cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-18 of 18 results.

A356283 a(n) = Sum_{k=0..n} binomial(3*n, n-k) * q(k), where q(k) is the number of partitions into distinct parts (A000009).

Original entry on oeis.org

1, 4, 22, 131, 807, 5066, 32188, 206242, 1329733, 8614685, 56024538, 365491218, 2390613557, 15671221522, 102925324569, 677110860689, 4460956827127, 29427611146335, 194348311824025, 1284856925961827, 8502252246841668, 56309476194587377, 373220349572126265
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 01 2022

Keywords

Crossrefs

Cf. A000009, A188675, A356281, A356282.

Programs

  • Mathematica
    Table[Sum[PartitionsQ[k]*Binomial[3*n, n-k], {k, 0, n}], {n, 0, 30}]

Formula

a(n) ~ c * 3^(3*n + 1/2) / (sqrt(Pi*n) * 2^(2*n + 1)), where c = Sum_{j>=0} q(j)/2^j = A079555 = 2.384231029031371724149899288678...

A382976 Expansion of Product_{k>=1} (1 + (2^k + 1) * x^k).

Original entry on oeis.org

1, 3, 5, 24, 44, 129, 384, 897, 2220, 5706, 15268, 35178, 89829, 212982, 526222, 1294263, 3087570, 7300896, 17726100, 41705904, 98782950, 236059794, 551697495, 1293417672, 3033232130, 7081297146, 16430673765, 38347412562, 88762751808, 204970377366, 473719894598
Offset: 0

Views

Author

Seiichi Manyama, Apr 11 2025

Keywords

Comments

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1, g(n) = -(2^n + 1).

Links

Crossrefs

Cf. A322199, A382977, A382978.
Cf. A000051, A266964, A284593.
Cf. A079555.

Programs

  • Mathematica
    n=30; CoefficientList[Normal@Series[Product[1+(2^k+1) x^k,{k,1,n}],{x,0,n}],x] (* Vincenzo Librandi, Apr 11 2025 *)
  • PARI
    f(n) = -1;
g(n) = -(2^n+1);
a_vector(n) = my(b=vector(n, k, sumdiv(k, d, d*f(d)*g(d)^(k/d))), v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, b[j]*v[i-j+1])/i); v;

Formula

a(n) = Sum_{k=0..n} 2^k * A284593(k,n-k).
a(n) ~ A079555 * 2^(n-1) * exp(Pi*sqrt(n/3)) / (3^(5/4) * n^(3/4)). - Vaclav Kotesovec, Apr 11 2025

A319919 Expansion of Product_{k>=1} (1 + x^k)^(2^k-1).

Original entry on oeis.org

1, 1, 3, 10, 25, 70, 182, 476, 1220, 3122, 7883, 19794, 49340, 122237, 301114, 737923, 1799597, 4369204, 10563800, 25441377, 61048713, 145988775, 347981713, 826921992, 1959363778, 4629903905, 10911757432, 25652950459, 60165831361, 140792215037, 328750398275, 766041930160, 1781452975346
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 01 2018

Keywords

Comments

Convolution of A081362 and A102866.
Weigh transform of A000225.

Links

Crossrefs

Cf. A000225, A007070, A081362, A098407, A102866, A319918.

Programs

  • Maple
    a:=series(mul((1+x^k)^(2^k-1),k=1..100),x=0,33): seq(coeff(a,x,n),n=0..32); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 32; CoefficientList[Series[Product[(1 + x^k)^(2^k - 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[Exp[Sum[(-1)^(k + 1) x^k/(k (1 - x^k) (1 - 2 x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (2^d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 32}]

Formula

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)*(1 - 2*x^k))).
a(n) ~ c * exp(2*sqrt(n) - 1/2) * 2^(n-1) / (A079555 * sqrt(Pi) * n^(3/4)), where c = exp(Sum_{k>=2} (-1)^(k-1)/(k*(2^(k-1)-1))) = 0.6602994483152065685... - Vaclav Kotesovec, Sep 15 2021

A082472 a(1) = 1, a(n) = Sum_{k=1..n-1} a(k)*2^k.

