A081295 -id:A081295 - cp's OEIS frontend

A081294 Expansion of (1-2x)/(1-4x).

1, 2, 8, 32, 128, 512, 2048, 8192, 32768, 131072, 524288, 2097152, 8388608, 33554432, 134217728, 536870912, 2147483648, 8589934592, 34359738368, 137438953472, 549755813888, 2199023255552, 8796093022208, 35184372088832

Offset: 0

Paul Barry, Mar 17 2003

Binomial transform of A046717. Second binomial transform of A000302 (with interpolated zeros). Partial sums are A007583.
Counts closed walks of length 2n at a vertex of the cyclic graph on 4 nodes C_4. With interpolated zeros, counts closed walks of length n at a vertex of the cyclic graph on 4 nodes C_4. - Paul Barry, Mar 10 2004
In general, Sum_{k=0..n} Sum_{j=0..n} C(2*(n-k), j)*C(2*k, j)*r^j has expansion (1 - (r+1)*x)/(1 - (r+3)*x - (r-1)*(r+3)*x^2 + (r-1)^3*x^3). - Paul Barry, Jun 04 2005 [corrected by Jason Yuen, Jan 20 2025]
a(n) is the number of binary strings of length 2n with an even number of 0's (and hence an even number of 1's). - Toby Gottfried, Mar 22 2010
Number of compositions of n where there are 2 sorts of part 1, 4 sorts of part 2, 8 sorts of part 3, ..., 2^k sorts of part k. - Joerg Arndt, Aug 04 2014
a(n) is also the number of permutations simultaneously avoiding 231 and 321 in the classical sense which can be realized as labels on an increasing strict binary tree with 2n-1 nodes. See A245904 for more information on increasing strict binary trees. - Manda Riehl Aug 07 2014
INVERT transform of powers of 2 (A000079). - Alois P. Heinz, Feb 11 2021
a(n) is the number of elements in an n-interval of the binomial poset of even-sized subsets of positive integers, cf. Stanley reference and second formula by Paul Barry.  Each multichain 0 = x_0 <= x_1 <= x_2 = 1 in such an n-interval corresponds to a closed walk described above by Paul Barry.  More generally, each multichain 0 = x_0 <= x_1 <= ... <= x_k = 1 corresponds to a closed walk of length 2n on the k-dimensional hypercube, cf. A054879, A092812, A121822. - Geoffrey Critzer, Apr 21 2023

			G.f. = 1 + 2*x + 8*x^2 + 32*x^3 + 128*x^4 + 512*x^5 + 2048*x^6 + 8192*x^7 + ...

Richard P. Stanley, Enumerative Combinatorics, Vol 1, second edition, Example 3.18.3-f, page 323.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
M. Paukner, L. Pepin, M. Riehl, and J. Wieser, Pattern Avoidance in Task-Precedence Posets, arXiv:1511.00080 [math.CO], 2015-2016.
Index to divisibility sequences.
Index entries for linear recurrences with constant coefficients, signature (4).

Row sums of triangle A136158.

Cf. A000079, A081295, A009117, A016742, A054879, A092812, A121822. Essentially the same as A004171.

[(4^n+0^n)/2: n in [0..30]]; // Vincenzo Librandi, Jul 26 2011

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (1-2*x)/(1-4*x))); // Marius A. Burtea, Jan 20 2020

a:= n-> 2^max(0, (2*n-1)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 20 2017

CoefficientList[Series[(1-2x)/(1-4x),{x,0,40}],x] (* or *)
Join[{1}, NestList[4 # &, 2, 40]] (* Harvey P. Dale, Apr 22 2011 *)

