A007758 -id:A007758 - cp's OEIS frontend

A198479 a(n) = 10^n * n^10.

0, 10, 102400, 59049000, 10485760000, 976562500000, 60466176000000, 2824752490000000, 107374182400000000, 3486784401000000000, 100000000000000000000, 2593742460100000000000, 61917364224000000000000

Offset: 0

Vincenzo Librandi, Oct 27 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..900

Cf. A008454, A011557.

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

Magma
```
[10^n*n^10: n in [0..20]]
```

Table[10^n*n^10, {n,0,20}] (* G. C. Greubel, May 17 2022 *)

[10^n*n^10 for n in (0..20)] # G. C. Greubel, May 17 2022

G.f.: 10*x*(1 + 10*x)*(1 + 10120*x + 4682800*x^2 + 408364000*x^3 + 9019900000*x^4 + 40836400000*x^5 + 46828000000*x^6 + 10120000000*x^7 + 100000000*x^8)/ (1-10*x)^11. - Colin Barker, May 01 2013

a(n) = A011557(n)*A008454(n). - Michel Marcus, May 18 2022

A248917 a(n) = 2^n * n^2 + 1.

Original entry on oeis.org

1, 3, 17, 73, 257, 801, 2305, 6273, 16385, 41473, 102401, 247809, 589825, 1384449, 3211265, 7372801, 16777217, 37879809, 84934657, 189267969, 419430401, 924844033, 2030043137, 4437573633, 9663676417, 20971520001, 45365592065, 97844723713, 210453397505, 451508436993

Offset: 0

Paul Curtz, Oct 22 2014

Binomial transform of A118239 (Engel expansion of cosh(1)).
Table of successive differences of a(n):
1,   3,   17,  73, 257, 801, 2305,...
2,   14,  56, 184, 544, 1504,...
12,  42, 128, 360, 960,...
30,  86, 232, 600,...
56, 146, 368,...
90, 222,...
132,...
etc.
Via b(n) = 0, 0, 0 followed by A055580(n), i.e., 0, 0, 0, 1, 7, 31, 111, ... (the main sequence for the recurrence), a link can be found between a(n) and A002064(n): c(n) = b(n+1) - 2*b(n) = 0, 0, 1, 5, 17, 49, 129, ... (the main sequence for the signature (5, -8, 4)).

			a(3) = 9 * 8 + 1 = 73.
a(4) = 16 * 16 + 1 = 257.
a(5) = 25 * 32 + 1 = 801.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-18,20,-8).

Cf. A000225, A002064 (Cullen numbers), A006784, A007758, A055580, A118239, A168298.

[2^n*n^2+1: n in [0..30]]; // Vincenzo Librandi, Oct 29 2016

Table[n^2 * 2^n + 1, {n, 0, 31}] (* Alonso del Arte, Oct 22 2014 *)
LinearRecurrence[{7,-18,20,-8}, {1,3,17,73}, 25] (* G. C. Greubel, Oct 28 2016 *)

Vec(-(12*x^3-14*x^2+4*x-1)/((x-1)*(2*x-1)^3) + O(x^100)) \\ Colin Barker, Oct 22 2014

a(n)=n^2<Charles R Greathouse IV, Oct 22 2014

a(n) = 4*a(n-1) - 4*a(n-2) + 2^(n+1) + 1.
a(n) = A007758(n) + 1.
a(n) = 7*a(n-1) - 18*a(n-2) + 20*a(n-3) - 8*a(n-4). - Jean-François Alcover, Oct 22 2014
G.f.: -(12*x^3-14*x^2+4*x-1) / ((x-1)*(2*x-1)^3). - Colin Barker, Oct 22 2014
E.g.f.: exp(x) + 2*x*(1 + 2*x)*exp(2*x). - G. C. Greubel, Oct 28 2016

A062189 a(n) = 2 * 3^(n-2)n(1+2*n).

