 1, 3, 7, 30, 166, 798, 4117, 27660, 196756, 1328448, 9866407, 86205210, 759842266, 6460661028, 60841732777, 651349676280, 6795873687496, 67981177154688, 770224145659627, 9854500496860470, 116983085896035646, 1301594922821009028, 17440543467561038557

Offset: 0

Seiichi Manyama, Aug 16 2022

sign

a(39) is negative. - Vaclav Kotesovec, Aug 17 2022

Seiichi Manyama, Table of n, a(n) for n = 0..486

Cf. A000248, A003727, A215518, A354517, A354520.

my(N=30, x='x+O('x^N)); Vec(serlaplace(cosh(x)^exp(x)))

a354520(n) = sum(k=1, n\2, (16^k-4^k)*bernfrac(2*k)/(2*k)*binomial(n, 2*k));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354520(j)*binomial(i-1, j-1)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} A354520(k) * binomial(n-1,k-1) * a(n-k).

A381273 Expansion of e.g.f. exp(x * cosh(2*x)).

Original entry on oeis.org

1, 1, 1, 13, 49, 201, 2161, 12629, 102817, 1118161, 9109921, 105660765, 1223720785, 13461561881, 186666204817, 2406325357861, 33607592404033, 516511765519521, 7658010172957249, 126206019752173997, 2115466479287184241, 36218229615683409001, 666810643855970901937

Offset: 0

Seiichi Manyama, Feb 18 2025

sign

As stated in the comment of A185951, A185951(n,0) = 0^n.

Cf. A003727, A381274.

Cf. A185951.

a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, 2^(n-k)*a185951(n, k));

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 4^k * (2*k+1) * binomial(n-1,2*k) * a(n-2*k-1).

a(n) = Sum_{k=0..n} 2^(n-k) * A185951(n,k).

A381274 Expansion of e.g.f. exp(x * cosh(3*x)).

Original entry on oeis.org

1, 1, 1, 28, 109, 676, 10261, 65584, 881497, 11930896, 122708521, 2186539840, 30542901445, 477545743936, 9168255077437, 149358238356736, 3043023842477233, 61000460650291456, 1225825910880514129, 28395625697194028032, 621110654837608378141, 14936817377079335166976

Offset: 0

Seiichi Manyama, Feb 18 2025

nonn

As stated in the comment of A185951, A185951(n,0) = 0^n.

Cf. A003727, A381273.

Cf. A185951.

a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, 3^(n-k)*a185951(n, k));

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 9^k * (2*k+1) * binomial(n-1,2*k) * a(n-2*k-1).

a(n) = Sum_{k=0..n} 3^(n-k) * A185951(n,k).

A191509 E.g.f. exp(x*sqrt(1+sin(x)^2)).

Original entry on oeis.org

1, 1, 1, 4, 13, -4, -59, 848, 1625, -57968, -82679, 5307072, 3378277, -761466432, -178851763, 155538255616, 13323839409, -43026868334336, -1145167641071, 15502018794828800, 110592144624061, -7038075176027079680, -12523284027203883, 3925127762389637074944, 1643266949074714633, -2635567108489125092225024

Offset: 0

Vladimir Kruchinin, Jun 04 2011

sign

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A003727.

CoefficientList[Series[E^(x*Sqrt[1+Sin[x]^2]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 04 2014 *)

a(n):=2*sum(binomial(n,n-2*j)*sum(4^(j-k)*binomial((n-2*j)/2,k)*sum((i-k)^(2*j)*binomial(2*k,i)*(-1)^(j+k-i),i,0,k-1),k,0,j),j,1,(n-1)/2)+1;

a(n)=2*sum(j=1..(n-1)/2, binomial(n,n-2*j)*sum(k=0..j, 4^(j-k)*binomial((n-2*j)/2,k)*sum(i=0..k-1, (i-k)^(2*j)*binomial(2*k,i)*(-1)^(j+k-i))))+1.

If n is odd, then a(n) ~ -sin(Pi*n/2) * 2^(5/4) * log(1+sqrt(2))^(3/2-n) * n^(n-1) / exp(n). If n is even, then limit n->infinity (|a(n)| / (n! * exp(w*cosh(w)) / w^n))^(1/n) = 1, where w = 2*LambertW(sqrt(n/2)). - Vaclav Kotesovec, Aug 05 2014

A352254 Expansion of e.g.f. exp( x * sinh(x) / 2 ) (even powers only).

Original entry on oeis.org

1, 1, 5, 48, 753, 16880, 507579, 19509042, 927229553, 53126200872, 3597373129635, 283321938437318, 25614466939850169, 2629191169850594388, 303549146372282854883, 39103024746814973908890, 5581172267077778765676129, 877211696663645448333041072, 151002471269513108372760683523

Offset: 0

Ilya Gutkovskiy, Mar 09 2022

nonn

Cf. A000248, A000807, A003724, A003727, A005046, A009233, A351937, A352253.

nmax = 36; Take[CoefficientList[Series[Exp[x Sinh[x]/2], {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[2 n - 1, 2 k - 1] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

my(x='x+O('x^40), v=Vec(serlaplace(exp(x*sinh(x)/2)))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Mar 10 2022

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

A381140 Expansion of e.g.f. exp( -LambertW(-x * cosh(x)) ).

Original entry on oeis.org

1, 1, 3, 19, 161, 1781, 24667, 409991, 7959233, 176920489, 4432942931, 123648692795, 3800647961761, 127654261471517, 4651982506605995, 182824074836850991, 7708128977570816129, 347059689259637711441, 16621016953663100702755, 843658152872351669816675

Offset: 0

Seiichi Manyama, Feb 15 2025

nonn

As stated in the comment of A185951, A185951(n,0) = 0^n.

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A003727, A162649, A381143.

Cf. A185951.

a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, (k+1)^(k-1)*a185951(n, k));

E.g.f. A(x) satisfies A(x) = exp( x * cosh(x) * A(x) ).

a(n) = Sum_{k=0..n} (k+1)^(k-1) * A185951(n,k).

A381143 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * cosh(x)) ).

Original entry on oeis.org

1, 1, 3, 19, 185, 2381, 38227, 739271, 16752465, 435437209, 12772234211, 417396070235, 15040805940745, 592531894182437, 25336144876513395, 1168670193628654351, 57845446906144852769, 3058248577410499021361, 172007282950136451003331, 10255035157348348977955619

Offset: 0

Seiichi Manyama, Feb 15 2025

nonn

As stated in the comment of A185951, A185951(n,0) = 0^n.

Index entries for reversions of series

Cf. A003727, A162649, A381140.

Cf. A185951.

a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, (n+1)^(k-1)*a185951(n, k));

E.g.f. A(x) satisfies A(x) = exp( x * A(x) * cosh(x * A(x)) ).

a(n) = Sum_{k=0..n} (n+1)^(k-1) * A185951(n,k).

cp's OEIS Frontend

A009189 Expansion of e.g.f.: exp(cos(x)*x).

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A274804 The exponential transform of sigma(n).

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A351937 Expansion of e.g.f. exp( (sinh(x) + x*cosh(x)) / 2 ).

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A354518 Expansion of e.g.f. cosh(x)^exp(x).

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A381273 Expansion of e.g.f. exp(x * cosh(2*x)).

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A381274 Expansion of e.g.f. exp(x * cosh(3*x)).

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A191509 E.g.f. exp(x*sqrt(1+sin(x)^2)).

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A352254 Expansion of e.g.f. exp( x * sinh(x) / 2 ) (even powers only).

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