A335643 -id:A335643 - cp's OEIS frontend

A247551 Decimal expansion of Product_{k>=2} 1/(1-1/k!).

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

Offset: 1

Vaclav Kotesovec, Sep 19 2014

			2.5294774720791526481801161542539542411787023948459973375849349825...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 269.
A. Knopfmacher, A. M. Odlyzko, B. Pittel, L. B. Richmond, D. Stark, G. Szekeres, and N. C. Wormald, The Asymptotic Number of Set Partitions with Unequal Block Sizes, The Electronic Journal of Combinatorics, Volume 6 (1999), Research Paper #R2.

Cf. A005651, A035341, A072605, A140585, A183239, A209668, A261600, A282529, A292713, A320566, A327827, A335643.

evalf(1/product(1-1/k!,k=2..infinity),100); # Vaclav Kotesovec, Sep 19 2014

digits = 105;
RealDigits[NProduct[1/(1-1/k!), {k, 2, Infinity}, WorkingPrecision -> digits+10, NProductFactors -> digits], 10, digits][[1]] (* Jean-François Alcover, Nov 02 2020 *)

default(realprecision,150); 1/prodinf(k=2,1 - 1/k!) \\ Vaclav Kotesovec, Sep 21 2014

Product_{k>=2} 1/(1-1/k!).
Equals lim_{n -> oo} A005651(n) / n!.
Equals 1/A282529. - Amiram Eldar, Sep 15 2023

A335636 Expansion of e.g.f. Product_{k>0} 1/(1 - tan(x)^k / k).

Original entry on oeis.org

1, 1, 3, 13, 80, 560, 4972, 48060, 552632, 6813560, 95846728, 1435488184, 23855755040, 419889384096, 8048166402304, 162616435301824, 3531256457687168, 80497793591765120, 1953028123616286592, 49561115477458450560, 1328614915154244276224, 37134707962379971432448

Offset: 0

Seiichi Manyama, Oct 03 2020

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..430

Cf. A000182, A007841, A335627, A335635, A335638, A335643.

max = 21; Range[0, max]! * CoefficientList[Series[Product[1/(1 - Tan[x]^k/k), {k, 1, max}], {x, 0, max}], x] (* Amiram Eldar, Oct 03 2020 *)

N=40; x='x+O('x^N); Vec(serlaplace(1/prod(k=1, N, 1-tan(x)^k/k)))

N=40; x='x+O('x^N); Vec(serlaplace(exp(sum(i=1, N, sum(j=1, N\i, tan(x)^(i*j)/(i*j^i))))))

E.g.f.: exp( Sum_{i>0} Sum_{j>0} tan(x)^(i*j)/(i*j^i) ).

Conjecture: a(n) ~ A080130 * n * 2^(2*n+2) * n! / Pi^(n+2). - Vaclav Kotesovec, Oct 04 2020

A335642 Expansion of e.g.f. Product_{k>0} 1/(1 - sin(x)^k / k!).

Original entry on oeis.org

1, 1, 3, 9, 35, 147, 710, 3780, 21391, 136063, 932190, 6887232, 55902274, 497726270, 4711586833, 47692742905, 528539419087, 6093676850975, 73010887114406, 943925266298096, 12740929019736310, 175037826035276730, 2561985529052306447, 39817440376814520907, 622315443336146270858

Offset: 0

Seiichi Manyama, Oct 03 2020

sign

a(74) is negative. - Vaclav Kotesovec, Oct 04 2020

Cf. A005651, A335626, A335635, A335643, A335644.

max = 24; Range[0, max]! * CoefficientList[Series[Product[1/(1 - Sin[x]^k/k!), {k, 1, max}], {x, 0, max}], x] (* Amiram Eldar, Oct 04 2020 *)

N=40; x='x+O('x^N); Vec(serlaplace(1/prod(k=1, N, 1-sin(x)^k/k!)))

N=40; x='x+O('x^N); Vec(serlaplace(exp(sum(i=1, N, sum(j=1, N\i, sin(x)^(i*j)/(i*j!^i))))))

E.g.f.: exp( Sum_{i>0} Sum_{j>0} sin(x)^(i*j)/(i*(j!)^i) ).

A336046 Expansion of e.g.f. Product_{k>0} (1 + tan(x)^k / k!).

Original entry on oeis.org

1, 1, 1, 6, 13, 112, 418, 4025, 23773, 237256, 2022526, 20878803, 236842838, 2567676659, 36410743437, 419956671339, 7116408372829, 87937527652592, 1724613303370022, 22889017703271151, 507452662263001722, 7236316297556572973, 178035555403835890935, 2728137658918521763201

Offset: 0

Seiichi Manyama, Oct 03 2020

nonn

Cf. A007837, A335630, A335638, A335643, A335644.

max = 23; Range[0, max]! * CoefficientList[Series[Product[1 + Tan[x]^k/k!, {k, 1, max}], {x, 0, max}], x] (* Amiram Eldar, Oct 04 2020 *)

N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, 1+tan(x)^k/k!)))

N=40; x='x+O('x^N); Vec(serlaplace(exp(sum(i=1, N, sum(j=1, N\i, (-1)^(i+1)*tan(x)^(i*j)/(i*j!^i))))))

E.g.f.: exp( Sum_{i>0} Sum_{j>0} (-1)^(i+1)*tan(x)^(i*j)/(i*(j!)^i) ).

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A247551 Decimal expansion of Product_{k>=2} 1/(1-1/k!).

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A335636 Expansion of e.g.f. Product_{k>0} 1/(1 - tan(x)^k / k).

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A335642 Expansion of e.g.f. Product_{k>0} 1/(1 - sin(x)^k / k!).

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A336046 Expansion of e.g.f. Product_{k>0} (1 + tan(x)^k / k!).

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