Original entry on oeis.org

0, 2, 20, 126, 648, 2970, 12636, 51030, 198288, 747954, 2755620, 9959598, 35429400, 124357194, 431530092, 1482720390, 5050815264, 17075199330, 57338232372, 191385721566, 635369601960, 2099044209402, 6903833113980

Offset: 0

Henry Bottomley, Jun 13 2001

Define a triangle with left (first) column T(n,0)=n^2 for n=0,1,2,3.. and the remaining terms T(r,c) = T(r-1,c-1) + 2*T(r,c-1). Then T(n,n) = a(n) on the diagonal. T(n,1) = A056105(n). - J. M. Bergot, Jan 26 2013

Harry J. Smith, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

List([0..30], n-> 2*3^(n-2)*n*(1+2*n)); # G. C. Greubel, Jun 06 2019

[2*3^(n-2)*n*(1+2*n): n in [0..30]]; // G. C. Greubel, Jun 06 2019

Table[2*3^(n-2)*n*(1+2*n), {n,0,30}] (* G. C. Greubel, Jun 06 2019 *)
LinearRecurrence[{9,-27,27},{0,2,20},30] (* Harvey P. Dale, Jun 08 2022 *)

{ for (n=0, 200, write("b062189.txt", n, " ", n*(4*n + 2)*3^(n - 2)) ) } \\ Harry J. Smith, Aug 02 2009

[2*3^(n-2)*n*(1+2*n) for n in (0..30)] # G. C. Greubel, Jun 06 2019

a(n) = A002943(n)*A000244(n-2). Binomial transform of A007758.
G.f.: 2*x*(1+x)/(1-3*x)^3. - Ralf Stephan, Mar 13 2003
a(n) = 2*A077616(n). - R. J. Mathar, Jan 29 2013
E.g.f.: 2*x*(1+2*x)*exp(3*x). - G. C. Greubel, Jun 06 2019

A116138 a(n) = 3^n * n*(n + 1).

Original entry on oeis.org

0, 6, 54, 324, 1620, 7290, 30618, 122472, 472392, 1771470, 6495390, 23383404, 82904796, 290166786, 1004423490, 3443737680, 11708708112, 39516889878, 132497807238, 441659357460, 1464449448420, 4832683179786, 15878816162154

Offset: 0

Mohammad K. Azarian, Apr 08 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

Cf. A007758, A036289, A128796, A027472.

List([0..30], n-> 3^n*n*(n+1)); # G. C. Greubel, May 10 2019

[(n^2+n)*3^n: n in [0..30]]; // Vincenzo Librandi, Feb 28 2013

I:=[0,6,54]; [n le 3 select I[n] else 9*Self(n-1)-27*Self(n-2)+27*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Feb 28 2013

Table[(n^2 + n) 3^n, {n, 0, 30}] (* Vincenzo Librandi, Feb 28 2013 *)

a(n)=(n^2+n)*3^n \\ Charles R Greathouse IV, Feb 28 2013

[3^n*n*(n+1) for n in (0..30)] # G. C. Greubel, May 10 2019

From R. J. Mathar, Dec 19 2008: (Start)
G.f.: 6*x/(1-3*x)^3.
a(n) = 6 * A027472(n+2). (End)
a(n) = 9*a(n-1) -27*a(n-2) +27*a(n-3). - Vincenzo Librandi, Feb 28 2013
E.g.f.: 3*x*(2 + 3*x)*exp(3*x). - G. C. Greubel, May 10 2019
From Amiram Eldar, Jul 20 2020: (Start)
Sum_{n>=1} 1/a(n) = 1 - 2*log(3/2).
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(4/3) - 1. (End)

A168298 a(n) = 1 - n^2*2^n.

Original entry on oeis.org

1, -1, -15, -71, -255, -799, -2303, -6271, -16383, -41471, -102399, -247807, -589823, -1384447, -3211263, -7372799, -16777215, -37879807, -84934655, -189267967, -419430399, -924844031, -2030043135, -4437573631, -9663676415, -20971519999, -45365592063

Offset: 0

Vladimir Joseph Stephan Orlovsky, Nov 22 2009

Numerator of 2^(-n) - n^2.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-18,20,-8).

Cf. A024012

[1-n^2*2^n: n in [0..30]]; // Vincenzo Librandi, Jul 18 2016

f[n_]:=2^n-n^2; Table[Numerator[f[n]],{n,0,-50,-1}]
LinearRecurrence[{7,-18,20,-8},{1,-1,-15,-71},30] (* Harvey P. Dale, May 14 2019 *)

Vec(-(4*x^3-10*x^2+8*x-1)/((x-1)*(2*x-1)^3) + O(x^100)) \\ Colin Barker, Feb 10 2015

a(n)= 7*a(n-1) -18*a(n-2) +20*a(n-3) -8*a(n-4) = 1-A007758(n). - R. J. Mathar, Nov 24 2009
G.f.: -(4*x^3-10*x^2+8*x-1) / ((x-1)*(2*x-1)^3). - Colin Barker, Feb 10 2015
E.g.f.: exp(x) - 2*x*(1 + 2*x)*exp(2*x). - G. C. Greubel, Jul 17 2016

Offset corrected, keyword:sign added, and definition simplified by R. J. Mathar, Nov 23 2009

A198402 a(n) = 5^n * n^5.

