A036289 -id:A036289 - cp's OEIS frontend

A378780 a(n) = n * 2^n * binomial(3*n, n).

0, 6, 120, 2016, 31680, 480480, 7128576, 104186880, 1506244608, 21596889600, 307660953600, 4359995228160, 61522462310400, 865005820084224, 12124867905454080, 169509237023047680, 2364380454476316672, 32913250644698726400, 457355892992216924160, 6345297974846973542400

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. (3.9).
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, A036289, A228484.

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

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

a(n) = A036289(n) * A005809(n).
a(n) = n * A228484(n).
a(n) == 0 (mod 6).
Sum_{n>=1} 1/a(n) = Pi/10 - log(2)/5 (Borwein et al., 2004; Borwein and Girgensohn, 2005; Batir, 2005).

A378804 a(n) = n * 2^n * binomial(4*n, n).

Original entry on oeis.org

0, 8, 224, 5280, 116480, 2480640, 51684864, 1060899840, 21541478400, 433812234240, 8680043806720, 172774871965696, 3424347806171136, 67626404043161600, 1331466198928588800, 26145958720005734400, 512257621575157678080, 10016204637370583089152, 195501127311163895316480

Offset: 0

Amiram Eldar, Dec 07 2024

Amiram Eldar, Table of n, a(n) for n = 0..500
Jonathan M. Borwein and Roland Girgensohn, Evaluations of binomial series, aequationes mathematicae, Vol. 70, No. 1 (2005), pp. 25-36.

Cf. A005810, A036289, A378802.

a[n_] := n * 2^n * Binomial[4*n, n]; Array[a, 20, 0]

PARI
```
a(n) = n * 2^n * binomial(4*n, n);
```

a(n) = A036289(n) * A005810(n).
a(n) = 2^n * A378802(n).
a(n) == 0 (mod 8).
Sum_{n>=1} (-1)^n/a(n) = (log(2) - 6*log(3))/7 + Sum_{r: 2*r^3 + 12*r + 13 = 0} log(r+2)/(r+3) = -0.120716907732393305... (Borwein and Girgensohn, 2005, p. 32, eq. (43)).

A046688 Antidiagonals of square array in which k-th row (k>0) is an arithmetic progression of difference 2^(k-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 5, 1, 1, 5, 7, 9, 9, 1, 1, 6, 9, 13, 17, 17, 1, 1, 7, 11, 17, 25, 33, 33, 1, 1, 8, 13, 21, 33, 49, 65, 65, 1, 1, 9, 15, 25, 41, 65, 97, 129, 129, 1, 1, 10, 17, 29, 49, 81, 129, 193, 257, 257, 1, 1, 11, 19, 33, 57, 97, 161, 257, 385, 513, 513, 1

Offset: 0

N. J. A. Sloane

			From _Gus Wiseman_, May 08 2021: (Start):
Array A(m,n) = 1 + n*2^(m-1) begins:
       n=0: n=1: n=2: n=3: n=4: n=5: n=6: n=7: n=8: n=9:
  m=0:   1    1    1    1    1    1    1    1    1    1
  m=1:   1    2    3    5    9   17   33   65  129  257
  m=2:   1    3    5    9   17   33   65  129  257  513
  m=3:   1    4    7   13   25   49   97  193  385  769
  m=4:   1    5    9   17   33   65  129  257  513 1025
  m=5:   1    6   11   21   41   81  161  321  641 1281
  m=6:   1    7   13   25   49   97  193  385  769 1537
  m=7:   1    8   15   29   57  113  225  449  897 1793
  m=8:   1    9   17   33   65  129  257  513 1025 2049
  m=9:   1   10   19   37   73  145  289  577 1153 2305
Triangle T(n,k) = 1 + (n-k)*2^(k-1) begins:
   1
   1   1
   1   2   1
   1   3   3   1
   1   4   5   5   1
   1   5   7   9   9   1
   1   6   9  13  17  17   1
   1   7  11  17  25  33  33   1
   1   8  13  21  33  49  65  65   1
   1   9  15  25  41  65  97 129 129   1
   1  10  17  29  49  81 129 193 257 257   1
   1  11  19  33  57  97 161 257 385 513 513   1
(End)

G. H. Hardy, A Theorem Concerning the Infinite Cardinal Numbers, Quart. J. Math., 35 (1904), p. 90 = Collected Papers, Vol. VII, p. 430.