Original entry on oeis.org

1, 2, 10, 90, 1530, 50490, 3281850, 423358650, 108803173050, 55816027774650, 57211428469016250, 117226216933014296250, 480275810774559571736250, 3934899717675966571235096250, 64473331874120712269687052056250
Offset: 1

Views

Author

Benoit Cloitre, Apr 27 2003

Keywords

Links

Crossrefs

Cf. A005329, A079555.

Programs

  • Mathematica
    Join[{1},RecurrenceTable[{a[1]==2,a[n]==(1+2^n) a[-1+n]},a[n], {n,15}]] (* Harvey P. Dale, May 11 2011 *)
  • Sage
    from ore_algebra import *
R. = QQ['x']; A. = OreAlgebra(R, 'Qx', q=2)
print((Qx - x - 1).to_list([0,1,2], 10)) # Ralf Stephan, Apr 24 2014

Formula

a(n+1) = (2^n+1)*a(n) for n>=2.
a(n) is asymptotic to c*2^(n*(n-1)/2) where c = Product_{k>=1} (1+1/(2*2^k)) = 1.5894873526.....
c = 2*A079555/3. - Vaclav Kotesovec, Jun 05 2020
G.f. A(x) satisfies: A(x) = x * (1 + A(2*x) / (1 - x)). - Ilya Gutkovskiy, Jun 04 2020

A330863 Decimal expansion of Product_{k>=1} (1 + 1/(-2)^k).

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Apr 28 2020

Keywords

Examples

			(1 - 1/2) * (1 + 1/2^2) * (1 - 1/2^3) * (1 + 1/2^4) * (1 - 1/2^5) * ... = 0.568698946265428505954976737...

Crossrefs

Cf. A000593, A027637, A048651, A062510, A065446, A079555, A081845, A085011, A100221, A132020, A330862, A216206.

Programs

  • Mathematica
    RealDigits[QPochhammer[-1, -1/2]/2, 10, 110] [[1]]
N[3/QPochhammer[-2, 1/4], 120] (* Vaclav Kotesovec, Apr 28 2020 *)
  • PARI
    prodinf(k=1, 1 + 1/(-2)^k) \\ Michel Marcus, Apr 28 2020

Formula

Equals Product_{k>=1} 1/(1 + 1/2^(2*k-1)).
Equals exp(Sum_{k>=1} A000593(k)/(k*(-2)^k)).
From Peter Bala, Dec 15 2020: (Start)
Constant C = (2/3) - (1/3)*Sum_{n >= 0} (-1)^n * 2^(n^2)/( Product_{k = 1..n+1} 4^k - 1 ).
C = Sum_{n >= 0} 1/( Product_{k = 1..n} (-2)^k - 1 ) = 1 - 1/3 - 1/9 + 1/81 + 1/1215 - - + + ... = Sum_{n >= 0} 1/A216206(n).
C = 1 + Sum_{n >= 0} (-1/2)^(n+1)*Product_{k = 1..n} (1 + (-1/2)^k).
3*C = 2 - Sum_{n >= 0} (1/4)^(n+1)*Product_{k = 1..n} (1 + (-1/2)^k).
9*C = 5 - Sum_{n >= 0} (-1/8)^(n+1)*Product_{k = 1..n} (1 + (-1/2)^k).
81*C = 46 + Sum_{n >= 0} (1/16)^(n+1)*Product_{k = 1..n} (1 + (-1/2)^k).
1215*C = 691 + Sum_{n >= 0} (-1/32)^(n+1)*Product_{k = 1..n} (1 + (-1/2)^k).
The sequence [1, 2, 5, 46, 691, ...] is the sequence of numerators of the partial sums of the series Sum_{n >= 0} 1/A216206(n). (End)

A347829 a(n) = Sum_{k=0..n} 2^k * A000041(k) * A000009(n-k).