Original entry on oeis.org

0, 5, 800, 30375, 640000, 9765625, 121500000, 1313046875, 12800000000, 115330078125, 976562500000, 7863818359375, 60750000000000, 453238525390625, 3282617187500000, 23174285888671875, 160000000000000000, 1083264923095703125

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 (30,-375,2500,-9375,18750,-15625).

Cf. A000351, A000584, A097206.

Sequences of the form n^m*m^n: A001477 (m=1), A007758 (m=2), A062074 (m=3), A062075 (m=4), this sequence (m=5), A198403 (m=6), A098803 (m=7), A198404 (m=8), A198478 (m=9), A198479 (m=10), A098880 (m=11).

Magma
```
[5^n*n^5: n in [0..20]]
```

With[{m = 5}, Table[n^m*m^n, {n, 0, 30}]] (* G. C. Greubel, May 18 2022 *)

a(n)=5^n*n^5 \\ Charles R Greathouse IV, Jul 06 2017

m=5; [n^m*m^n for n in (0..30)] # G. C. Greubel, May 18 2022

G.f.: 5*x*(1 + 130*x + 1650*x^2 + 3250*x^3 + 625*x^4)/(1-5*x)^6. - Colin Barker, Apr 30 2013
E.g.f.: 5*x*(1 + 75*x + 625*x^2 + 1250*x^3 + 625*x^4)*exp(5*x). - G. C. Greubel, May 18 2022
a(n) = A000351(n)*A000584(n). - Michel Marcus, May 19 2022

A198403 a(n) = 6^n * n^6.

Original entry on oeis.org

0, 6, 2304, 157464, 5308416, 121500000, 2176782336, 32934190464, 440301256704, 5355700839936, 60466176000000, 642717115324416, 6499837226778624, 63041475422674944, 590045794670739456, 5355700839936000000

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 (42,-756,7560,-45360,163296,-326592,279936).

Cf. A000400, A001014.

Sequences of the form n^m*m^n: A001477 (m=1), A007758 (m=2), A062074 (m=3), A062075 (m=4), A198402 (m=5), this sequence (m=6), A098803 (m=7), A198404 (m=8), A198478 (m=9), A198479 (m=10), A098880 (m=11).

Magma
```
[6^n*n^6: n in [0..20]]
```

With[{m=6}, Table[n^m*m^n, {n,0,30}]] (* G. C. Greubel, May 18 2022 *)

a(n)=6^n*n^6 \\ Charles R Greathouse IV, Jul 06 2017

m=6; [n^m*m^n for n in (0..30)] # G. C. Greubel, May 18 2022

G.f.: 6*x*(1 + 336*x + 8856*x^2 + 12096*x^3 + 1296*x^4)/(1-6*x)^7. - Colin Barker, Apr 30 2013
E.g.f.: 6*x*(1 + 186*x + 3240*x^2 + 14040*x^3 + 19440*x^4 + 7776*x^5)*exp(6*x). - G. C. Greubel, May 18 2022
a(n) = A000400(n)*A001014(n). - Michel Marcus, May 19 2022
a(n) = 42*a(n-1) - 756*a(n-2) + 7560*a(n-3) - 45360*a(n-4) + 163296*a(n-5) - 326592*a(n-6) + 279936*a(n-7). - Wesley Ivan Hurt, Sep 04 2022

A240983 Integers of the form 2^p*p^2 where p is the lesser of a pair of twin primes.

Original entry on oeis.org

72, 800, 247808, 37879808, 451508436992, 3696558092582912, 2006659878768217161728, 11902724720072940761120768, 25862607545856336249335738796081152, 1857706460417663797470176639788777472, 3270020989306416138620967939526071071138643968

Offset: 1

Zak Seidov, Aug 21 2014

nonn

Subsequence of A071837.

			a(1)=2^3*3^2=72=A071837(3), a(4)=2^17*17^2=37879808=A071837(10).

