A007613 -id:A007613 - cp's OEIS frontend

A083322 a(n) = 2^n - A081374(n).

1, 2, 6, 11, 22, 42, 85, 170, 342, 683, 1366, 2730, 5461, 10922, 21846, 43691, 87382, 174762, 349525, 699050, 1398102, 2796203, 5592406, 11184810, 22369621, 44739242, 89478486, 178956971, 357913942, 715827882, 1431655765, 2863311530, 5726623062

Offset: 1

David Applegate, Aug 22 2003

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,2).

Cf. A081374.

Trisections: A082365, A007613, A132804.

I:=[1,2,6,11]; [n le 4 select I[n] else 2*Self(n-1)-Self(n-3)+2*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 08 2016

CoefficientList[Series[(1 + 2 x^2) / ((1 - 2 x) (1 + x) (1 - x + x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 08 2016 *)
LinearRecurrence[{2,0,-1,2},{1,2,6,11},40] (* Harvey P. Dale, Jan 30 2024 *)

G.f.: x*(1+2*x^2) / ( (1-2*x)*(1+x)*(1-x+x^2) ). - R. J. Mathar, May 27 2011
From Paul Curtz, May 27 2011: (Start)
a(n) = 2*a(n-1) - a(n-3) + 2*a(n-4).
a(n)+a(n+3) = 3*2^(n+1) = A007283(n+1).
a(n+6)-a(n) = 21*2^(n+1) = A175805(n+1).
(End)

A139459 Triangle read by rows: binomial(3n,3k), 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 20, 1, 1, 84, 84, 1, 1, 220, 924, 220, 1, 1, 455, 5005, 5005, 455, 1, 1, 816, 18564, 48620, 18564, 816, 1, 1, 1330, 54264, 293930, 293930, 54264, 1330, 1, 1, 2024, 134596, 1307504, 2704156, 1307504, 134596, 2024, 1, 1, 2925, 296010, 4686825, 17383860, 17383860, 4686825, 296010, 2925, 1

Offset: 0

Gary W. Adamson, Apr 22 2008

ConvOffsStoT transform of the dodecahedral numbers A006566 starting (1, 20, 84, 220,...).
Row sums give A007613.
The matrix inverse starts:
  1;
  -1,1;
  19,-20,1;
  -1513,1596,-84,1;
  315523,-332860,17556,-220,1;
  -136085041,143562965,-7572565,95095,-455,1;
  105261234643,-111045393456,5857368972,-73562060,352716,-816,1; - R. J. Mathar, Mar 22 2013

			First few rows of the triangle are:
  [0] 1;
  [1] 1,   1;
  [2] 1,  20,     1;
  [3] 1,  84,    84,     1;
  [4] 1, 220,   924,   220,     1;
  [5] 1, 455,  5005,  5005,   455,   1;
  [6] 1, 816, 18564, 48620, 18564, 816, 1;
  ...
Row 5 = (1, 220, 924, 220, 1) = ConvOffs transform of (1, 20, 84, 220); where A006566 = (0, 1, 20, 84, 220, 455, ...).

Cf. A006566, A007613, A086645, A034839, A070775, A177808.

Table[Binomial[3*n, 3*k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 01 2025 *)

More terms from Amiram Eldar, Jun 01 2025

A326474 A(n, k) = (m*k)! [x^k] MittagLefflerE(m, x)^n, for m = 3, n >= 0, k >= 0; square array read by descending antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 22, 3, 1, 0, 1, 170, 63, 4, 1, 0, 1, 1366, 2187, 124, 5, 1, 0, 1, 10922, 59535, 7732, 205, 6, 1, 0, 1, 87382, 1594323, 599548, 18485, 306, 7, 1, 0, 1, 699050, 43033599, 39945364, 2416045, 36126, 427, 8, 1

