A355234 -id:A355234 - cp's OEIS frontend

A007758 a(n) = 2^n*n^2.

0, 2, 16, 72, 256, 800, 2304, 6272, 16384, 41472, 102400, 247808, 589824, 1384448, 3211264, 7372800, 16777216, 37879808, 84934656, 189267968, 419430400, 924844032, 2030043136, 4437573632, 9663676416, 20971520000, 45365592064, 97844723712, 210453397504

Offset: 0

David J. Snook (ua532(AT)freenet.victoria.bc.ca)

"The traveling salesman problem can be solved in time O(n^2 2^n) (where n is the size of the network to visit)." [Wikipedia] - Jonathan Vos Post, Apr 10 2006

Satisfies Benford's law [Theodore P. Hill, Personal communication, Feb 06, 2017]. - N. J. A. Sloane, Feb 08 2017

Arno Berger and Theodore P. Hill. An Introduction to Benford's Law. Princeton University Press, 2015.
Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 269.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original german edition of "Theory and Application of Infinite Series")
Wikipedia, Complexity.
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).
Index entries for sequences related to Benford's law.

Cf. A014477, A248917, A355234.

[2^n*n^2: n in [0..30]]; // Vincenzo Librandi, Oct 27 2011

seq(seq(k^n*n^k, k=2..2), n=0..25); and seq(2^n*n^2, n=0..25); # Zerinvary Lajos, Jul 01 2007

Table[n^2 * 2^n, {n, 0, 31}] (* Alonso del Arte, Oct 22 2014 *)
LinearRecurrence[{6,-12,8},{0,2,16},30] (* Harvey P. Dale, Jan 27 2017 *)

a(n)=n^2<Charles R Greathouse IV, Oct 28 2014

From Henry Bottomley, Jun 13 2001: (Start)
a(n) = 2*A014477(n-1).
G.f.: 2*x(1+2*x)/(1-2*x)^3.
Binomial transform of A002939.
Inverse binomial transform of A062189. (End)
Sum_{n>=1} 1/a(n) = Pi^2/12 - (1/2)*(log(2))^2. - Benoit Cloitre, Apr 05 2002
a(n) = Sum_{k=1..n} k*2^k. - Zerinvary Lajos, Oct 09 2006
E.g.f.: exp(2*x)*(2*x + 4*x^2). - Geoffrey Critzer, Aug 28 2013
Sum_{n>=1} (-1)^(n+1)/a(n) = -Li_2(-1/2) (A355234). - Amiram Eldar, Jun 28 2022

A076788 Decimal expansion of Sum_{m>=1} (1/(2^m*m^2)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jun 05 2003

Dilog function Li_2(1/2).

			0.5822405264650125059026563201596801087441984748...

Calvin C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Springer, 2013. See p. 221.
L. B. W. Jolley, Summation of Series, Dover (1961), eq. (116) on page 22 and eq. (360c) on page 68.
L. Lewin, Polylogarithms and Associated Functions, North Holland (1981), A2.1(4).

G. C. Greubel, Table of n, a(n) for n = 0..5000
R. Barbieri, J. A. Mignaco, and E. Remiddi, Electron form factors up to fourth order. I., Il Nuovo Cim. 11A (4) (1972) 824-864 Table I (6).
Eugène-Charles Catalan, Mémoire sur la transformation des séries et sur quelques intégrales définies, Mémoires de l'Académie royale de Belgique, 1867, Vol. 33, pp. 1-50.
Eric Weisstein's World of Mathematics, Dilogarithm

Cf. A001008, A002805, A355234.

