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.
nmax = 50; CoefficientList[Series[Product[Product[1/(1-x^(k*j))^j, {j, 1, Min[k, nmax/k]}], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Product[Product[1/(1-x^(k*j))^k, {j, 1, Min[k, nmax/k]}], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Product[Product[1/(1-x^(k*j))^(k*j), {j, 1, Min[k, nmax/k]}], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[Product[1/(1 - x^k)^Floor[DivisorSigma[0, k]/2], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 02 2018 *)
a:= proc(n) option remember; `if`(n<1, 1, add(a(n-i)*
ceil(numtheory[sigma][0](i)/2), i=1..n))
end:
seq(a(n), n=0..34); # Alois P. Heinz, Sep 23 2019
nmax = 33; CoefficientList[Series[1/(1 - Sum[x^(k^2)/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Floor[(DivisorSigma[0, k] + 1)/2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 33}]
a(0) = 1: the empty sum. a(6) = 2: 1*6 = 2*3. a(8) = 2: 1*8 = 2*4. a(10) = 3: 1*10 = 2*5 = 1*4+2*3. a(11) = 3: 1*11 = 1*5+2*3 = 2*4+1*3. a(12) = 4: 1*12 = 2*6 = 1*6+2*3 = 3*4. a(13) = 4: 1*13 = 1*7+2*3 = 2*5+1*3 = 1*5+2*4. a(14) = 6: 1*14 = 1*8+2*3 = 2*7 = 1*6+2*4 = 2*5+1*4 = 3*4+1*2. a(15) = 5: 1*15 = 1*9+2*3 = 1*7+2*4 = 2*6+1*3 = 3*5. a(25) = 14: 1*25 = 1*19+2*3 = 1*17+2*4 = 1*15+2*5 = 1*13+2*6 = 1*13+3*4 = 2*11+1*3 = 1*11+2*7 = 2*10+1*5 = 1*10+3*5 = 2*9+1*7 = 1*9+2*8 = 3*7+1*4 = 1*7+3*6.
h:= proc(n) option remember;
(((2*n+3)*n-2)*n-`if`(n::odd, 3, 0))/12
end:
g:= (n, i, s)-> `if`(n=0, 1, `if`(n>h(i), 0,
b(n, i, select(x-> x<=i, s)))):
b:= proc(n, i, s) option remember; g(n, i-1, s)+
`if`(i in s, 0, add(`if`(j in s, 0, g(n-i*j,
min(n-i*j, i-1), s union {j})), j=1..min(i-1, n/i)))
end:
a:= n-> g(n$2, {}):
seq(a(n), n=0..100);
h[n_] := h[n] = (((2*n + 3)*n - 2)*n - If[OddQ[n], 3, 0])/12;
g[n_, i_, s_] := If[n==0, 1, If[n>h[i], 0, b[n, i, Select[s, # <= i&]]]];
b[n_, i_, s_] := b[n, i, s] = g[n, i - 1, s] + If[MemberQ[s, i], 0, Sum[If[MemberQ[s, j], 0, g[n - i*j, Min[n - i*j, i - 1], s ~Union~ {j}]], {j, 1, Min[i - 1, n/i]}]];
a[n_] := g[n, n, {}];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 01 2018, after Alois P. Heinz *)
a(4) = 2: 1*4 = 2*2. a(5) = 2: 1*5 = 2*2+1*1. a(6) = 2: 1*6 = 2*3. a(7) = 3: 1*7 = 2*3+1*1 = 1*3+2*2. a(8) = 3: 1*8 = 2*4 = 1*4+2*2. a(9) = 4: 1*9 = 1*5+2*2 = 2*4+1*1 = 3*3. a(10) = 5: 1*10 = 1*6+2*2 = 2*5 = 1*4+2*3 = 3*3+1*1. a(11) = 6: 1*11 = 1*7+2*2 = 2*5+1*1 = 1*5+2*3 = 2*4+1*3 = 3*3+1*2. a(12) = 5: 1*12 = 1*8+2*2 = 2*6 = 1*6+2*3 = 3*4.
h:= proc(n) option remember;
n*(n+1)*(2*n+1)/6
end:
g:= (n, i, s)-> `if`(n=0, 1, `if`(n>h(i), 0,
b(n, i, select(x-> x<=i, s)))):
b:= proc(n, i, s) option remember; g(n, i-1, s)+
`if`(i in s, 0, add(`if`(j in s, 0, g(n-i*j,
min(n-i*j, i-1), s union {j})), j=1..min(i, n/i)))
end:
a:= n-> g(n$2, {}):
seq(a(n), n=0..100);
h[n_] := h[n] = n(n+1)(2n+1)/6;
g[n_, i_, s_ ] := If[n == 0, 1, If[n > h[i], 0,
b[n, i, Select[s, # <= i&]]]];
b[n_, i_, s_] := b[n, i, s] = g[n, i - 1, s] +
If[MemberQ[s, i], 0, Sum[If[MemberQ[s, j], 0, g[n - i*j,
Min[n - i*j, i - 1], s ~Union~ {j}]], {j, 1, Min[i, n/i]}]];
a[n_] := g[n, n, {}];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Aug 01 2021, after Alois P. Heinz *)
Comments