A009393 -id:A009393 - cp's OEIS frontend

A296439 Expansion of e.g.f. log(1 + arctanh(x))*exp(x).

0, 1, 1, 4, 0, 53, -155, 2364, -15288, 216817, -2147215, 32932700, -433435816, 7431919285, -120703007451, 2326504612964, -44614898438480, 963118686971137, -21195404220321151, 508991484878443860, -12604990423335824688, 334199905021923072597, -9181752759370241656699, 266806716890671639953964

Offset: 0

Ilya Gutkovskiy, Dec 12 2017

sign

			E.g.f.: A(x) = x/1! + x^2/2! + 4*x^3/3! + 53*x^5/5! - 155*x^6/6! + 2364*x^7/7! - 15288*x^8/8! + ...

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A009371, A009390, A009393, A202139, A291484, A296438.

a:=series(log(1+arctanh(x))*exp(x),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 27 2019

nmax = 23; CoefficientList[Series[Log[1 + ArcTanh[x]] Exp[x], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Log[1 + (Log[1 + x] - Log[1 - x])/2] Exp[x], {x, 0, nmax}], x] Range[0, nmax]!

my(ox=O(x^30)); Vecrev(Pol(serlaplace(log(1 + atanh(x + ox)) * exp(x + ox)))) \\ Andrew Howroyd, Dec 12 2017

E.g.f.: log(1 + (log(1 + x) - log(1 - x))/2)*exp(x).

a(n) ~ -(-1)^n * (n-1)! * exp((1-exp(2))/(1+exp(2))) * ((exp(2)+1)/(exp(2)-1))^n. - Vaclav Kotesovec, Dec 21 2017

A297213 Expansion of e.g.f. log(1 + arctanh(x))*exp(-x).

Original entry on oeis.org

0, 1, -3, 10, -40, 213, -1383, 11002, -100616, 1062625, -12508067, 164543938, -2368224032, 37311284645, -634900302775, 11658800863330, -229004281334768, 4804124787023265, -106986109080667043, 2524701174424967130, -62860054802079553016, 1648303843512405478485

Offset: 0

Ilya Gutkovskiy, Dec 27 2017

sign

			log(1 + arctanh(x))*exp(-x) = x/1! - 3*x^2/2! + 10*x^3/3! - 40*x^4/4! + 213*x^5/5! - 1383*x^6/6! + ...

Robert Israel, Table of n, a(n) for n = 0..432

Cf. A009337, A009356, A009374, A009393, A202139, A296439, A297209, A297210, A297211.

S:= series(log(1+arctanh(x))*exp(-x),x,51):
seq(coeff(S,x,j)*j!,j=0..50); # Robert Israel, Jul 09 2018

nmax = 21; CoefficientList[Series[Log[1 + ArcTanh[x]] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Log[1 + (Log[1 + x] - Log[1 - x])/2] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!

A297209 Expansion of e.g.f. log(1 + arcsin(x))*exp(-x).

Original entry on oeis.org

0, 1, -3, 9, -32, 148, -853, 6027, -49576, 470624, -5005137, 59454923, -774282632, 11035740844, -169997137269, 2826070412955, -50256453936368, 954657085889760, -19247168446169665, 411277539407862707, -9269937746437524256, 220085825544691181500, -5483977295221312280757

Offset: 0

Ilya Gutkovskiy, Dec 27 2017

sign

			log(1 + arcsin(x))*exp(-x) = x/1! - 3*x^2/2! + 9*x^3/3! - 32*x^4/4! + 148*x^5/5! - 853*x^6/6! + ...

Cf. A009337, A009356, A009374, A009393, A189815, A296436, A297210, A297211, A297213.

a:=series(log(1+arcsin(x))*exp(-x),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019

nmax = 22; CoefficientList[Series[Log[1 + ArcSin[x]] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Log[1 - I Log[I x + Sqrt[1 - x^2]]] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!

x='x+O('x^99); concat([0], Vec(serlaplace(exp(-x)*log(1+asin(x))))) \\ Altug Alkan, Dec 28 2017

A297210 Expansion of e.g.f. log(1 + arcsinh(x))*exp(-x).

Original entry on oeis.org

0, 1, -3, 7, -16, 48, -213, 1027, -4856, 32512, -309377, 2527963, -16805072, 179877332, -2916171997, 32511289795, -227822369168, 3575741575680, -98643332014049, 1352701143217491, -6534261348983096, 168508582018012980, -9094443640555413357, 143341194607564099595

Offset: 0

Ilya Gutkovskiy, Dec 27 2017

sign

			log(1 + arcsinh(x))*exp(-x) = x/1! - 3*x^2/2! + 7*x^3/3! - 16*x^4/4! + 48*x^5/5! - 213*x^6/6! + ...

