A297120 Number of compositions derived from the overpartitions of n.

1, 2, 5, 14, 36, 92, 234, 586, 1452, 3562, 8674, 20956, 50290, 119922, 284308, 670458, 1573250, 3674700, 8546282, 19796234, 45681908, 105041402, 240723618, 549919604, 1252492674, 2844551866, 6442833156, 14555300218, 32801922154, 73749649900, 165443000338

Offset

0,2

Comments

Start by enumerating the overpartitions of n, then allow the parts to vary their arrangements.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..3000 (terms 0..1000 from Alois P. Heinz)

Example

The A015128(4) = 14 overpartitions of 4 are: 4; 4'; 3,1; 3,1'; 3'1; 3',1', 2,2; 2',2; 2,1,1; 2,1',1; 2',1,1; 2',1',1; 1,1,1,1; and 1',1,1,1.  The corresponding 36 compositions are 4; 4'; 3,1; 1,3; 3,1'; 1',3; 3',1; 1,3'; 3',1'; 1',3'; 2,2; 2,2'; 2',2; 2,1,1; 1,2,1; 1,1,2; 2,1,1'; 2,1',1; 1,2,1'; 1,1',2'; 1',1,2; 1',2,1; 2',1,1; 1,2',2; 1,1,2'; 2',1,1'; 2',1',1; 1,2',1'; 1,1',2'; 1',2',1; 1',1,2'; 1,1,1,1; 1,1,1,1'; 1,1,1',1; 1,1',1,1; and 1',1,1,1. Note: For a sequence of like parts p,p,...p, an overcomposition of n will only recognize p,p...p and p',p...,p; the p' is not allowed to be other than the initial p term.

Maple

b:= proc(n, i, p) option remember; `if`(n=0 or i=1, (p+n)!*
      (1+n)/n!, add(b(n-i*j, i-1, p+j)*(1+j)/j!, j=0..n/i))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 26 2017

Mathematica

b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, (p + n)!*(n + 1)/n!, Sum[b[n - i*j, i - 1, p + j]*(j + 1)/j!, {j, 0, n/i}]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Dec 27 2017, after Alois P. Heinz *)

Prog

(PARI) {my(n=30); apply(p->subst(serlaplace(p), y, 1), Vec(prod(k=1, n, (1+y*x^k)*exp(y*x^k + O(x*x^n)))))} \\ Andrew Howroyd, Dec 26 2017
(Python)
from sympy.core.cache import cacheit
from sympy import factorial
@cacheit
def b(n, i, p):  return factorial(p + n)*(n + 1)//factorial(n) if n==0 or i==1 else sum(b(n - i*j, i - 1, p + j)*(j + 1)//factorial(j) for j in range(n//i + 1))
def a(n): return b(n, n, 0)
print([a(n) for n in range(41)]) # Indranil Ghosh, Dec 29 2017, after Maple code

Crossrefs

Cf. A015128, A236002.

Sequence in context: A244099 A201371 A244061 * A299167 A182890 A102714

Adjacent sequences: A297117 A297118 A297119 * A297121 A297122 A297123

Keyword

nonn

Author

Gregory L. Simay, Dec 25 2017

Extensions

More terms from Alois P. Heinz, Dec 26 2017

Status

approved