RealDigits[ PolyLog[2, 1/2] , 10, 105] // First (* Jean-François Alcover, Feb 20 2013 *)

PARI
```
\p 200 dilog(1/2)
```
PARI
```
Pi^2/12-1/2*(log(2))^2
```

lerchphi(.5,2,1)/2 \\ Charles R Greathouse IV, Jan 30 2025

Equals 1 - (1+1/2)/2 + (1+1/2+1/3)/3 - ... [Jolley].
Equals Pi^2/12 - 1/2*(log(2))^2 [Lewin]. - Rick L. Shepherd, Jul 21 2004
From Amiram Eldar, Aug 15 2020: (Start)
Equals Sum_{k>=1} (-1)^(k+1)*H(k)/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.
Equals Integral_{x=0..1} log(1+x)/(x*(1+x)) dx. (End)
From Peter Bala, Aug 18 2024: (Start)
Equals Integral_{x = 0..1} (log(2) - log(1 + x))/(1 - x) dx. See Catalan, Section 51, but note error in equation 94.
Note that Pi^2/12 + 1/2*(log(2))^2 = Integral_{x >= 1} log(1 + x)/(x*(1 + x)) dx. (End)

A014477 Expansion of (1 + 2x)/(1 - 2x)^3.

Original entry on oeis.org

1, 8, 36, 128, 400, 1152, 3136, 8192, 20736, 51200, 123904, 294912, 692224, 1605632, 3686400, 8388608, 18939904, 42467328, 94633984, 209715200, 462422016, 1015021568, 2218786816, 4831838208, 10485760000, 22682796032, 48922361856, 105226698752, 225754218496

Offset: 0

N. J. A. Sloane

The sequence 0,1,8,... has a(n) = n^2*2^(n-1) and is the binomial transform of the hexagonal numbers A000384 (with leading 0). - Paul Barry, Jun 09 2003
As 0,1,8,... this is n^2*2^(n-1), the binomial transform of the hexagonal numbers A000384 (include the leading 0). Partial sums are A036826. - Paul Barry, Jun 10 2003
Sequence gives total value of all possible sums of distinct odd integers with maximum term less than 2n+1. E.g., for a(3) we can have the sums 1, 3, 5, 1+3, 1+5, 3+5, 1+3+5, which sum to 1+3+5+4+6+8+9 = 36. - Jon Perry, Feb 06 2004
Number of edges on a partially truncated (n+1)-cube (column 2 of A271316).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A007758, A036826, A118414, A355234.

[(n+1)^2*2^n: n in [0..35]]; // Vincenzo Librandi, Aug 21 2011

a:=n->sum(binomial(n,j)*n*j,j=0..n): seq(a(n), n=0..25); # Zerinvary Lajos, Oct 19 2006
a:=n->sum(n*numbcomb(n)/2, j=1..n): seq(a(n), n=1..25); # Zerinvary Lajos, Apr 25 2007

f[n_]:=(n^2*2^n)/2;Table[f[n],{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Dec 05 2009 *)

a(n)=(n+1)^2*2^n \\ Charles R Greathouse IV, Apr 07 2016

O.g.f.: (1 + 2*x)/(1 - 2*x)^3 (see the name).
a(n) = (n+1)^2*2^n = A007758(n+1)/2. - Henry Bottomley, Jun 13 2001
The binomial transform of 0, 1, 8, ... is A077616. - Paul Barry, Jul 24 2003
a(1)=1, a(n) = 2a(n-1) + (2n-1)*2^(n-1). - Jon Perry, Feb 06 2004
a(n) = sum of (n+1)-th row of the triangle in A118416. - Reinhard Zumkeller, Apr 27 2006
a(n) = Sum_{j=0..n} binomial(n,j)*n*j. - Zerinvary Lajos, Oct 19 2006
E.g.f.: exp(2*x)*(1 + 6*x + 8*x^2/2!). - Wolfdieter Lang, Jul 29 2017
Sum_{n>=0} 1/a(n) = Pi^2/6 - log(2)^2. - Daniel Suteu, Oct 31 2017
Sum_{n>=0} (-1)^n/a(n) = -2 *  Li_2(-1/2) = -2 * A355234. - Amiram Eldar, Oct 01 2022

A370337 Expansion of Product_{n>=1} (1 - 2^(n-1)x^n) (1 + 2^(n-1)*x^n)^2.

