A007758 -id:A007758 - cp's OEIS frontend

A128796 a(n) = n(n-1)2^n.

0, 0, 8, 48, 192, 640, 1920, 5376, 14336, 36864, 92160, 225280, 540672, 1277952, 2981888, 6881280, 15728640, 35651584, 80216064, 179306496, 398458880, 880803840, 1937768448, 4244635648, 9261023232, 20132659200, 43620761600, 94220845056, 202937204736, 435939180544

Offset: 0

Mohammad K. Azarian, Apr 07 2007

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

Cf. A001788, A007758, A036289.

[(n^2-n)*2^n: n in [0..30]]; // Vincenzo Librandi, Feb 10 2013

CoefficientList[Series[8 x^2/(1 - 2 x)^3, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 10 2013 *)

a(n)=n*(n-1)<Charles R Greathouse IV, Sep 24 2015

G.f.: 8*x^2/(1 - 2*x)^3. - Vincenzo Librandi, Feb 10 2013
a(n) = 8*A001788(n-1). - R. J. Mathar, Apr 26 2015
From Amiram Eldar, Jul 11 2020: (Start)
Sum_{n>=2} 1/a(n) = (1 - log(2))/2.
Sum_{n>=2} (-1)^n/a(n) = (3*log(3/2) - 1)/2. (End)
E.g.f.: 4*exp(2*x)*x^2. - Stefano Spezia, Sep 02 2024

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

A098803 a(n) = n^7 * 7^n.

Original entry on oeis.org

0, 7, 6272, 750141, 39337984, 1313046875, 32934190464, 678223072849, 12089663946752, 193010051319183, 2824752490000000, 38532504363714053, 495958345459089408, 6079641716636816419, 71493870602660352896

Offset: 0

Parthasarathy Nambi, Oct 05 2004

			a(1) = 1^7 * 7^1 = 7.
a(2) = 2^7 * 7^2 = 6272.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (56,-1372,19208,-168070,941192,-3294172,6588344,-5764801).

Cf. A007758, A062074, A062075.

[7^n*n^7: n in [0..20]]; // Vincenzo Librandi, Oct 27 2011

Table[n^7*7^n, {n, 1, 20}] (* Stefan Steinerberger, Mar 06 2006 *)

a(n)=n^7*7^n \\ Charles R Greathouse IV, Oct 07 2015

G.f.: 7*x*(117649*x^6 +2016840*x^5 +2859591*x^4 +828688*x^3 +58359*x^2 +840*x +1) / (7*x -1)^8. - Colin Barker, Apr 30 2013

More terms from Stefan Steinerberger, Mar 06 2006

Offset changed from 1 to 0 by Vincenzo Librandi, Oct 27 2011

A355234 Decimal expansion of Li_2(-1/2), the dilogarithm of (-1/2) (negated).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jun 25 2022

			-0.44841420692364620244306440591577432083426994134919...

Michael Ian Shamos, Shamos's Catalog of the Real Numbers, 2011, p. 456.
Eric Weisstein's World of Mathematics, Dilogarithm, eq. (26).
Wikipedia, Spence's function.

Cf. A001008, A002805, A007758.

Other values of Li_2: A072691, A076788, A152115, A242599, A242600.

RealDigits[PolyLog[2, -1/2], 10, 100][[1]]

-dilog(-1/2) \\ Michel Marcus, Jun 25 2022

From Shamos (2011):
Equals -Li_2(1/3) - log(3/2)^2/2.
Equals Li_2(2/3) + log(3)^2/2 - log(2)^2/2 - Pi^2/6.
Equals Li_2(1/4)/2 + log(2)^2/2 - Pi^2/12.
Equals -Sum_{k>=1} (-1)^(k+1)/(2^k*k^2) = -Sum_{k>=1} (-1)^(k+1)/A007758(k).
Equals -Sum_{k>=1} H(k)/(k*3^k), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.
Equals -Integral_{x=0..1} log(x)^2/(x+2)^2 dx.
Equals -Integral_{x>=1} log(x)^2/(2*x+1)^2 dx.
Equals Integral_{x=0..1} log(x)/(x+2) dx.
Equals -Integral_{x>=0} log(1 + exp(-x)/2) dx.