Offset: 0

Peter Luschny, Jul 08 2019

			Array starts:
[0] 1, 0,   0,     0,       0,          0,            0, ... A000007
[1] 1, 1,   1,     1,       1,          1,            1, ... A000012
[2] 1, 2,  22,   170,    1366,      10922,        87382, ... A007613
[3] 1, 3,  63,  2187,   59535,    1594323,     43033599, ...
[4] 1, 4, 124,  7732,  599548,   39945364,   2556712828, ...
[5] 1, 5, 205, 18485, 2416045,  352060805,  46660373965, ...
[6] 1, 6, 306, 36126, 6673266, 1544907006, 379696000626, ...
      A051874,

Rows include: A000007, A000012, A007613.
Columns include: A051874.
Cf. A326476 (m=2, p>=0), A326327 (m=2, p<=0), this sequence (m=3, p>=0), A326475 (m=3, p<=0).

(* The function MLPower is defined in A326327. *)
For[n = 0, n < 8, n++, Print[MLPower[3, n, 8]]]

# uses[MLPower from A326327]
for n in (0..6): print(MLPower(3, n, 9))

A361043 Array read by descending antidiagonals. A(n, k) is, if n > 0, the number of multiset permutations of {0, 1} of length n * k where the number of occurrences of 1 are multiples of n. A(0, k) = k + 1.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 4, 2, 1, 5, 8, 8, 2, 1, 6, 16, 32, 22, 2, 1, 7, 32, 128, 170, 72, 2, 1, 8, 64, 512, 1366, 992, 254, 2, 1, 9, 128, 2048, 10922, 16512, 6008, 926, 2, 1, 10, 256, 8192, 87382, 261632, 215766, 37130, 3434, 2, 1, 11, 512, 32768, 699050, 4196352, 6643782, 2973350, 232562, 12872, 2, 1

Offset: 0

Peter Luschny, Mar 18 2023

Because of the interchangeability of 0 and 1 in the definition, A(n, k) is even if n, k >= 1. In other words, if the binary representation of a permutation of the defined type is counted, then so is the 1's complement of that representation.

			Array A(n, k) starts:
 [0] 1, 2,    3,      4,        5,          6,            7, ...  A000027
 [1] 1, 2,    4,      8,       16,         32,           64, ...  A000079
 [2] 1, 2,    8,     32,      128,        512,         2048, ...  A081294
 [3] 1, 2,   22,    170,     1366,      10922,        87382, ...  A007613
 [4] 1, 2,   72,    992,    16512,     261632,      4196352, ...  A070775
 [5] 1, 2,  254,   6008,   215766,    6643782,    215492564, ...  A070782
 [6] 1, 2,  926,  37130,  2973350,  174174002,  11582386286, ...  A070967
 [7] 1, 2, 3434, 232562, 42484682, 4653367842, 644032289258, ...  A094211
.
Triangle T(n, k) starts:
 [0]  1;
 [1]  2,   1;
 [2]  3,   2,    1;
 [3]  4,   4,    2,     1;
 [4]  5,   8,    8,     2,      1;
 [5]  6,  16,   32,    22,      2,      1;
 [6]  7,  32,  128,   170,     72,      2,     1;
 [7]  8,  64,  512,  1366,    992,    254,     2,    1;
 [8]  9, 128, 2048, 10922,  16512,   6008,   926,    2, 1;
 [9] 10, 256, 8192, 87382, 261632, 215766, 37130, 3434, 2, 1;
.
A(2, 2) = 8 = card(0000, 1100, 1010, 1001, 0110, 0101, 0011, 1111).
A(1, 3) = 8 = card(000, 100, 010, 001, 110, 101, 011, 111).

Rows: A000027 (n=0), A000079 (n=1), A081294 (n=2), A007613 (n=3), A070775 (n=4), A070782 (n=5), A070967 (n=6), A094211 (n=7), A070832 (n=8), A094213 (n=9), A070833 (n=10).
Variant: A308500 (upwards antidiagonals).
Cf. A167009 (main diagonal).

