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%I #41 Sep 23 2023 15:00:16
%S 1,1,5,25,173,1297,12225,124997,1492765,19452389,284145077,4500039733,
%T 78159312233,1460072616929,29459406350773,634783708448137,
%U 14613962109584749,356957383060502945,9241222160142506097,252390723655315856437,7260629936987794508973
%N Sum of ordered factorizations over all prime signatures with n factors.
%C 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.
%C When the unordered spectrum A035310 is so ordered the sequences A000041 A000070 ...A035098 A000110 yield A000079 A001792 ... A005649 A000670 respectively.
%C Row sums of A095705. - _David Wasserman_, Feb 22 2008
%C From _Ludovic Schwob_, Sep 23 2023: (Start)
%C 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:
%C [1 1 1] [1 2] [2 1] [3]
%C .
%C [1 1] [1 1] [1 1 0] [1 0 1] [0 1 1] [2] [0 2] [2 0]
%C [1 0] [0 1] [0 0 1] [0 1 0] [1 0 0] [1] [1 0] [0 1]
%C .
%C [1] [1 0] [0 1] [1 0] [0 1] [1 0 0] [1 0 0] [0 1] [1 0]
%C [1] [1 0] [0 1] [0 1] [1 0] [0 1 0] [0 0 1] [1 0] [0 1]
%C [1] [0 1] [1 0] [1 0] [0 1] [0 0 1] [0 1 0] [1 0] [0 1]
%C .
%C [0 1 0] [0 1 0] [0 0 1] [0 0 1]
%C [1 0 0] [0 0 1] [1 0 0] [0 1 0]
%C [0 0 1] [1 0 0] [0 1 0] [1 0 0] (End)
%H Alois P. Heinz, <a href="/A035341/b035341.txt">Table of n, a(n) for n = 0..250</a> (first 36 terms from David Wasserman)
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PerfectPartition.html">Perfect Partition</a>
%F 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
%e 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.
%p b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p add(b(n-i*j, i-1, k)*binomial(i+k-1, k-1)^j, j=0..n/i)))
%p end:
%p a:= n->add(add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k), k=0..n):
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Aug 29 2015
%t 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}]]];
%t a[n_] := Sum[Sum[b[n, n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}], {k, 0, n}];
%t Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Oct 26 2015, after _Alois P. Heinz_, updated Dec 15 2020 *)
%o (PARI)
%o R(n,k)=Vec(-1 + 1/prod(j=1, n, 1 - binomial(k+j-1,j)*x^j + O(x*x^n)))
%o 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
%Y Cf. A005651, A025487, A035310, A255906, A365961.
%Y Row sums of A261719.
%K nonn,nice
%O 0,3
%A _Alford Arnold_
%E More terms from _Erich Friedman_.
%E More terms from _David Wasserman_, Feb 22 2008
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