Chai Wah Wu, Table of n, a(n) for n = 1..86

Cf. A001359, A071837.

forprime(p=3,100,isprime(2+p) && print1(p","))

from sympy import prime, isprime
A240983 = [2**p*p*p for p in (prime(n) for n in range(1,2*10**3)) if isprime(p+2)] # Chai Wah Wu, Aug 27 2014

a(n) = 2^p*p^2, with p=A001359(n).

a(n) = A007758(A001359(n)). - Michel Marcus, Aug 21 2014

More terms from Chai Wah Wu, Aug 27 2014

A378778 a(n) = n^2 * 2^n * binomial(3*n, n).

Original entry on oeis.org

0, 6, 240, 6048, 126720, 2402400, 42771456, 729308160, 12049956864, 194372006400, 3076609536000, 47959947509760, 738269547724800, 11245075661094912, 169748150676357120, 2542638555345715200, 37830087271621066752, 559525260959878348800, 8232406073859904634880, 120560661522092497305600

Offset: 0

Amiram Eldar, Dec 07 2024

Jonathan Borwein, David Bailey, and Roland Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A K Peters, Natick, MA, 2004. See p. 26.

Amiram Eldar, Table of n, a(n) for n = 0..500
Necdet Batir, On the series Sum_{k=1..oo} binomial(3k,k)^{-1} k^{-n} x^k, Proc. Indian Acad. Sci. (Math. Sci.), Vol. 115, No. 4 (2005), pp. 371-381; arXiv preprint, arXiv:math/0512310 [math.AC], 2005. See p. 379, eq. (3.6).
Jonathan M. Borwein and Roland Girgensohn, Evaluations of binomial series, aequationes mathematicae, Vol. 70, No. 1 (2005), pp. 25-36. See p. 32, eq. (43).

Cf. A005809, A007758, A228484, A378780.

a[n_] := n^2 * 2^n * Binomial[3*n, n]; Array[a, 25, 0]

PARI
```
a(n) = n^2 * 2^n * binomial(3*n, n);
```

a(n) = A007758(n) * A005809(n).
a(n) = n^2 * A228484(n).
a(n) = n * A378780(n).
a(n) == 0 (mod 6).
Sum_{n>=1} 1/a(n) = Pi^2/24 - log(2)^2/2 (Borwein et al., 2004; Borwein and Girgensohn, 2005; Batir, 2005).

A112327 Triangle read by rows: T(n,k)=k^32^kbinomial(2n-k,n-k)/(2n-k) (1<=k<=n).

Original entry on oeis.org

2, 2, 16, 4, 32, 72, 10, 80, 216, 256, 28, 224, 648, 1024, 800, 84, 672, 2016, 3584, 4000, 2304, 264, 2112, 6480, 12288, 16000, 13824, 6272, 858, 6864, 21384, 42240, 60000, 62208, 43904, 16384, 2860, 22880, 72072, 146432, 220000, 253440, 219520, 131072

Offset: 1

Emeric Deutsch, Sep 04 2005

T(n,1) = 2*Catalan(n-1) = 2*A000108(n-1); T(n,n) = 2^n*n^2 = A007758(n).

Row sums yield A112328.

			Triangle starts:
2;
2,16;
4,32,72;
10,80,216,256;

F. Ruskey, Average shape of binary trees, SIAM J. Alg. Disc. Meth., 1, 1980, 43-50 (Eq. (8)).

Cf. A000108, A007758, A112328.

T:=proc(n,k) if k<2*n then k^3*2^k*binomial(2*n-k,n-k)/(2*n-k) else 0 fi end: for n from 1 to 10 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form

cp's OEIS Frontend

A198479 a(n) = 10^n * n^10.

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A248917 a(n) = 2^n * n^2 + 1.

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A062189 a(n) = 2 * 3^(n-2)*n*(1+2*n).

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A116138 a(n) = 3^n * n*(n + 1).

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A168298 a(n) = 1 - n^2*2^n.

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A198402 a(n) = 5^n * n^5.

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A198403 a(n) = 6^n * n^6.

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A062189 a(n) = 2 * 3^(n-2)n(1+2*n).

A112327 Triangle read by rows: T(n,k)=k^32^kbinomial(2n-k,n-k)/(2n-k) (1<=k<=n).