A062275 Array A(n, k) = n^k * k^n, n, k >= 0, read by antidiagonals.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 16, 3, 0, 0, 4, 72, 72, 4, 0, 0, 5, 256, 729, 256, 5, 0, 0, 6, 800, 5184, 5184, 800, 6, 0, 0, 7, 2304, 30375, 65536, 30375, 2304, 7, 0, 0, 8, 6272, 157464, 640000, 640000, 157464, 6272, 8, 0, 0, 9, 16384, 750141, 5308416, 9765625

Offset: 0

Henry Bottomley, Jul 02 2001

Here 0^0 is defined to be 1. - Wolfdieter Lang, May 27 2018

			A(3, 2) = 3^2 * 2^3 = 9*8 = 72.
The array A(n, k) begins:
n\k 0 1   2   3    4     5      6      7       8        9       10 ...
0:  1 0   0   0    0     0      0      0       0        0        0 ...
1:  0 1   2   3    4     5      6      7       8        9       10 ...
2:  0 2  16  72  256   800   2304   6272   16384    41472   102400 ...
3:  0 3  72 729 5184 30375 157464 750141 3359232 14348907 59049000 ...
...
The triangle T(n, k) begins:
n\k  0  1    2      3      4      5      6    7  8  9 ...
0:   1
1:   0  0
2:   0  1    0
3:   0  2    2      0
4:   0  3   16      3      0
5:   0  4   72     72      4      0
6:   0  5  256    729    256      5      0
7:   0  6  800   5184   5184    800      6    0
8:   0  7 2304  30375  65536  30375   2304    7  0
9:   0  8 6272 157464 640000 640000 157464 6272  8  0
... - _Wolfdieter Lang_, May 22 2018

Eric Chen, Table of n, a(n) for n = 0..5049 (first 100 antidiagonals)

Columns and rows of A, or columns and diagonals of T, include A000007, A001477, A007758, A062074, A062075 etc. Diagonals of A include A062206, A051443, A051490. Sum of rows of T are A062817(n), for n >= 1

Cf. A055651, A055652, A303990.

{{1}}~Join~Table[(#^k k^#) &[n - k], {n, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, May 24 2018 *)

t1(n)=n-binomial(round(sqrt(2+2*n)), 2)
t2(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1)
a(n)=t1(n)^t2(n)*t2(n)^t1(n) \\ Eric Chen, Jun 09 2018

From Wolfdieter Lang, May 22 2018: (Start)
As a sequence: a(n) = A003992(n)*A004248(n).
As a triangle: T(n, k) = (n-k)^k * k^(n-k), for n >= 1 and k = 1..n. (End)

A128782 a(n) = n^2*4^n.

Original entry on oeis.org

0, 4, 64, 576, 4096, 25600, 147456, 802816, 4194304, 21233664, 104857600, 507510784, 2415919104, 11341398016, 52613349376, 241591910400, 1099511627776, 4964982194176, 22265110462464, 99230924406784, 439804651110400, 1939538511396864, 8514618045497344, 37225065669984256

Offset: 0

Mohammad K. Azarian, Apr 07 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-48,64).

Cf. A036289, A007758, A086952.

[n^2*4^n: n in [0..20]]; // Vincenzo Librandi, Feb 06 2013

Table[n^2 * 4^n, {n, 0, 30}] (* Vincenzo Librandi, Feb 06 2013 *)
LinearRecurrence[{12,-48,64},{0,4,64},30] (* Harvey P. Dale, May 13 2025 *)

From R. J. Mathar, Sep 20 2011: (Start)
G.f.: 4*x*(1 + 4*x)/(1 - 4*x)^3 .
a(n) = 4*A086952(n). (End)
E.g.f.: 4*exp(4*x)*x*(1 + 4*x). - Stefano Spezia, Oct 09 2022

A128789 n^3*2^n.

Original entry on oeis.org

0, 2, 32, 216, 1024, 4000, 13824, 43904, 131072, 373248, 1024000, 2725888, 7077888, 17997824, 44957696, 110592000, 268435456, 643956736, 1528823808, 3596091392, 8388608000, 19421724672, 44660948992, 102064193536, 231928233984

Offset: 0

Mohammad K. Azarian, Apr 07 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Cf. A036289, A007758.

