A027471 -id:A027471 - cp's OEIS frontend

A086972 a(n) = n*3^(n-1) + (3^n + 1)/2.

1, 3, 11, 41, 149, 527, 1823, 6197, 20777, 68891, 226355, 738113, 2391485, 7705895, 24712007, 78918989, 251105873, 796364339, 2518233179, 7942120025, 24988621541, 78452649023, 245818300271, 768835960421, 2400651060089

Offset: 0

Paul Barry, Jul 26 2003

Binomial transform of A057711 (without leading zero). Second binomial transform of (1,1,3,3,5,5,7,7,9,9,11,11,...).

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (7,-15,9).

Cf. A007051, A027471, A057711, A081038.

Partial sums of A199923.

[n*3^(n-1) + (3^n+1)/2: n in [0..30]]; // Vincenzo Librandi, Jun 09 2011

Table[((2*n+3)*3^(n-1) +1)/2, {n,0,30}] (* G. C. Greubel, Nov 24 2023 *)

Vec((1-4*x+5*x^2)/((1-x)*(1-3*x)^2) + O(x^40)) \\ Michel Marcus, Mar 08 2016

[((2*n+3)*3^(n-1) +1)//2 for n in range(31)] # G. C. Greubel, Nov 24 2023

a(n) = (1/2)*(A081038(n) + 1).
G.f.: (1-4*x+5*x^2)/((1-x)*(1-3*x)^2).
a(n) = A027471(n) + A007051(n).
E.g.f.: (1/2)*( exp(x) + (2*x+1)*exp(3*x) ). - G. C. Greubel, Nov 24 2023

A104002 Triangle T(n,k) read by rows: number of permutations in S_n avoiding all k-length patterns that start with 1 except one fixed pattern and containing it exactly once.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 12, 6, 1, 5, 32, 27, 8, 1, 6, 80, 108, 48, 10, 1, 7, 192, 405, 256, 75, 12, 1, 8, 448, 1458, 1280, 500, 108, 14, 1, 9, 1024, 5103, 6144, 3125, 864, 147, 16, 1, 10, 2304, 17496, 28672, 18750, 6480, 1372, 192, 18, 1, 11, 5120, 59049, 131072

Offset: 2

Ralf Stephan, Feb 26 2005

T(n+k,k+1) = total number of occurrences of any given letter in all possible n-length words on a k-letter alphabet. For example, with the 2 letter alphabet {0,1} there are 4 possible 2-length words: {00,01,10,11}. The letter 0 occurs 4 times altogether, as does the letter 1. T(4,3) = 4. - Ross La Haye, Jan 03 2007

Table T(n,k) = k*n^(k-1) n,k > 0 read by antidiagonals. - Boris Putievskiy, Dec 17 2012

			Triangle begins:
  1;
  2,   1;
  3,   4,    1;
  4,  12,    6,    1;
  5,  32,   27,    8,   1;
  6,  80,  108,   48,  10,   1;
  7, 192,  405,  256,  75,  12,  1;
  8, 448, 1458, 1280, 500, 108, 14, 1;

Michael De Vlieger, Table of n, a(n) for n = 2..11176 (rows 2 <= n <= 150).
T. Mansour, Permutations containing and avoiding certain patterns, arXiv:math/9911243 [math.CO], 1999-2000.
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.

Cf. Left-edge columns include A001787, A027471, A002697, A053464, A053469, A027473, A053539, A053540, A053541, A081127, A081128.

Table[(n - k + 1) (k - 1)^(n - k), {n, 2, 12}, {k, 2, n}] // Flatten (* Michael De Vlieger, Aug 22 2018 *)

T(n, k) = (n-k+1) * (k-1)^(n-k), k<=n.

As a linear array, the sequence is a(n) = A004736(n)*A002260(n)^(A004736(n)-1) or a(n) = ((t*t+3*t+4)/2-n)*(n-(t*(t+1)/2))^((t*t+3*t+4)/2-n-1), where t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 17 2012

A174719 Triangle T(n, k, q) = (1-q^n)*( binomial(n, k) - 1 ) + 1, with q = 3, read by rows.