T := (n, k) -> add(binomial((n - k)*k, j*k), j = 0 .. n-k):
seq(print(seq(T(n, k), k = 0..n)), n = 0..7);

# In Python use this import:
# from sympy.utilities.iterables import multiset_permutations
def A(n: int, k: int) -> int:
    if n == 0: return k + 1
    count = 0
    for a in range(0, n * k + 1, n):
        S = [i < a for i in range(n * k)]
        count += Permutations(S).cardinality()
    return count
def ARow(n: int, size: int) -> list[int]:
    return [A(n, k) for k in range(size)]
for n in range(6): print(ARow(n, 5))

A(n, k) = Sum_{j=0..k} binomial(n*k, n*j).

T(n, k) = Sum_{j=0..n-k} binomial((n - k)*k, j*k).

A090409 a(n) = (78^n + 2(-1)^n)/9.

Original entry on oeis.org

1, 6, 50, 398, 3186, 25486, 203890, 1631118, 13048946, 104391566, 835132530, 6681060238, 53448481906, 427587855246, 3420702841970, 27365622735758, 218924981886066, 1751399855088526, 14011198840708210, 112089590725665678, 896716725805325426, 7173733806442603406

Offset: 0

Paul Barry, Nov 29 2003

Index entries for linear recurrences with constant coefficients, signature (7,8).

First differences of A015565.

Cf. A007613, A082311, A082365.

LinearRecurrence[{7,8},{1,6},20] (* Harvey P. Dale, Aug 15 2016 *)

a(n) = Sum_{j=0..2} Sum_{k=0..n} C(3*n+j, 3*k)/3.
a(n) = (A007613(n) + A082311(n) + A082365(n))/3.
G.f.: (-1+x)/((1+x)*(8*x-1)). - R. J. Mathar, Dec 10 2014
From Elmo R. Oliveira, Aug 18 2024: (Start)
E.g.f.: exp(-x)*(7*exp(9*x) + 2)/9.
a(n) = 7*a(n-1) + 8*a(n-2) for n > 1. (End)

a(20)-a(21) from Elmo R. Oliveira, Aug 18 2024

A216317 G.f.: 1/( (1-8x)(1+x)^2 )^(1/6).

Original entry on oeis.org

1, 1, 6, 34, 217, 1449, 9996, 70512, 505674, 3672682, 26943748, 199284540, 1483955746, 11113108930, 83628685440, 631963708200, 4793067536265, 36469419494985, 278278625232510, 2128794954411930, 16322188021280505, 125405739232800585, 965313101906567700

Offset: 0

Paul D. Hanna, Sep 03 2012

nonn

			G.f.: A(x) = 1 + x + 6*x^2 + 34*x^3 + 217*x^4 + 1449*x^5 + 9996*x^6 +...
where 1/A(x)^6 = 1 - 6*x - 15*x^2 - 8*x^3.
The logarithm of the g.f. begins:
log(A(x)) = x + 11*x^2/2 + 85*x^3/3 + 683*x^4/4 + 5461*x^5/5 + 43691*x^6/6 +...+ A007613(n)/2*x^n/n +...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A216316, A007613.

CoefficientList[Series[1/((1-8*x)*(1+x)^2)^(1/6), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

{a(n)=polcoeff(1/( (1-8*x)*(1+x)^2 +x*O(x^n) )^(1/6),n)}

{a(n)=local(A=1+x); A=exp(sum(m=1, n+1, sum(j=0, m, binomial(3*m, 3*j))/2*x^m/m +x*O(x^n))); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))

G.f.: exp( Sum_{n>=1} A007613(n)/2 * x^n/n ) where A007613(n) = Sum_{k=0..n}  binomial(3*n,3*k).
Recurrence: n*a(n) = (7*n-6)*a(n-1) + 4*(2*n-3)*a(n-2). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ Gamma(5/6)*8^n/(3^(2/3)*Pi*n^(5/6)). - Vaclav Kotesovec, Oct 20 2012

A308500 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = Sum_{j=0..n} binomial(kn,kj).