[n^3*2^n: n in [0..30]]; // Vincenzo Librandi, Feb 07 2013

CoefficientList[Series[2 x (1 + 8 x + 4 x^2)/(1 - 2 x)^4, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 07 2013 *)
Table[n^3 2^n,{n,0,30}] (* or *) LinearRecurrence[{8,-24,32,-16},{0,2,32,216},30] (* Harvey P. Dale, Jun 14 2013 *)

G.f.: 2*x*(1 + 8*x + 4*x^2)/(1 - 2*x)^4. - Vincenzo Librandi, Feb 07 2013
a(0)=0, a(1)=2, a(2)=32, a(3)=216, a(n)=8*a(n-1)-24*a(n-2)+ 32*a(n-3)- 16*a(n-4). - Harvey P. Dale, Jun 14 2013
E.g.f.: exp(2*x)*(2*x + 12*x^2 + 8*x^3). - Geoffrey Critzer, Aug 28 2013
Sum_{n>=1} 1/a(n) = (log(2))^3/6 - Pi^2*log(2)/12 + 7*Zeta(3)/8 = 0.53721319360804020094... . - Vaclav Kotesovec, Feb 15 2015

A146748 Numbers of the form n^k * k^n, where n,k > 1.

Original entry on oeis.org

16, 72, 256, 729, 800, 2304, 5184, 6272, 16384, 30375, 41472, 65536, 102400, 157464, 247808, 589824, 640000, 750141, 1384448, 3211264, 3359232, 5308416, 7372800, 9765625, 14348907, 16777216, 37879808, 39337984, 59049000, 84934656

Offset: 1

Howard Berman (howard_berman(AT)hotmail.com), Nov 01 2008

nonn

			2^2 * 2^2 = 16,
2^3 * 3^2 = 72.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007758, A062074, A062075, A062275.

N:= 10^20: # for terms <= N
S:= {}:
for n from 2 to ilog2(N) do
  for k from n do
    v:= n^k * k^n;
    if v > N then break fi;
    S:= S union {v};
od od:
sort(convert(S,list)); # Robert Israel, Oct 31 2023

A198404 8^n*n^8.

Original entry on oeis.org

0, 8, 16384, 3359232, 268435456, 12800000000, 440301256704, 12089663946752, 281474976710656, 5777633090469888, 107374182400000000, 1841328767004311552, 29548117155177824256, 448452706436800053248, 6490588908866265677824, 90173697372979200000000

Offset: 0

Vincenzo Librandi, Oct 27 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (72,-2304,43008,-516096,4128768,-22020096,75497472,-150994944,134217728).

Cf. A007758, A062074, A062075, A097206, A098880.

Magma
```
[8^n*n^8: n in [0..20]]
```

Table[8^n n^8,{n,0,20}] (* or *) LinearRecurrence[{72,-2304,43008,-516096,4128768,-22020096,75497472,-150994944,134217728},{0,8,16384,3359232,268435456,12800000000,440301256704,12089663946752,281474976710656},20] (* Harvey P. Dale, Apr 28 2018 *)

a(n)=8^n*n^8 \\ Charles R Greathouse IV, Jul 06 2017

G.f.: -8*x*(8*x +1)*(262144*x^6 +8060928*x^5 +16576512*x^4 +5924864*x^3 +259008*x^2 +1968*x +1) / (8*x -1)^9. - Colin Barker, Apr 30 2013

A198478 a(n) = 9^n * n^9.

Original entry on oeis.org

0, 9, 41472, 14348907, 1719926784, 115330078125, 5355700839936, 193010051319183, 5777633090469888, 150094635296999121, 3486784401000000000, 73994897046174912819, 1457274373159131021312, 26955214582765006137717

Offset: 0

Vincenzo Librandi, Oct 27 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A001017, A001019.

Cf. A007758, A062074, A062075, A097206, A098803, A098880.

Magma
```
[9^n*n^9: n in [0..20]]
```

Table[9^n*n^9, {n, 0, 20}] (* G. C. Greubel, May 17 2022 *)

[9^n*n^9 for n in (0..20)] # G. C. Greubel, May 17 2022

G.f.: 9*x*(1 + 4518*x + 1183248*x^2 + 64322586*x^3 + 1024762590*x^4 + 5210129466*x^5 + 7763290128*x^6 + 2401050438*x^7 + 43046721*x^8)/(1 - 9*x)^10. - Colin Barker, Apr 30 2013

a(n) = A001019(n)*A001017(n). - Michel Marcus, May 18 2022

cp's OEIS Frontend

A128796 a(n) = n*(n-1)*2^n.

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

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A098803 a(n) = n^7 * 7^n.

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A355234 Decimal expansion of Li_2(-1/2), the dilogarithm of (-1/2) (negated).

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A062275 Array A(n, k) = n^k * k^n, n, k >= 0, read by antidiagonals.

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A128782 a(n) = n^2*4^n.

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Magma

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A128789 n^3*2^n.

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Magma

Mathematica

A128796 a(n) = n(n-1)2^n.

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