Original entry on oeis.org

1, 1, 1, 1, -7, 1, 1, -51, -51, 1, 1, -239, -399, -239, 1, 1, -967, -2177, -2177, -967, 1, 1, -3639, -10191, -13831, -10191, -3639, 1, 1, -13115, -43719, -74323, -74323, -43719, -13115, 1, 1, -45919, -177119, -360799, -452639, -360799, -177119, -45919, 1

Offset: 0

Roger L. Bagula, Mar 28 2010

The row sums of this class of sequences, for varying q, is given by Sum_{k=0..n} T(n, k, q) = q^n * (n+1) + 2^n * (1 - q^n). - G. C. Greubel, Feb 09 2021

			Triangle begins as:
  1;
  1,      1;
  1,     -7,       1;
  1,    -51,     -51,       1;
  1,   -239,    -399,    -239,       1;
  1,   -967,   -2177,   -2177,    -967,       1;
  1,  -3639,  -10191,  -13831,  -10191,   -3639,       1;
  1, -13115,  -43719,  -74323,  -74323,  -43719,  -13115,      1;
  1, -45919, -177119, -360799, -452639, -360799, -177119, -45919, 1;

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A000012 (q=1), A174718 (q=2), this sequence (q=3), A174720 (q=4).

Cf. A027471, A248216.

T:= func< n,k,q | 1 + (1-q^n)*(Binomial(n,k) -1) >;
[T(n,k,3): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 09 2021

T[n_, k_, q_]:= 1 +(1-q^n)*(Binomial[n, k] -1);
Table[T[n,k,3], {n,0,12}, {k,0,n}]//Flatten

def T(n,k,q): return 1 + (1-q^n)*(binomial(n,k) - 1)
flatten([[T(n,k,3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 09 2021

T(n, k, q) = (1-q^n)*( binomial(n, k) - 1 ) + 1, with q=3.

Sum_{k=0..n} T(n, k, 3) = 3^n*(n+1) + 2^n*(1 - 3^n) = A027471(n+2) - A248216(n). - G. C. Greubel, Feb 09 2021

Edited by G. C. Greubel, Feb 09 2021

A192083 Arithmetic derivative of squares of prime powers: a(n) = A003415(A056798(n)).

Original entry on oeis.org

0, 4, 6, 32, 10, 14, 192, 108, 22, 26, 1024, 34, 38, 46, 500, 1458, 58, 62, 5120, 74, 82, 86, 94, 1372, 106, 118, 122, 24576, 134, 142, 146, 158, 17496, 166, 178, 194, 202, 206, 214, 218, 226, 5324, 18750, 254, 114688, 262, 274, 278, 298, 302, 314, 326, 334

Offset: 1

Reinhard Zumkeller, Jun 26 2011

nonn

A001787 and A024622 give record values and where they occur.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A003415, A024622, A025473, A025474, A056798, A192084.

Subsequences: A001787, A027471, A051674, A079705, A100484.

s[n_] := If[PrimePowerQ[n], f = FactorInteger[n][[1]]; 2*f[[2]]*n^(2 - 1/f[[2]]), Nothing]; s[1] = 0; Array[s, 200] (* Amiram Eldar, Apr 06 2025 *)

a(n) = 2 * A025474(n) * A025473(n)^(2*A025474(n) - 1).

A192084(n) = A003415(a(n)).

A230539 a(n) = 3n2^(3*n-1).

Original entry on oeis.org

0, 12, 192, 2304, 24576, 245760, 2359296, 22020096, 201326592, 1811939328, 16106127360, 141733920768, 1236950581248, 10720238370816, 92358976733184, 791648371998720, 6755399441055744, 57420895248973824, 486388759756013568, 4107282860161892352

Offset: 0

Bruno Berselli, Oct 23 2013

Arithmetic derivative of 8^n: a(n) = A003415(8^n).

Sum of reciprocals of a(n), for n>0: (2/3)*log(8/7).

Bruno Berselli, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (16,-64).

Cf. A001018, A003415.
Cf. arithmetic derivative of k^n: A001787 (k=2), A027471 (k=3), A018215 (k=4), A053464 (k=5), A212700 (k=6), A027473 (k=7), this sequence, A230540 (k=9), A085708 (k=10), A081127 (k=11).
Row n=8 of A258997.

Magma
```
[3*n*2^(3*n-1): n in [0..20]];
```

A230539:=n->3*n*2^(3*n-1): seq(A230539(n), n=0..30); # Wesley Ivan Hurt, May 03 2017

Table[3 n 2^(3 n - 1), {n,0,20}]
LinearRecurrence[{16,-64},{0,12},20] (* Harvey P. Dale, Dec 25 2022 *)

a(n) = 3*n*2^(3*n-1); \\ Michel Marcus, Oct 23 2013

G.f.: 12*x/(1-8*x)^2.

a(n) = 12*A053539(n).

A230540 a(n) = 2n3^(2*n-1).