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 4, 4, 1, 2, 8, 8, 5, 1, 2, 22, 32, 16, 6, 1, 2, 72, 170, 128, 32, 7, 1, 2, 254, 992, 1366, 512, 64, 8, 1, 2, 926, 6008, 16512, 10922, 2048, 128, 9, 1, 2, 3434, 37130, 215766, 261632, 87382, 8192, 256, 10, 1, 2, 12872, 232562, 2973350, 6643782, 4196352, 699050, 32768, 512, 11

Offset: 0

Seiichi Manyama, Jun 01 2019

			Square array begins:
   1,  1,    1,     1,       1,         1, ...
   2,  2,    2,     2,       2,         2, ...
   3,  4,    8,    22,      72,       254, ...
   4,  8,   32,   170,     992,      6008, ...
   5, 16,  128,  1366,   16512,    215766, ...
   6, 32,  512, 10922,  261632,   6643782, ...
   7, 64, 2048, 87382, 4196352, 215492564, ...

Seiichi Manyama, Antidiagonals n = 0..115, flattened

Columns k=0..10 give A000027(n+1), A000079, A081294, A007613, A070775, A070782, A070967, A094211, A070832, A094213, A070833.

Main diagonal gives A167009.

T[n_, k_] := Sum[Binomial[k*n, k*j], {j, 0, n}] ; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 12 2021 *)

A378031 Cogrowth sequence for the 18-element group C6 X C3 = .

Original entry on oeis.org

1, 1, 2, 85, 926, 5461, 37130, 349525, 2973350, 22369621, 174174002, 1431655765, 11582386286, 91625968981, 729520967450, 5864062014805, 47006639297270, 375299968947541, 2999857885752002, 24019198012642645, 192222214478506046, 1537228672809129301

Offset: 0

Sean A. Irvine, Nov 14 2024

Sequence gives terms for n = 0 (mod 3), all other terms are 0.

Cf. A095364 (D9), A377627 (C6 X C2), A007613 (C3 X C3), A378109 (S3 X C3), A378110 (S3:C3).

G.f.: (36*x^4+99*x^3-14*x^2+6*x-1) / ((8*x-1) * (x+1) * (27*x^2+1)).

A191370 a(n) = 2(1+(-1)^n)/3 + 2A010892(n-1).

Original entry on oeis.org

1, 2, 4, 2, 4, 8, 22, 44, 88, 170, 340, 680, 1366, 2732, 5464, 10922, 21844, 43688, 87382, 174764, 349528, 699050, 1398100, 2796200, 5592406, 11184812, 22369624, 44739242

Offset: 0

Paul Curtz, Jun 01 2011

a(n) and successive differences define an infinite array:
    1,   2,   4,   2,   4,   8, ...
    1,   2,  -2,   2,   4,  14, ...
    1,  -4,   4,   2,  10,   8, ...
   -5,   8,  -2,   8,  -2,  14, ...
   13, -10,  10, -10,  16,   2, ...
  -23,  20, -20,  26, -14,  32, ...
  ...
Its main diagonal consists of the powers 2^n. The first upper diagonal is a signed sequence of 2's. The second upper diagonal contains essentially A135440.

Muniru A Asiru, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,2).

Cf. A010892, A131531, A007613, A001045.

Cf. A007283, A024495, A113405.

A010892 := proc(n) op( 1+(n mod 6),[1,1,0,-1,-1,0]) ; end proc:
A191370 := proc(n) 2^n/3+2*(-1)^n/3+2*A010892(n-1) ; end proc:
seq(A191370(n),n=0..30) ; # R. J. Mathar, Jun 06 2011

