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 1-4 of 4 results.

A292190 Sum of n-th powers of products of terms in all partitions of n into distinct parts.

Original entry on oeis.org

1, 1, 4, 35, 337, 11925, 371081, 49032439, 3545396034, 3416952655320, 749189363202730, 598250899004413536, 2383502427069445040595, 1729793152213690218766715, 131751643363739706679145099315, 271212858254426215135033141804302
Offset: 0

Views

Author

Seiichi Manyama, Sep 11 2017

Keywords

Examples

			5 = 4 + 1 = 3 + 2. So a(5) = 5^5 + (4*1)^5 + (3*2)^5 = 11925.

Links

Crossrefs

Main diagonal of A292189.
Cf. A265949, A292167, A292194, A292305, A292306.

Programs

  • Maple
    b:= proc(n, i, k) option remember; (m->
      `if`(mn, 0, i^k*b(n-i, i-1, k)))))(i*(i+1)/2)
    end:
a:= n-> b(n$3):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 11 2017
  • Mathematica
    nmax = 15; Table[SeriesCoefficient[Product[(1 + k^n*x^k), {k, 1, nmax}], {x, 0, n}], {n, 0, nmax}] (* Vaclav Kotesovec, Sep 12 2017 *)
  • PARI
    {a(n) = polcoeff(prod(k=1, n, 1+k^n*x^k+x*O(x^n)), n)}

Formula

a(n) = [x^n] Product_{k=1..n} (1 + k^n*x^k).

A292193 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - j^k*x^j).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 6, 5, 1, 1, 9, 14, 14, 7, 1, 1, 17, 36, 46, 25, 11, 1, 1, 33, 98, 164, 107, 56, 15, 1, 1, 65, 276, 610, 505, 352, 97, 22, 1, 1, 129, 794, 2324, 2531, 2474, 789, 198, 30, 1, 1, 257, 2316, 8986, 13225, 18580, 7273, 2314, 354, 42
Offset: 0

Views

Author

Seiichi Manyama, Sep 11 2017

Keywords

Examples

			Square array begins:
   1,  1,  1,   1,   1, ...
   1,  1,  1,   1,   1, ...
   2,  3,  5,   9,  17, ...
   3,  6, 14,  36,  98, ...
   5, 14, 46, 164, 610, ...

Links

Crossrefs

Columns k=0..5 give A000041, A006906, A077335, A265837, A265838, A265839.
Rows 0+1, 2 give A000012, A000051.
Main diagonal gives A292194.
Cf. A292166.

Programs

  • Maple
    b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
      `if`(i>n, 0, i^k*b(n-i, i, k))+b(n, i-1, k))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 11 2017
  • Mathematica
    m = 12;
col[k_] := col[k] = Product[1/(1 - j^k*x^j), {j, 1, m}] + O[x]^(m+1) // CoefficientList[#, x]&;
A[n_, k_] := col[k][[n+1]];
Table[A[n, d-n], {d, 0, m}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 11 2021 *)

Formula

A(0,k) = 1 and A(n,k) = (1/n) * Sum_{j=1..n} (Sum_{d|j} d^(1+k*j/d)) * A(n-j,k) for n > 0. - Seiichi Manyama, Nov 02 2017

A301655 a(n) = [x^n] 1/(1 - Sum_{k>=1} k^n*x^k).

Original entry on oeis.org

1, 1, 5, 44, 723, 24655, 1715816, 239697569, 69557364821, 41297123651644, 49900451628509015, 125141540794392423599, 641579398300246011553552, 6729809577032172543373047313, 146355880526667013027682326650073, 6505380999057202235872595196799580684
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 25 2018

Keywords

Comments

Number of compositions (ordered partitions) of n where there are k^n sorts of part k.
a(n) is the n-th term of invert transform of n-th powers.

Links

Crossrefs

Main diagonal of A320251.
Cf. A001906, A033453, A053506, A144109, A193678, A221460, A252782, A292194.

Programs

  • Mathematica
    Table[SeriesCoefficient[1/(1 - Sum[k^n x^k, {k, 1, n}]), {x, 0, n}], {n, 0, 15}]
Table[SeriesCoefficient[1/(1 - PolyLog[-n, x]), {x, 0, n}], {n, 0, 15}]

Formula

a(n) = [x^n] 1/(1 - PolyLog(-n,x)), where PolyLog() is the polylogarithm function.
From Vaclav Kotesovec, Mar 27 2018: (Start)
a(n) ~ 3^(n^2/3) if mod(n,3)=0
a(n) ~ 3^(n*(n-4)/3-2)*2^(2*n-1)*(n-1)*(n+8) if mod(n,3)=1
a(n) ~ 3^((n+1)*(n-3)/3)*2^n*(n+1) if mod(n,3)=2
(End)

A292072 Main diagonal of A292068.

Original entry on oeis.org

1, -1, -3, -20, 66, 4439, 454420, 4873175, -3803048954, -7320203267692, -1403057989033446, 6669491545211096686, 78492109668913945526447, 69591502229308312804788424, -6243846072108996200105800383026, -604234376454072219680822138902122079
Offset: 0

Views

Author

Seiichi Manyama, Sep 12 2017

Keywords

Links

Crossrefs

Cf. A292190, A292194.

Programs

  • Maple
    b:= proc(n, i, k) option remember; (m->
      `if`(mn, 0, i^k*b(n-i, i-1, k)))))(i*(i+1)/2)
    end:
g:= proc(n,k) option remember; `if`(n=0, 1,
      -add(b(n-i$2, k)*g(i, k), i=0..n-1))
    end:
a:= n-> g(n$2):
seq(a(n), n=0..15);  # Alois P. Heinz, Sep 12 2017
  • Mathematica
    b[n_, i_, k_] := b[n, i, k] = Function[m, If[m < n, 0, If[n == m, i!^k, b[n, i - 1, k] + If[i > n, 0, i^k*b[n - i, i - 1, k]]]]][i*(i + 1)/2];
g[n_, k_] := g[n, k] = If[n == 0, 1, -Sum[b[n-i, n-i, k]*g[i, k], {i, 0, n-1}]];
a[n_] := g[n, n];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jun 03 2018, after Alois P. Heinz *)
  • PARI
    {a(n) = polcoeff(1/prod(k=1, n, 1+k^n*x^k+x*O(x^n)), n)}
  • Python
    from sympy.core.cache import cacheit
from sympy import factorial as f
@cacheit
def b(n, i, k):
    m=i*(i + 1)/2
    return 0 if mn else i**k*b(n - i, i - 1, k))
@cacheit
def g(n, k): return 1 if n==0 else -sum([b(n - i, n - i, k)*g(i, k) for i in range(n)])
def a(n): return g(n, n)
print([a(n) for n in range(16)]) # Indranil Ghosh, Sep 14 2017, after Maple program

Formula

a(n) = [x^n] Product_{k=1..n} 1/(1 + k^n*x^k).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.