a(n)=1<Charles R Greathouse IV, Jul 25 2011

x='x+O('x^100); Vec((1-2*x)/(1-4*x)) \\ Altug Alkan, Dec 21 2015

G.f.: (1-2*x)/(1-4*x).
a(n) = 4*a(n-1) n > 1, with a(0)=1, a(1)=2.
a(n) = (4^n+0^n)/2 (i.e., 1 followed by 4^n/2, n > 0).
E.g.f.: exp(2*x)*cosh(2*x) = (exp(4*x)+exp(0))/2. - Paul Barry, May 10 2003
a(n) = Sum_{k=0..n} C(2*n, 2*k). - Paul Barry, May 20 2003
a(n) = A001045(2*n+1) - A001045(2*n-1) + 0^n/2. - Paul Barry, Mar 10 2004
a(n) = 2^n*A011782(n); a(n) = gcd(A011782(2n), A011782(2n+1)). - Paul Barry, Jan 12 2005
a(n) = Sum_{k=0..n} Sum_{j=0..n} C(2*(n-k), j)*C(2*k, j). - Paul Barry, Jun 04 2005
a(n) = Sum_{k=0..n} A038763(n,k). - Philippe Deléham, Sep 22 2006
a(n) = Integral_{x=0..4} p(n,x)^2/(Pi*sqrt(x(4-x))) dx, where p(n,x) is the sequence of orthogonal polynomials defined by C(2*n,n): p(n,x) = (2*x-4)*p(n-1,x) - 4*p(n-2,x), with p(0,x)=1, p(1,x)=-2+x. - Paul Barry, Mar 01 2007
a(n) = ((2+sqrt(4))^n + (2-sqrt(4))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Nov 22 2008
a(n) = A000079(n) * A011782(n). - Philippe Deléham, Dec 01 2008
a(n) = A004171(n-1) = A028403(n) - A000079(n) for n >= 1. - Jaroslav Krizek, Jul 27 2009
a(n) = Sum_{k=0..n} A201730(n,k)*3^k. - Philippe Deléham, Dec 06 2011
a(n) = Sum_{k=0..n} A134309(n,k)*2^k = Sum_{k=0..n} A055372(n,k). - Philippe Deléham, Feb 04 2012
G.f.: Q(0), where Q(k) = 1 - 2*x/(1 - 2/(2 - 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 29 2013
E.g.f.: 1/2 + exp(4*x)/2 = (Q(0)+1)/2, where Q(k) = 1 + 4*x/(2*k+1 - 2*x*(2*k+1)/(2*x + (k+1)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 29 2013
a(n) = ceiling( 2^(2n-1) ). - Wesley Ivan Hurt, Jun 30 2013
G.f.: 1 + 2*x/(1 + x)*( 1 + 5*x/(1 + 4*x)*( 1 + 8*x/(1 + 7*x)*( 1 + 11*x/(1 + 10*x)*( 1 + ... )))). - Peter Bala, May 27 2017
Sum_{n>=0} 1/a(n) = 5/3. - Amiram Eldar, Aug 18 2022
Sum_{n>=0} a(n)*x^n/A000680(n) = E(x)^2 where E(x) = Sum_{n>=0} x^n/A000680(n). - Geoffrey Critzer, Apr 21 2023

A344817 a(n) = Sum_{k=1..n} floor(n/k) * (-2)^(k-1).

Original entry on oeis.org

1, 0, 5, -4, 13, -16, 49, -88, 173, -324, 701, -1384, 2713, -5416, 10989, -21916, 43621, -87224, 174921, -349872, 698773, -1397356, 2796949, -5593872, 11183361, -22366976, 44742149, -89483716, 178951741, -357903312, 715838513, -1431678040, 2863290285

Offset: 1

Seiichi Manyama, May 29 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Column k=2 of A344824.
Cf. A081295.
Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): A344820 (q=-n), A344819 (q=-4), A344818 (q=-3), this sequence (q=-2), A059851 (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), A344815 (q=4), A344816 (q=5), A332533 (q=n).

A344817:= func< n | (&+[Floor(n/k)*(-2)^(k-1): k in [1..n]]) >;
[A344817(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024

a[n_] := Sum[(-2)^(k - 1) * Quotient[n, k], {k, 1, n}]; Array[a, 30] (* Amiram Eldar, May 29 2021 *)

PARI
```
a(n) = sum(k=1, n, n\k*(-2)^(k-1));
```

a(n) = sum(k=1, n, sumdiv(k, d, (-2)^(d-1)));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1+2*x^k))/(1-x))

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (-2)^(k-1)*x^k/(1-x^k))/(1-x))

def A344817(n): return sum((n//k)*(-2)^(k-1) for k in range(1,n+1))
[A344817(n) for n in range(1,41)] # G. C. Greubel, Jun 26 2024

a(n) = Sum_{k=1..n} Sum_{d|k} (-2)^(d-1).
G.f.: (1/(1 - x)) * Sum_{k>=1} x^k/(1 + 2*x^k).
G.f.: (1/(1 - x)) * Sum_{k>=1} (-2)^(k-1) * x^k/(1 - x^k).
a(n) ~ -(-1)^n * 2^n / 3. - Vaclav Kotesovec, Jun 05 2021

A128315 Inverse Moebius transform of signed A007318.