LinearRecurrence[{2,0,-1,2},{1,2,4,2},30] (* Harvey P. Dale, Sep 06 2022 *)

a(n+3) = 3*2^n - a(n), n >= 0.
a(n+1) - 2*a(n) = -6*A131531(n+1).
a(3*n) = A007613(n), a(1+3*n) = 2*A007613(n), a(2+3*n) = 4*A007613(n).
a(n+6) = a(n) + 21*2^n.
a(n) = ((2^n + 2*(-1)^n)*2^n - 2*i*sqrt(3)*((1+i*sqrt(3))^n - (1-i*sqrt(3))^n))/(3*2^n), where i=sqrt(-1); a(n+1) = 2*(A001045(n) + A010892(n)). - Bruno Berselli, Jun 06 2011
G.f.: ( -1+5*x^3 ) / ( (2*x-1)*(1+x)*(x^2-x+1) ). - R. J. Mathar, Jun 06 2011
a(n) = 2*a(n-1) - a(n-3) + 2*a(n-4). - Paul Curtz, Jun 07 2011
a(n) = A113405(n+3) - 5*A113405(n). - R. J. Mathar, Jun 24 2011

A191566 a(n) = 7a(n-1) + (-1)^n6*2^(n-1).

Original entry on oeis.org

1, 1, 19, 109, 811, 5581, 39259, 274429, 1921771, 13450861, 94159099, 659107549, 4613765131, 32296331341, 226074368539, 1582520481469, 11077643566891, 77543504575021, 542804532811579, 3799631728108189

Offset: 0

Paul Curtz, Jun 06 2011

A007283(n) = 3*2^n. A091629(n+1) = 6*2^n.
a(n) + a(n+2) = 10 * (b(n) = 2, 11, 83, 569, 4007, ...).
b(n+1) = 7*b(n) - (-1)^n*3*2^n.
Inverse binomial transform of A007613(n).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (5,14).

[(7^n+2*(-2)^n)/3: n in [0..30]]; // Vincenzo Librandi, Jun 07 2011

LinearRecurrence[{5,14},{1,1},40] (* Harvey P. Dale, Mar 01 2017 *)
CoefficientList[Series[(1 - 4*x)/(1 - 5*x - 14*x^2), {x, 0, 20}], x] (* Stefano Spezia, Sep 12 2018 *)

a(n)=(7^n+2*(-2)^n)/3 \\ Charles R Greathouse IV, Jun 06 2011

a(n+1) - a(n) = 18 * (0 followed by A053573(n)).
a(n) = (7^n + 2*(-2)^n)/3. - Charles R Greathouse IV, Jun 06 2011
G.f.: (1-4*x)/(1 - 5*x - 14*x^2). - Bruno Berselli, Jun 07 2011
a(n) = 5*a(n-1) + 14*a(n-2).

cp's OEIS Frontend

A083322 a(n) = 2^n - A081374(n).

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Magma

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A139459 Triangle read by rows: binomial(3*n,3*k), 0 <= k <= n.

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A326474 A(n, k) = (m*k)! [x^k] MittagLefflerE(m, x)^n, for m = 3, n >= 0, k >= 0; square array read by descending antidiagonals.

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A361043 Array read by descending antidiagonals. A(n, k) is, if n > 0, the number of multiset permutations of {0, 1} of length n * k where the number of occurrences of 1 are multiples of n. A(0, k) = k + 1.

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Maple

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A090409 a(n) = (7*8^n + 2*(-1)^n)/9.

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A216317 G.f.: 1/( (1-8*x)*(1+x)^2 )^(1/6).

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PARI

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A308500 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = Sum_{j=0..n} binomial(k*n,k*j).

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Mathematica

A378031 Cogrowth sequence for the 18-element group C6 X C3 = .

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A191370 a(n) = 2*(1+(-1)^n)/3 + 2*A010892(n-1).

A139459 Triangle read by rows: binomial(3n,3k), 0 <= k <= n.

A090409 a(n) = (78^n + 2(-1)^n)/9.

A216317 G.f.: 1/( (1-8x)(1+x)^2 )^(1/6).

A308500 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = Sum_{j=0..n} binomial(kn,kj).

A191370 a(n) = 2(1+(-1)^n)/3 + 2A010892(n-1).

A191566 a(n) = 7a(n-1) + (-1)^n6*2^(n-1).