Original entry on oeis.org

0, 6, 108, 1458, 17496, 196830, 2125764, 22320522, 229582512, 2324522934, 23245229340, 230127770466, 2259436291848, 22029503845518, 213516729579636, 2058911320946490, 19765548681086304, 189008059262887782, 1801135623563989452, 17110788423857899794

Offset: 0

Bruno Berselli, Oct 23 2013

Arithmetic derivative of 9^n: a(n) = A003415(9^n).

Sum of reciprocals of a(n), for n>0: (3/2)*log(9/8).

Bruno Berselli, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (18,-81).

Cf. A001019, A003415.

Cf. arithmetic derivative of k^n: A001787 (k=2), A027471 (k=3), A018215 (k=4), A053464 (k=5), A212700 (k=6), A027473 (k=7), A230539 (k=8), this sequence, A085708 (k=10), A081127 (k=11).

Magma
```
[2*n*3^(2*n-1): n in [0..20]];
```
Mathematica
```
Table[2 n 3^(2 n - 1), {n, 0, 20}]
```

a(n) = 2*n*3^(2*n-1); \\ Michel Marcus, Oct 23 2013

G.f.: 6*x/(1-9*x)^2.

a(n) = 6*A053540(n), with A053540(0)=0.

A361608 a(n) = 7^n(n+1)(81n^4+684n^3+1401n^2+434n+40)/40.

Original entry on oeis.org

1, 924, 48804, 1337014, 26622288, 437049228, 6295986235, 82489361052, 1005444707211, 11576481361732, 127278262644918, 1346951022678114, 13803666582387682, 137633164619393268, 1340161331495822661, 12782144706910135480, 119711031072135899781, 1103157160378734314700, 10019811250265958667288

Offset: 0

R. J. Mathar, Mar 17 2023

The sequences A(n,k) = Sum_{j=0..n} Sum_{i=0..j} (-1)^(j-i) * binomial(n,j) * binomial(j,i) * binomial(j+k+(k+1)*i,j+k) are C-sequences for fixed integer k, here A(n,k=5) = a(n).

Project Euler, Problem 831. Triple Product
R. J. Mathar, On an alternating double sum of a triple product of aerated binomial coefficients, arXiv:2306.08022 (2023)
Index entries for linear recurrences with constant coefficients, signature (42,-735,6860,-36015,100842,-117649).

Cf. A027471 (k=1), A361609 (k=2), A361610 (k=3).

LinearRecurrence[{42,-735,6860,-36015,100842,-117649},{1,924,48804,1337014,26622288,437049228},20] (* Harvey P. Dale, May 29 2023 *)

def A361608(n): return 7**n*(n*(n*(n*(n*(81*n + 765) + 2085) + 1835) + 474) + 40)//40 # Chai Wah Wu, Mar 17 2023

G.f.: ( 1+882*x+10731*x^2-40474*x^3+36015*x^4 ) / (7*x-1)^6 .
a(n) = +42*a(n-1) -735*a(n-2) +6860*a(n-3) -36015*a(n-4) +100842*a(n-5) -117649*a(n-6).
D-finite with recurrence n*(81*n^4+360*n^3-165*n^2-640*n+404)*a(n) -7*(n+1)*(81*n^4+684*n^3+1401*n^2+434*n+40)*a(n-1)=0.

A361609 a(n) = 4^n(1 + (23/8)n + (9/8)*n^2).

Original entry on oeis.org

1, 20, 180, 1264, 7808, 44544, 240640, 1249280, 6291456, 30932992, 149159936, 707788800, 3313500160, 15334375424, 70262980608, 319169757184, 1438814044160, 6442450944000, 28673201668096, 126924873531392, 559101662724096, 2451910929940480, 10709243254538240, 46601700831657984

Offset: 0

R. J. Mathar, Mar 17 2023

The sequences A(n,k) = Sum_{j=0..n} Sum_{i=0..j} (-1)^(j-i) * binomial(n,j) *binomial(j,i) * binomial(j+k+(k+1)*i,j+k) are C-sequences for fixed integer k, here A(n,k=2) = a(n).

Winston de Greef, Table of n, a(n) for n = 0..1640
Project Euler, Problem 831. Triple Product
Index entries for linear recurrences with constant coefficients, signature (12,-48,64).

Cf. A027471 (k=1), A361610 (k=3), A361608 (k=5).