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Row sums are A000079.
Diagonal n = m + 1 of the array is A002064.
Diagonal n = m of the array is A005183.
Column m = 1 of the array is A094373.
Diagonal n = m - 1 of the array is A131056.
A002109 gives hyperfactorials (sigma: A260146, omega: A303281).
A009998(k,n) = n^k.
A009999(n,k) = n^k.
A057156 = (2^n)^(2^n).
A062319 counts divisors of n^n.
Cf. A000169, A000272, A000312, A036289, A343656, A343658.

Table[If[k==0,1,n*2^(k-1)+1],{n,0,9},{k,0,9}] (* ARRAY, Gus Wiseman, May 08 2021 *)
Table[If[k==0,1,1+(n-k)*2^(k-1)],{n,0,10},{k,0,n}] (* TRIANGLE, Gus Wiseman, May 08 2021 *)

A(m,n)={if(m>0, 1+n*2^(m-1), 1)}
{ for(m=0, 10, for(n=0, 10, print1(A(m,n), ", ")); print) } \\ Andrew Howroyd, Mar 07 2020

A(m,n) = 1 + n*2^(m-1) for m > 1. - Andrew Howroyd, Mar 07 2020

As a triangle, T(n,k) = A(k,n-k) = 1 + (n-k)*2^(k-1). - Gus Wiseman, May 08 2021

More terms from Larry Reeves (larryr(AT)acm.org), Apr 06 2000

A069229 a(n) = n*(2^n + 1).

Original entry on oeis.org

0, 3, 10, 27, 68, 165, 390, 903, 2056, 4617, 10250, 22539, 49164, 106509, 229390, 491535, 1048592, 2228241, 4718610, 9961491, 20971540, 44040213, 92274710, 192938007, 402653208, 838860825, 1744830490, 3623878683, 7516192796

Offset: 0

Vladeta Jovovic, Apr 12 2002

Odd terms are multiples of 3. - Dario Piazzalunga, Jan 10 2013

Index entries for linear recurrences with constant coefficients, signature (6, -13, 12, -4).

Cf. A066524.

Table[n*2^n+n,{n,0,3*4!}] (* Vladimir Joseph Stephan Orlovsky, Apr 25 2010 *)
LinearRecurrence[{6,-13,12,-4},{0,3,10,27},30] (* Harvey P. Dale, May 06 2021 *)

Recurrence: a(n) = 6*a(n-1) - 13*a(n-2) + 12*a(n-3) - 4*a(n-4).
G.f.: x*(3 - 8*x + 6*x^2)/(1-x)^2/(1 - 2*x)^2.
a(n) = n+A036289(n). - R. J. Mathar, Jun 17 2020

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)

A128988 a(n) = (n^3 - n^2)*5^n.

Original entry on oeis.org

0, 100, 2250, 30000, 312500, 2812500, 22968750, 175000000, 1265625000, 8789062500, 59082031250, 386718750000, 2475585937500, 15551757812500, 96130371093750, 585937500000000, 3527832031250000, 21011352539062500

Offset: 1

Mohammad K. Azarian, Apr 30 2007

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (20,-150,500,-625).

Cf. A128796; A036289.

[(n^3-n^2)*5^n: n in [1..30]]; // Vincenzo Librandi, Oct 26 2011

Table[(n^3-n^2)5^n,{n,20}] (* or *) LinearRecurrence[{20,-150,500,-625},{0,100,2250,30000},20]

a(1)=0, a(2)=100, a(3)=2250, a(4)=30000, a(n)=20*a(n-1)- 150*a(n-2)+ 500*a(n-3)- 625*a(n-4). - Harvey P. Dale, Oct 25 2011

G.f.: (50*(5*x^2+2*x))/(5*x-1)^4. - Harvey P. Dale, Oct 25 2011

Offset corrected by Mohammad K. Azarian, Nov 19 2008

A171647 a(1) = 1; for n > 1, a(n) = 2a(n-1) if n is even, a(n) = ((n+1)/(n-1))a(n-1) if n is odd.

