A035341 Sum of ordered factorizations over all prime signatures with n factors.

1, 1, 5, 25, 173, 1297, 12225, 124997, 1492765, 19452389, 284145077, 4500039733, 78159312233, 1460072616929, 29459406350773, 634783708448137, 14613962109584749, 356957383060502945, 9241222160142506097, 252390723655315856437, 7260629936987794508973

Offset: 0

Author

Alford Arnold

Keywords

• nonn
• nice

Comments

Let f(n) = number of ordered factorizations of n (A074206(n)); a(n) = sum of f(k) over all terms k in A025487 that have n factors.

When the unordered spectrum A035310 is so ordered the sequences A000041 A000070 ...A035098 A000110 yield A000079 A001792 ... A005649 A000670 respectively.

Row sums of A095705. - David Wasserman, Feb 22 2008

From Ludovic Schwob, Sep 23 2023: (Start)

a(n) is the number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums. The a(3) = 25 matrices:

[1 1 1] [1 2] [2 1] [3]

.

[1 1] [1 1] [1 1 0] [1 0 1] [0 1 1] [2] [0 2] [2 0]

[1 0] [0 1] [0 0 1] [0 1 0] [1 0 0] [1] [1 0] [0 1]

.

[1] [1 0] [0 1] [1 0] [0 1] [1 0 0] [1 0 0] [0 1] [1 0]

[1] [1 0] [0 1] [0 1] [1 0] [0 1 0] [0 0 1] [1 0] [0 1]

[1] [0 1] [1 0] [1 0] [0 1] [0 0 1] [0 1 0] [1 0] [0 1]

.

[0 1 0] [0 1 0] [0 0 1] [0 0 1]

[1 0 0] [0 0 1] [1 0 0] [0 1 0]

[0 0 1] [1 0 0] [0 1 0] [1 0 0] (End)

Examples

			a(3) = 25 because there are 3 terms in A025487 with 3 factors, namely 8, 12, 30; and f(8)=4, f(12)=8, f(30)=13 and 4+8+13 = 25.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..250 (first 36 terms from David Wasserman)
• Eric Weisstein's World of Mathematics, Perfect Partition

Crossrefs

Cf. A005651, A025487, A035310, A255906, A365961.

Row sums of A261719.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)*binomial(i+k-1, k-1)^j, j=0..n/i)))
    end:
a:= n->add(add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 29 2015

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, k]*If[j == 0, 1, Binomial[i + k - 1, k - 1]^j], {j, 0, n/i}]]];
a[n_] := Sum[Sum[b[n, n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}], {k, 0, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Oct 26 2015, after Alois P. Heinz, updated Dec 15 2020 *)

PARI

R(n,k)=Vec(-1 + 1/prod(j=1, n, 1 - binomial(k+j-1,j)*x^j + O(x*x^n)))
seq(n) = {concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ))} \\ Andrew Howroyd, Sep 23 2023

Formula

a(n) ~ c * n! / log(2)^n, where c = 1/(2*log(2)) * Product_{k>=2} 1/(1-1/k!) = A247551 / (2*log(2)) = 1.8246323... . - Vaclav Kotesovec, Jan 21 2017

Extensions

More terms from Erich Friedman.

More terms from David Wasserman, Feb 22 2008