Cf. A009337, A009356, A009374, A009393, A296435, A296437, A297209, A297211, A297213.

a:=series(log(1+arcsinh(x))*exp(-x),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 26 2019

nmax = 23; CoefficientList[Series[Log[1 + ArcSinh[x]] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Log[1 + Log[x + Sqrt[1 + x^2]]] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!

A297211 Expansion of e.g.f. log(1 + arctan(x))*exp(-x).

Original entry on oeis.org

0, 1, -3, 6, -8, 13, -103, 462, 824, -8239, -147747, 1233518, 12148288, -127674419, -2090702391, 24495009510, 410685350032, -5514147250815, -111860639828131, 1673006899192118, 37306857729115304, -619246417449233555, -15476404474443728487, 281907759055194714206

Offset: 0

Ilya Gutkovskiy, Dec 27 2017

sign

			log(1 + arctan(x))*exp(-x) = x/1! - 3*x^2/2! + 6*x^3/3! - 8*x^4/4! + 13*x^5/5! - 103*x^6/6! + ...

Cf. A009337, A009356, A009374, A009393, A110708, A296438, A297209, A297210, A297213.

a:=series(log(1+arctan(x))*exp(-x),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 26 2019

nmax = 23; CoefficientList[Series[Log[1 + ArcTan[x]] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Log[1 + (I/2) (Log[1 - I x] - Log[1 + I x])] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!

A176668 Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial sum_{k=0..infinity} (2k+1)^nbinomial(x,k) / 2^x.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 8, 21, 10, 1, 1, 5, 45, 55, 15, 1, 1, 7, 30, 185, 120, 21, 1, 1, 70, -77, 245, 595, 231, 28, 1, 1, 72, 490, -756, 1435, 1596, 406, 36, 1, 1, -1311, 3762, -546, -2625, 6111, 3738, 666, 45, 1, 1, -1309, -11325, 35130, -20895, -1743, 20685

Offset: 0

Roger L. Bagula, Apr 23 2010

Row sums are A007051(n).

Exponential Riordan array [exp(x), log((exp(2x)+1)/2)]=[exp(x),x+log(cosh(x))]. - Paul Barry Jan 10 2011

			1;
1, 1;
1, 3, 1;
1, 6, 6, 1;
1, 8, 21, 10, 1;
1, 5, 45, 55, 15, 1;
1, 7, 30, 185, 120, 21, 1;
1, 70, -77, 245, 595, 231, 28, 1;
1, 72, 490, -756, 1435, 1596, 406, 36, 1;
1, -1311, 3762, -546, -2625, 6111, 3738, 666, 45, 1;
1, -1309, -11325, 35130, -20895, -1743, 20685, 7890, 1035, 55, 1;
Production matrix begins
1, 1,
0, 2, 1,
0, -1, 3, 1,
0, 1, -3, 4, 1,
0, -1, 4, -6, 5, 1,
0, 1, -5, 10, -10, 6, 1,
0, -1, 6, -15, 20, -15, 7, 1,
0, 1, -7, 21, -35, 35, -21, 8, 1,
0, -1, 8, -28, 56, -70, 56, -28, 9, 1
- _Paul Barry_ Jan 10 2011

Cf. A007051, A009393.

A176668 := proc(n,k) sum( (2*l+1)^n*binomial(x,l),l=0..infinity) ; simplify(%/2^x) ; coeftayl(%,x=0,k) ; end proc: # R. J. Mathar, Jan 15 2011

p[x_, n_] = Sum[(2*k + 1)^n*Binomial[x, k], {k, 0, Infinity}]/2^x ;
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];
Flatten[%]

From Peter Bala, Mar 16 2012. (Start)
The row polynomials of this triangle may be obtained by applying the operator x*d/dx repeatedly to x*(1+x^2)^n = sum {k = 0..n} binomial(n,k)*x^(2*k+1) and evaluating the result at x = 1. The first few results are:
sum {k = 0..n} (2*k+1)*binomial(n,k) = (n+1)*2^n
sum {k = 0..n} (2*k+1)^2*binomial(n,k) = (n^2+3*n^+1)*2^n
sum {k = 0..n} (2*k+1)^3*binomial(n,k) = (n^3+6*n^2+6*n+1)*2^n.
Compare with A209849. (End)

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A296439 Expansion of e.g.f. log(1 + arctanh(x))*exp(x).

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A297213 Expansion of e.g.f. log(1 + arctanh(x))*exp(-x).

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A297209 Expansion of e.g.f. log(1 + arcsin(x))*exp(-x).

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A297210 Expansion of e.g.f. log(1 + arcsinh(x))*exp(-x).

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A297211 Expansion of e.g.f. log(1 + arctan(x))*exp(-x).

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Maple

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A176668 Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial sum_{k=0..infinity} (2*k+1)^n*binomial(x,k) / 2^x.

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A176668 Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial sum_{k=0..infinity} (2k+1)^nbinomial(x,k) / 2^x.