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.

Showing 1-3 of 3 results.

A175788 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of n that do not contain k as a part.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 1, 1, 3, 1, 1, 1, 1, 5, 1, 1, 2, 2, 2, 7, 1, 1, 2, 2, 3, 2, 11, 1, 1, 2, 3, 4, 4, 4, 15, 1, 1, 2, 3, 4, 5, 6, 4, 22, 1, 1, 2, 3, 5, 6, 8, 8, 7, 30, 1, 1, 2, 3, 5, 6, 9, 10, 11, 8, 42, 1, 1, 2, 3, 5, 7, 10, 12, 15, 15, 12, 56
Offset: 0

Views

Author

Alois P. Heinz, Dec 04 2010

Keywords

Examples

			Square array A(n,k) begins:
  1, 1, 1, 1, 1, 1, ...
  1, 0, 1, 1, 1, 1, ...
  2, 1, 1, 2, 2, 2, ...
  3, 1, 2, 2, 3, 3, ...
  5, 2, 3, 4, 4, 5, ...
  7, 2, 4, 5, 6, 6, ...

Links

Crossrefs

Columns k=0-10 give: A000041, A002865, A027336, A027337, A027338, A027339, A027340, A027341, A027342, A027343, A027344.
Rows n=0-1 give: A000012, A060576.
Main diagonal gives A000065 (for n>0).

Programs

  • Maple
    A41:= n-> `if`(n<0, 0, combinat[numbpart](n)):
A:= (n,k)-> A41(n) -`if`(k>0, A41(n-k), 0):
seq(seq(A(n,d-n), n=0..d), d=0..11);
  • Mathematica
    A41[n_] := If[n<0, 0, PartitionsP[n]]; A[n_, k_] := A41[n]-If[k>0, A41[n-k], 0]; Table[A[n, d-n], {d, 0, 11}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 18 2017, translated from Maple *)

Formula

G.f. of column 0: Product_{m>0} 1/(1-x^m).
G.f. of column k>0: (1-x^k) * Product_{m>0} 1/(1-x^m).
A(n,0) = A000041(n); A(n,k) = A000041(n) - A000041(n-k) for k>0.
For fixed k>0, A(n,k) ~ k*Pi * exp(sqrt(2*n/3)*Pi) / (12*sqrt(2)*n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + Pi/(24*sqrt(6)) + k*Pi/(2*sqrt(6)))/sqrt(n) + (1/8 + 3*k/2 + 9/(2*Pi^2) + Pi^2/6912 + k*Pi^2/288 + k^2*Pi^2/36)/n). - Vaclav Kotesovec, Nov 04 2016

A343666 Number of partitions of an n-set without blocks of size 6.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 202, 870, 4084, 20727, 112825, 654546, 4026487, 26145511, 178550986, 1278168860, 9564026947, 74615547996, 605593775899, 5103054929621, 44564754448972, 402677613100491, 3759094788129312, 36205919126040190, 359340174509911325, 3670825700549853053
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 25 2021

Keywords

Crossrefs

Cf. A000110, A000296, A027340, A097514, A124504, A343664, A343665, A343667, A343668, A343669.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=6, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 25 2021
  • Mathematica
    nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^6/6!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 6 k]/((n - 6 k)! k! (6!)^k), {k, 0, Floor[n/6]}], {n, 0, 25}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 6, 0, Binomial[n - 1, k - 1]  a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 25}]

Formula

E.g.f.: exp(exp(x) - 1 - x^6/6!).
a(n) = n! * Sum_{k=0..floor(n/6)} (-1)^k * Bell(n-6*k) / ((n-6*k)! * k! * (6!)^k).

A132091 Expansion of psi(x^3) * chi(-x^9) / f(-x^2) in powers of x where psi(), chi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 3, 2, 5, 3, 7, 5, 10, 7, 14, 11, 20, 15, 27, 22, 37, 30, 49, 42, 66, 56, 86, 75, 113, 99, 146, 131, 189, 170, 241, 221, 308, 283, 389, 363, 492, 460, 616, 583, 771, 732, 958, 918, 1189, 1143, 1467, 1421, 1807, 1756, 2215, 2166, 2711, 2658, 3303, 3256
Offset: 0

Views

Author

Michael Somos, Aug 09 2007

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Also number of partitions of n into parts not divisible by 3 with every part repeated at least twice. Conjectured by R. H. Hardin, Jun 06 2009, proved by Max Alekseyev, Jun 06 2009.
The number of partitions of n into parts not divisible by 3 with every part repeated at least twice has g.f. f(x) = Product_{k>=1} (1 + x^(2k) + x^(3*k) + ...) = Product_{k>=1} (1/(1-x^k) - x^k) = Product_{k>=1} (1 - x^k + x^(2*k)) / (1 - x^k). Excluding parts divisible by 3, we have: f(x) / f(x^3) = Product_{k>=1} (1 - x^k + x^(2*k)) * (1 - x^(3*k)) / (1 - x^k) / (1 - x^(3*k) + x^(6*k)) = Product_{k>=1} (1 - x^k + x^(2*k)) * (1 + x^k + x^(2*k)) / (1 - x^(3*k) + x^(6*k)) = Product_{k>=1} (1 + x^(2*k) + x^(4*k)) / (1 - x^(3*k) + x^(6*k)), which matches the definition of this sequence. - Max Alekseyev, Jun 06 2009

Examples

			G.f. = 1 + x^2 + x^3 + 2*x^4 + x^5 + 3*x^6 + 2*x^7 + 5*x^8 + 3*x^9 + ...
G.f. = 1/q + q^23 + q^35 + 2*q^47 + q^59 + 3*q^71 + 2*q^83 + 5*q^95 + ...

Links

Crossrefs

Cf. A000701, A027340.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(3/2)] / (2 x^(3/8) QPochhammer[ -x^9, x^9] QPochhammer[ x^2]), {x, 0, n}]; (* Michael Somos, Aug 25 2015 *)
nmax=60; CoefficientList[Series[Product[(1-x^(6*k))^2 * (1-x^(9*k)) / ( (1-x^(2*k)) * (1-x^(3*k)) * (1-x^(18*k))) ,{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 14 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^2 * eta(x^9 + A )/ (eta(x^2+A) * eta(x^3 + A) * eta(x^18 + A)), n))};

Formula

Expansion of q^(1/12) * eta(q^6)^2 * eta(q^9) / ( eta(q^2) * eta(q^3) * eta(q^18)) in powers of q.
Euler transform of period 18 sequence [ 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, ...].
G.f.: Product_{k>0} (1 + x^(2*k) + x^(4*k)) / (1 - x^(3*k) + x^(6*k)).
G.f.: Sum_{k>=0} Product_{0
a(2*n - 1) = A000701(n). a(2*n) = A027340(n) = - Michael Somos, Aug 25 2015
a(n) ~ exp(2*Pi*sqrt(2*n/3)/3) / (2^(3/4) * 3^(5/4) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015

Extensions

Edited by N. J. A. Sloane, Jun 07 2009
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