Original entry on oeis.org

1, 1, 1, 5, 6, 22, 40, 108, 192, 536, 1072, 2528, 5344, 12288, 26624, 61312, 129024, 286720, 646656, 1389568, 3028992, 6717440, 14708736, 31604736, 69763072, 150110208, 329809920, 714473472, 1546649600, 3324772352, 7332954112, 15626403840, 33840693248, 73194799104, 158456610816

Offset: 0

Paul D. Hanna, Feb 26 2024

nonn

Compare to Product_{n>=1} (1 - 2^n*x^n) * (1 + 2^n*x^n)^2 = Sum_{n>=0} 2^(n*(n+1)/2) * x^(n*(n+1)/2).

			G.f.: A(x) = 1 + x + x^2 + 5*x^3 + 6*x^4 + 22*x^5 + 40*x^6 + 108*x^7 + 192*x^8 + 536*x^9 + 1072*x^10 + 2528*x^11 + 5344*x^12 + ...
where A(x) is the series expansion of the infinite product given by
A(x) = (1 - x)*(1 + x)^2 * (1 - 2*x^2)*(1 + 2*x^2)^2 * (1 - 4*x^3)*(1 + 4*x^3)^2 * (1 - 8*x^4)*(1 + 8*x^4)^2 * ... * (1 - 2^(n-1)*x^n)*(1 + 2^(n-1)*x^n)^2 * ...
Compare A(x) to the series that results from a similar infinite product:
(1 - 2*x)*(1 + 2*x)^2 * (1 - 4*x^2)*(1 + 4*x^2)^2 * (1 - 8*x^3)*(1 + 8*x^3)^2 * (1 - 16*x^4)*(1 + 16*x^4)^2 * ... = 1 + 2*x + 8*x^3 + 64*x^6 + 1024*x^10 + 32768*x^15 + 2097152*x^21 + 268435456*x^28 + 68719476736*x^36 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..1035

Cf. A006125, A075900, A304961, A337299, A355234, A370338.

CoefficientList[Series[8*QPochhammer[1/2, 2*x] * QPochhammer[-1/2, 2*x]^2/9, {x, 0, 40}], x] (* Vaclav Kotesovec, Feb 26 2024 *)

{a(n) = polcoeff( prod(k=1,n, (1 - 2^(k-1)*x^k) * (1 + 2^(k-1)*x^k)^2 +x*O(x^n)), n)}
for(n=0,40, print1(a(n),", "))

a(n) ~ c^(1/4) * 2^n * exp(sqrt(c*n)) / (3*sqrt(Pi)*n^(3/4)), where c = 2*log(2)^2 - Pi^2/3 - 8*polylog(2,-1/2) = Pi^2 + 6*log(2)^2 + 8*polylog(2,-2) = 1.258351549529119595933889966687474131697... - Vaclav Kotesovec, Feb 26 2024

A367732 Decimal expansion of Sum_{k>=1} (-1)^(k+1) / (k^2 * k!).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Nov 28 2023

			0.89121279811130237606985786245535...

Cf. A002775, A163931, A239069, A355234, A367731.

RealDigits[HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -1], 10, 100][[1]]

cp's OEIS Frontend

A007758 a(n) = 2^n*n^2.

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A076788 Decimal expansion of Sum_{m>=1} (1/(2^m*m^2)).

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A014477 Expansion of (1 + 2*x)/(1 - 2*x)^3.

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A370337 Expansion of Product_{n>=1} (1 - 2^(n-1)*x^n) * (1 + 2^(n-1)*x^n)^2.

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A367732 Decimal expansion of Sum_{k>=1} (-1)^(k+1) / (k^2 * k!).

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Mathematica

A014477 Expansion of (1 + 2x)/(1 - 2x)^3.

A370337 Expansion of Product_{n>=1} (1 - 2^(n-1)x^n) (1 + 2^(n-1)*x^n)^2.