A065446 -id:A065446 - cp's OEIS frontend

A333938 Decimal expansion of Product_{k>=1} (1 - k/2^k).

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

Offset: 0

Vaclav Kotesovec, Apr 11 2020

			0.0788983500212491810064185920122947774736728143458491250873967214687...

Cf. A048651, A065446, A277355.

evalf(Product(1 - k/2^k, k = 1..infinity), 100);

Join[{0},RealDigits[Product[1-n/2^n,{n,500}],10,120][[1]]] (* Harvey P. Dale, Jan 12 2024 *)

PARI
```
prodinf(k=1, 1 - k/2^k)
```

Equals exp(-Sum_{j>=1} polylog(-j, 1/2^j)/j).

A347830 a(n) = Sum_{k=0..n} 2^k * A000009(k) * A000041(n-k).

Original entry on oeis.org

1, 3, 8, 27, 67, 189, 509, 1329, 3344, 8694, 22062, 54756, 136741, 335103, 822277, 2016738, 4872787, 11711655, 28253743, 67319328, 160333627, 381350646, 901272326, 2121969771, 4991176893, 11689645776, 27305992220, 63705989106, 148106539514, 343371565449, 795524336390

Offset: 0

Vaclav Kotesovec, Sep 15 2021

nonn

Cf. A000009, A000041, A015128, A065446, A264686, A347829.

Table[Sum[2^k*PartitionsQ[k]*PartitionsP[n-k], {k, 0, n}], {n, 0, 50}]
nmax = 50; CoefficientList[Series[Product[(1 + 2^k*x^k) / (1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ A065446 * 2^n * A000009(n).
a(n) ~ 2^(n-2) * exp(Pi*sqrt(n/3)) / (3^(1/4) * QPochhammer(1/2) * n^(3/4)).
G.f.: Product_{k>=1} (1 + 2^k*x^k) / (1 - x^k).

A370468 Decimal expansion of Product_{k>=2} 1 / (1 - 1/k^k).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Mar 30 2024

			1.3905217844818216805690355050266316825...

Cf. A000312, A065446, A370466, A370467.

A276823 a(n) = 3 * [3n]_2! / ([2n+1]_2! * [n+1]_2!), where [n]_q! is the q-factorial.

Original entry on oeis.org

1, 9, 1241, 2634489, 87807053113, 46414431022602681, 390913823614809035461305, 52571422826552549403006580802745, 113007269646365312407427675894837602068665, 3884802624238339577626451297006421856376970743148729

Offset: 1

Vladimir Reshetnikov, Sep 18 2016

nonn

Eric Weisstein's World of Mathematics, q-Factorial, q-Binomial Coefficient.

Cf. A069777, A005329, A022166, A218449.

a:= n-> 3*mul((2^j-1), j=1..3*n)/
         (mul((2^j-1), j=1..2*n+1)*
          mul((2^j-1), j=1..n+1)):
seq(a(n), n=1..12);  # Alois P. Heinz, Sep 20 2016

Table[3 QFactorial[3 n, 2]/(QFactorial[2 n + 1, 2] QFactorial[n + 1, 2]), {n, 10}] (* or *)
Table[3 QBinomial[3 n, 2 n + 1, 2]/(1 - 3 * 2^n + 2^(2 n + 1)), {n, 10}]

a(n) ~ c * 2^((n-2)*(2*n+1)), where c = 3/QPochhammer(1/2, 1/2) = 3*A065446 = 3/A048651. - Vaclav Kotesovec, Sep 20 2016

A306947 Expansion of e.g.f. Product_{k>=0} 1/(1 - x^(2^k)/2^k).

Original entry on oeis.org

1, 1, 3, 9, 48, 240, 1620, 11340, 103320, 929880, 9865800, 108523800, 1362160800, 17708090400, 253361908800, 3800428632000, 63340477200000, 1076788112400000, 19769829743664000, 375626765129616000, 7640832869429760000, 160457490258024960000, 3559701523615997760000

Offset: 0

Ilya Gutkovskiy, Mar 17 2019

nonn

Cf. A007841, A319105.

nmax = 22; CoefficientList[Series[Product[1/(1 - x^2^k/2^k), {k, 0, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[Boole[IntegerQ[Log[2, d]]] d^(1 - k/d), {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[Sum[Boole[IntegerQ[Log[2, d]]] d^(1 - k/d), {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 22}]

a(n) ~ c * n!, where c = A065446 = 1/A048651 = 1/QPochhammer[1/2] = 3.4627466194550636115379573429244311645407579... - Vaclav Kotesovec, Mar 18 2019

cp's OEIS Frontend

A333938 Decimal expansion of Product_{k>=1} (1 - k/2^k).

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A347830 a(n) = Sum_{k=0..n} 2^k * A000009(k) * A000041(n-k).

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A370468 Decimal expansion of Product_{k>=2} 1 / (1 - 1/k^k).

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A276823 a(n) = 3 * [3*n]_2! / ([2*n+1]_2! * [n+1]_2!), where [n]_q! is the q-factorial.

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Maple

Mathematica

Formula

A306947 Expansion of e.g.f. Product_{k>=0} 1/(1 - x^(2^k)/2^k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A276823 a(n) = 3 * [3n]_2! / ([2n+1]_2! * [n+1]_2!), where [n]_q! is the q-factorial.