LinearRecurrence[{12, -48, 64}, {1, 20, 180}, 25] (* or *)
A361609[n_] := 4^n (1 + 23/8 n + 9/8 n^2);
Array[A361609, 25, 0] (* Paolo Xausa, Jan 18 2024 *)

def A361609(n): return (n*(9*n + 23) + 8)<<((n<<1)-3) if n > 1 else 19*n+1 # Chai Wah Wu, Mar 17 2023

G.f.: ( -1-8*x+12*x^2 ) / (4*x-1)^3.
a(n) = 12*a(n-1) -48*a(n-2) +64*a(n-3).
D-finite with recurrence (-9*n^2-5*n+6)*a(n) +4*(9*n^2+23*n+8)*a(n-1)=0.

A361610 a(n) = 5^n(n+1)(4n^2+14n+3)/3.

Original entry on oeis.org

1, 70, 1175, 13500, 128125, 1081250, 8421875, 61875000, 434765625, 2949218750, 19443359375, 125195312500, 790283203125, 4904785156250, 29998779296875, 181152343750000, 1081695556640625, 6394958496093750, 37471771240234375, 217819213867187500, 1257038116455078125

Offset: 0

R. J. Mathar, Mar 17 2023

The sequences A(n,k) = Sum_{j=0..n} Sum_{i=0..j} (-1)^(j-i) * binomial(n,j) * binomial(j,i) * binomial(j+k+(k+1)*i,j+k) are C-sequences for fixed integer k, here A(n,k=3) = a(n).

Winston de Greef, Table of n, a(n) for n = 0..1408
Project Euler, Problem 831. Triple Product
Index entries for linear recurrences with constant coefficients, signature (20,-150,500,-625).

Cf. A027471 (k=1), A361609 (k=2), A361608 (k=5).

LinearRecurrence[{20,-150,500,-625},{1,70,1175,13500},30] (* Harvey P. Dale, Aug 29 2024 *)

def A361610(n): return 5**n*(n*(n*(4*n + 18) + 17) + 3)//3 # Chai Wah Wu, Mar 17 2023

G.f.: (1 + 50*x - 75*x^2) / (5*x - 1)^4.
a(n) = 20*a(n-1) -150*a(n-2) +500*a(n-3) -625*a(n-4).
D-finite with recurrence n*(4*n^2+6*n-7)*a(n) -5*(n+1)*(4*n^2+14*n+3)*a(n-1)=0.

A380651 a(n) = 4^n - n*3^(n-1).

Original entry on oeis.org

1, 3, 10, 37, 148, 619, 2638, 11281, 48040, 203095, 851746, 3544765, 14651452, 60200131, 246114934, 1001997289, 4065384784, 16448074927, 66394953802, 267516917653, 1076266398436, 4324824038683, 17362058273950, 69646979806657, 279215540418808

Offset: 0

Enrique Navarrete, Jan 29 2025

a(n) is the number of words of length n defined on 4 letters where one of the letters is not used or is used any number of times except once.

			For n=2, the 10 words on {0, 1, 2, 3} that do not use 0 exactly once are 12, 21, 13, 31, 23, 32, 11, 22, 33, 00.

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-33,36).

Cf. A000302, A027471.

Table[4^n - n*3^(n - 1), {n, 0, 25}] (* Paolo Xausa, Feb 06 2025 *)

E.g.f.: exp(3*x)*(exp(x)-x).
From Alois P. Heinz, Jan 29 2025: (Start)
G.f.: -(13*x^2-7*x+1)/((4*x-1)*(3*x-1)^2).
a(n) = A000302(n) - A027471(n+1). (End)

cp's OEIS Frontend

A086972 a(n) = n*3^(n-1) + (3^n + 1)/2.

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A104002 Triangle T(n,k) read by rows: number of permutations in S_n avoiding all k-length patterns that start with 1 except one fixed pattern and containing it exactly once.

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A174719 Triangle T(n, k, q) = (1-q^n)*( binomial(n, k) - 1 ) + 1, with q = 3, read by rows.

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A192083 Arithmetic derivative of squares of prime powers: a(n) = A003415(A056798(n)).

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A230539 a(n) = 3*n*2^(3*n-1).

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A230540 a(n) = 2*n*3^(2*n-1).

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A361608 a(n) = 7^n*(n+1)*(81*n^4+684*n^3+1401*n^2+434*n+40)/40.

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A230539 a(n) = 3n2^(3*n-1).

A230540 a(n) = 2n3^(2*n-1).

A361608 a(n) = 7^n(n+1)(81n^4+684n^3+1401n^2+434n+40)/40.

A361609 a(n) = 4^n(1 + (23/8)n + (9/8)*n^2).

A361610 a(n) = 5^n(n+1)(4n^2+14n+3)/3.