Original entry on oeis.org

1, 2, 4, 8, 12, 24, 32, 64, 80, 160, 192, 384, 448, 896, 1024, 2048, 2304, 4608, 5120, 10240, 11264, 22528, 24576, 49152, 53248, 106496, 114688, 229376, 245760, 491520, 524288, 1048576, 1114112, 2228224, 2359296, 4718592, 4980736, 9961472

Offset: 1

Gary W. Adamson, Dec 13 2009

a(n) is the number of subsets of {1,2,...,n} that contain exactly one odd number. For example, for n=5, a(5)=12 and the 12 subsets are {1}, {3}, {5}, {1,2}, {1,4}, {2,3}, {2,5}, {3,4}, {4,5}, {1,2,4}, {2,3,4}, {2,4,5}. - Enrique Navarrete, Dec 15 2019

2*a(n-1) is the number of subsets of {1,2,...,n} that contain exactly one even number. For example, for n=5, 2*a(4)=16 and the 16 subsets are {2}, {4}, {1,2}, {1,4}, {2,3}, {2,5}, {3,4}, {4,5}, {1,2,3}, {1,2,5}, {1,3,4}, {1,4,5}, {2,3,5}, {3,4,5}, {1,2,3,5}, {1,3,4,5}. - Enrique Navarrete, Dec 16 2019

			a(6) = 2*a(5) = 2*12 = 24;
a(7) = (8/6)*a(6) = (4/3)*24 = 32.

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-4).

Cf. A001787, A036289 (bisections).

Cf. A016631.

[ n eq 1 select 1 else IsEven(n) select 2*Self(n-1) else ((n+1)/(n-1))*Self(n-1): n in [1..40] ];

a[n_] := If[ OddQ@ n, (n + 1)/(n - 1) a[n - 1] , 2 a[n - 1]]; a[1] = 1; Array[a, 38]
LinearRecurrence[{0,4,0,-4},{1,2,4,8},40] (* Harvey P. Dale, Jan 14 2015 *)

From R. J. Mathar, Dec 06 2010: (Start)
a(n) = 4*a(n-2) - 4*a(n-4).
G.f.: x*(1+2*x)/(-1+2*x^2)^2. (End)
a(n) = (2*n - (-1)^n+1)*2^((2*n + (-1)^n - 9)/4). - Bruno Berselli, Dec 07 2010
G.f.: G(0), where G(k) = 1 + 2*x*(k+1)/(k + 1 - x*(k+1)*(k+2)/(x*(k+2) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 27 2013
Sum_{n>=1} 1/a(n) = 3*log(2) (A016631). - Amiram Eldar, Aug 27 2022

A190730 Let b(n,0) = n and b(n,k) = 2*b(n,k-1) + 1 for k > 0. Then a(n) = b(n,1) + b(n,2) + ... + b(n,n).

Original entry on oeis.org

3, 16, 53, 146, 367, 876, 2025, 4582, 10211, 22496, 49117, 106458, 229335, 491476, 1048529, 2228174, 4718539, 9961416, 20971461, 44040130, 92274623, 192937916, 402653113, 838860726, 1744830387, 3623878576, 7516192685, 15569256362, 32212254631, 66571992996

Offset: 1

J. M. Bergot, May 17 2011

It turns out that b(n,k) = A087322(n,k) = (n + 1)*2^k - 1 for 1 <= k <= n (without the 0th column). - Petros Hadjicostas, Feb 15 2021

			One way to view it is to begin with n = 5, then 5 + 6 = 11 --> 11 + 12 = 23 --> 23 + 24 = 47 --> 47 + 48 = 95 --> 95 + 96 = 191. There are n steps, in this case 5, that give the sum 11 + 23 + 47 + 95 + 191 = 367. This is the same as (2*5+1) + (4*5+3) + (8*5+7) + (16*5+15) + (32*5+31). The formula gives (5+1)*2^(5+1) - 3*5 - 2 = 6*64 - 17 = 367.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Cf. A016789, A036289, A087322.

[(n+1) * 2^(n+1) - 3*n - 2 : n in [1..30]]; // Vincenzo Librandi, Sep 29 2011

LinearRecurrence[{6,-13,12,-4},{3,16,53,146},40] (* or *)
Array[(#+1)2^(#+1)-3#-2&,40] (* Paolo Xausa, Oct 17 2023 *)

a(n) = (n+1) * 2^(n+1) - 3*n - 2 = A036289(n+1) - A016789(n).
G.f.: -x*(-3 + 2*x + 4*x^2) / ( (2*x-1)^2*(x-1)^2 ). - R. J. Mathar, May 29 2011
E.g.f.: exp(x)*(2*exp(x)*(1 + 2*x) - 2 - 3*x). - Stefano Spezia, Oct 16 2023

A204207 Triangle based on (1,2,3) averaging array.