Original entry on oeis.org

1, 0, 1, 2, -2, 1, -1, 4, -3, 1, 2, -4, 6, -4, 1, 0, 4, -9, 10, -5, 1, 2, -6, 15, -20, 15, -6, 1, -2, 11, -24, 36, -35, 21, -7, 1, 3, -10, 29, -56, 70, -56, 28, -8, 1, 0, 6, -30, 80, -125, 126, -84, 36, -9, 1, 2, -10, 45, -120, 210, -252, 210, -120, 45, -10, 1, -2, 18, -67, 176, -335, 463, -462, 330, -165, 55, -11, 1

Offset: 1

Gary W. Adamson, Feb 25 2007

A128316 = A000012 * A128315.

			First few rows of the triangle:
   1;
   0,  1;
   2, -2,  1;
  -1,  4, -3,  1;
   2, -4,  6, -4,  1;
   0,  4, -9, 10, -5, 1;
  ...

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A000034, A000984, A007318, A010693, A048272, A051731, A128316.
Cf. A130595, A325940, A363615, A363616.
Cf. A000012 (row sums).

A128315:= func< n,k | (&+[0^(n mod j)*(-1)^(k+j)*Binomial(j-1, k-1): j in [k..n]]) >;
[A128315(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jun 22 2024

A128315[n_, k_]:= (-1)^k*DivisorSum[n, (-1)^#*Binomial[#-1, k-1] &];
Table[A128315[n,k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jun 22 2024 *)

def A128315(n,k): return sum( 0^(n%j)*(-1)^(k+j)*binomial(j-1,k-1) for j in range(k,n+1))
flatten([[A128315(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Jun 22 2024

T(n, k) = A051731(n, k) * A130595(n-1, k-1) as infinite lower triangular matrices.
T(n, 1) = A048272(n).
Sum_{k=1..n} T(n, k) = A000012(n) = 1 (row sums).
From G. C. Greubel, Jun 22 2024: (Start)
T(n, k) = (-1)^k * Sum_{d|n} (-1)^d * binomial(d-1, k-1).
T(n, 2) = A325940(n), n >= 2.
T(n, 3) = A363615(n), n >= 3.
T(n, 4) = A363616(n), n >= 4.
T(2*n-1, n) = (-1)^(n-1)*A000984(n-1), n >= 1.
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n-1)*A081295(n).
Sum_{k=1..n} k*T(n, k) = A000034(n-1), n >= 1.
Sum_{k=1..n} (k+1)*T(n, k) = A010693(n-1), n >= 1. (End)

a(43) = 28 inserted and more terms from Georg Fischer, Jun 05 2023

A101561 a(n) = (-1)^n * [x^n] Sum_{k>=1} x^(k-1)/(1+3*x^k).

Original entry on oeis.org

1, 2, 10, 29, 82, 236, 730, 2216, 6571, 19604, 59050, 177410, 531442, 1593596, 4783060, 14351123, 43046722, 129133838, 387420490, 1162281098, 3486785140, 10460294156, 31381059610, 94143358424, 282429536563, 847288078004, 2541865834900, 7625599078610

Offset: 0

Paul Barry, Dec 07 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A048272, A051731, A081295, A101562, A101563.