Original entry on oeis.org

1, 3, 11, 36, 118, 351, 1082, 3093, 8984, 25164, 70434, 191808, 525559, 1404672, 3755506, 9906111, 26057062, 67703310, 175745506, 451392114, 1157272780, 2944110060, 7468477985, 18821686554, 47337840114, 118344795738, 295156919969, 732694232394, 1814357671094
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 15 2021

Keywords

Crossrefs

Cf. A000009, A000041, A015128, A079555, A264685, A347830.

Programs

  • Mathematica
    Table[Sum[2^k*PartitionsP[k]*PartitionsQ[n-k], {k, 0, n}], {n, 0, 50}]
nmax = 50; CoefficientList[Series[Product[(1 + x^k) / (1 - 2^k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ A079555 * 2^n * A000041(n).
a(n) ~ QPochhammer(-1/2, 1/2) * 2^(n-2) * exp(Pi*sqrt(2*n/3)) / (sqrt(3)*n).
G.f.: Product_{k>=1} (1 + x^k) / (1 - 2^k*x^k).

A353196 Number of stabilizer states on n qubits.

Original entry on oeis.org

6, 60, 1080, 36720, 2423520, 315057600, 81284860800, 41780418451200, 42866709330931200, 87876754128408960000, 360118938418219918080000, 2950814581398894008747520000, 48352047730802277227336862720000, 1584496604138390624739828991334400000
Offset: 1

Views

Author

James Rayman, Apr 29 2022

Keywords

Comments

A stabilizer state is a quantum state on n qubits prepared by applying a series of Hadamard, CNOT, and S gates to the all-zero state. There are only a finite number of such states for any n.

Examples

			For n = 1, the a(1) = 6 states are |0>, |1>, |+>, |->, |i>, and |-i>.

Links

Crossrefs

Cf. A000079, A003956, A028362.

Programs

  • Mathematica
    Table[2^n * QPochhammer[-2, 2, n], {n, 13}] (* Amiram Eldar, Aug 17 2025 *)
  • Python
    def a(n):
    ans = 2 ** n
    for i in range(1, n+1):
        ans *= 2 ** i + 1
    return ans
  • Python
    from math import prod
def A353196(n): return prod((1<Chai Wah Wu, Jun 20 2022

Formula

a(n) = 2^n*Product_{i=1..n} (2^i+1).
a(n) = A000079(n)*A028362(n+1).
a(n) ~ c * 2^(n*(n+3)/2) where c = Product_{k>=1} (1 + 1/2^k) = A079555. - Amiram Eldar, Aug 17 2025

A079674 Continued fraction expansion of Product_{k>=1} (1 + 1/2^k) = 2.384231029031371...

Original entry on oeis.org

2, 2, 1, 1, 1, 1, 14, 1, 3, 1, 1, 6, 9, 18, 7, 1, 27, 1, 1, 4, 1, 1, 1, 1, 1, 2, 1, 1, 4, 7, 4, 36, 1, 6, 1, 28, 1, 1, 6, 1, 3, 2, 150, 2, 1, 1, 36, 3, 2, 6, 4, 1, 2, 1, 1, 9, 1, 12, 12, 6, 7, 7, 2, 4, 3, 56, 1, 22, 1, 7, 2, 1, 1, 36, 4, 1, 3, 1, 1, 2, 1, 10, 1, 1, 82, 16, 2, 1, 1, 6, 15, 1, 2, 1, 5, 1, 1
Offset: 0

Views

Author

Benoit Cloitre, Jan 25 2003

Keywords

Crossrefs

Cf. A028362.
Cf. A079555 (decimal expansion).

Programs

  • Mathematica
    ContinuedFraction[QPochhammer[-1/2,1/2],97] (* Stefano Spezia, Aug 19 2025 *)

Extensions

Offset changed by Andrew Howroyd, Jul 06 2024
Previous Showing 11-18 of 18 results.

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