Original entry on oeis.org

2, 3, 5, 5, 8, 11, 9, 13, 19, 23, 17, 22, 32, 42, 47, 33, 39, 54, 74, 89, 95, 65, 72, 93, 128, 163, 184, 191, 129, 137, 165, 221, 291, 347, 375, 383, 257, 266, 302, 386, 512, 638, 722, 758, 767, 513, 523, 568, 688, 898, 1150, 1360, 1480, 1525, 1535, 1025

Offset: 1

Clark Kimberling, Jan 12 2012

See A204201 for a discussion and guide to other averaging arrays.

			First six rows:
2
3....5
5....8....11
9....13...19...23
17...22...32...42...47
33...39...54...74...89...95

Cf. A204201.

a = 1; r = 2; b = 3;
t[1, 1] = r;
t[n_, 1] := (a + t[n - 1, 1])/2;
t[n_, n_] := (b + t[n - 1, n - 1])/2;
t[n_, k_] := (t[n - 1, k - 1] + t[n - 1, k])/2;
u[n_] := Table[t[n, k], {k, 1, n}]
Table[u[n], {n, 1, 5}]    (* averaging array *)
u = Table[2 (1/2) (1/r) 2^n*u[n], {n, 1, 12}];
TableForm[u] (* A204207 triangle *)
Flatten[u]   (* A204207 sequence *)

T(n,n) = A083329(n). - Philippe Deléham, Dec 24 2013
T(n,1) = A000051(n-1). - Philippe Deléham, Dec 24 2013
Sum_{k=1..n} T(n,k)=A036289(n). - Philippe Deléham, Dec 24 2013
T(n,k) = T(n-1,k) + 3*T(n-1,k-1) - 2*T(n-2,k-1) - 2*T(n-2,k-2), T(1,1)=2, T(2,1)=3, T(2,2)=5, T(n,k)=0 if k<1 or if k>n. - Philippe Deléham, Dec 24 2013

Example corrected by Philippe Deléham, Dec 22 2013

A344041 Decimal expansion of Sum_{k>=1} F(k)/(k*2^k), where F(k) is the k-th Fibonacci number (A000045).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, May 07 2021

This constant is a transcendental number (Adhikari et al., 2001).
A similar series is Sum_{k>=1} F(k)/2^k = 2.
The corresponding series with Lucas numbers (A000032) is Sum_{k>=1} L(k)/(k*2^k) = 2*log(2) (A016627).
In general, for m>=2, Sum_{k>=1} F(k)/(k*m^k) = log(1 - 2*sqrt(5)/(1 + sqrt(5) - 2*m)) / sqrt(5) and Sum_{k>=1} L(k)/(k*m^k) = log(m^2 / (m^2 - m - 1)). - Vaclav Kotesovec, May 08 2021

			0.86081788192800807777886646590121085084914136508057...

S. D. Adhikari, N. Saradha, T. N. Shorey and R. Tijdeman, Transcendental infinite sums, Indagationes Mathematicae, Vol. 12, No. 1 (2001), pp. 1-14.
István Mező, Several Generating Functions for Second-Order Recurrence Sequences, Journal of Integer Sequences, Vol. 12 (2009), Article 09.3.7.
Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, INTEGERS 6 (2006) #A27.
Index entries for transcendental numbers.

Cf. A000032, A000045, A002163, A002390, A002457, A036289.

RealDigits[Sum[Fibonacci[n]/n/2^n, {n, 1, Infinity}], 10, 100][[1]]

suminf(k=1, fibonacci(k)/(k*2^k)) \\ Michel Marcus, May 07 2021

Equals Sum_{k>=0} (-1)^k/A002457(k).
Equals 4*log(phi)/sqrt(5) = 4*arcsinh(1/2)/sqrt(5) = arccosh(7/2)/sqrt(5) = 4*A002390/A002163.
Equals Integral_{x>=2} 1/(x^2 - x - 1) dx.

cp's OEIS Frontend

A378780 a(n) = n * 2^n * binomial(3*n, n).

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A378804 a(n) = n * 2^n * binomial(4*n, n).

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Mathematica

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A046688 Antidiagonals of square array in which k-th row (k>0) is an arithmetic progression of difference 2^(k-1).

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Mathematica

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A069229 a(n) = n*(2^n + 1).

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

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A128988 a(n) = (n^3 - n^2)*5^n.

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A171647 a(1) = 1; for n > 1, a(n) = 2*a(n-1) if n is even, a(n) = ((n+1)/(n-1))*a(n-1) if n is odd.

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A171647 a(1) = 1; for n > 1, a(n) = 2a(n-1) if n is even, a(n) = ((n+1)/(n-1))a(n-1) if n is odd.