A101561:= func< n | (&+[(-1)^(n-k)*3^k*0^((n+1) mod (k+1)): k in [0..n]]) >;
[A101561(n): n in [0..40]]; // G. C. Greubel, Jun 25 2024

a[n_]:= Sum[(-1)^(n-k) * If[Mod[n+1, k+1]==0, 1, 0] * 3^k, {k, 0, n}];
Table[a[n], {n, 0, 25}] (* James C. McMahon, Jan 01 2024 *)
A101561[n_]:= (-1)^n*DivisorSum[n+1, (-3)^(#-1) &];
Table[A101561[n], {n,0,40}] (* G. C. Greubel, Jun 25 2024 *)

def A101561(n): return sum((-1)^(n+k)*3^k*0^((n+1)%(k+1)) for k in range(n+1))
[A101561(n) for n in range(41)] # G. C. Greubel, Jun 25 2024

a(n) = Sum_{k=0..n} (-1)^(n-k) * 3^k * A051731(n+1, k+1).

a(n) = (-1)^n * Sum_{d|n+1} (-3)^(d-1). - G. C. Greubel, Jun 25 2024

A101562 a(n) = (-1)^n * coefficient of x^n in Sum_{k>=1} x^(k-1)/(1+4*x^k).

Original entry on oeis.org

1, 3, 17, 67, 257, 1011, 4097, 16451, 65553, 261891, 1048577, 4195379, 16777217, 67104771, 268435729, 1073758275, 4294967297, 17179804659, 68719476737, 274878168899, 1099511631889, 4398045462531, 17592186044417, 70368748389427

Offset: 0

Paul Barry, Dec 07 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A048272, A051731, A081295, A101561, A101563.

A101562:= func< n | (&+[(-1)^(n-k)*4^k*0^((n+1) mod (k+1)): k in [0..n]]) >;
[A101562(n): n in [0..40]]; // G. C. Greubel, Jun 25 2024

A101562[n_]:= (-1)^n*DivisorSum[n+1, (-4)^(#-1) &];
Table[A101562[n], {n,0,40}] (* G. C. Greubel, Jun 25 2024 *)

def A101562(n): return sum((-1)^(n+k)*4^k*0^((n+1)%(k+1)) for k in range(n+1))
[A101562(n) for n in range(41)] # G. C. Greubel, Jun 25 2024

a(n) = Sum_{k=0..n} (-1)^(n-k) * 4^k * A051731(n+1, k+1).

a(n) = (-1)^n * Sum_{d|n+1} (-4)^(d-1). - G. C. Greubel, Jun 25 2024

A101563 a(n) = (-1)^n * coefficient of x^n in Sum_{k>=1} x^(k-1)/(1+10*x^k).

Original entry on oeis.org

1, 9, 101, 1009, 10001, 99909, 1000001, 10001009, 100000101, 999990009, 10000000001, 100000100909, 1000000000001, 9999999000009, 100000000010101, 1000000010001009, 10000000000000001, 99999999900099909

Offset: 0

Paul Barry, Dec 07 2004

G. C. Greubel, Table of n, a(n) for n = 0..990

Cf. A051731, A081295, A048272, A101561, A101562.

A101563:= func< n | (&+[(-1)^(n-k)*(10)^k*0^((n+1) mod (k+1)): k in [0..n]]) >;
[A101563(n): n in [0..40]]; // G. C. Greubel, Jun 25 2024

A101563[n_]:= (-1)^n*DivisorSum[n+1, (-10)^(#-1) &];
Table[A101563[n], {n,0,40}] (* G. C. Greubel, Jun 25 2024 *)

def A101563(n): return sum((-1)^(n+k)*(10)^k*0^((n+1)%(k+1)) for k in range(n+1))
[A101563(n) for n in range(41)] # G. C. Greubel, Jun 25 2024

a(n) = Sum_{k=0..n} (-1)^(n-k) * (10)^k * A051731(n+1, k+1).

a(n) = (-1)^n * Sum_{d|n+1} (-10)^(d-1). - G. C. Greubel, Jun 25 2024

cp's OEIS Frontend

A081294 Expansion of (1-2*x)/(1-4*x).

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A344817 a(n) = Sum_{k=1..n} floor(n/k) * (-2)^(k-1).

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A128315 Inverse Moebius transform of signed A007318.

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A101561 a(n) = (-1)^n * [x^n] Sum_{k>=1} x^(k-1)/(1+3*x^k).

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A101562 a(n) = (-1)^n * coefficient of x^n in Sum_{k>=1} x^(k-1)/(1+4*x^k).

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A101563 a(n) = (-1)^n * coefficient of x^n in Sum_{k>=1} x^(k-1)/(1+10*x^k).

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A081294 Expansion of (1-2x)/(1-4x).