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A124796 Coefficients in expansion of powers of the operator "multiplication by f(x) followed by differentiation", in the prime factorization order.

%I A124796 #18 Feb 28 2023 15:24:15
%S A124796 1,1,1,1,0,3,0,1,1,1,0,6,0,0,0,1,0,7,0,4,0,0,0,10,0,0,1,1,0,4,0,1,0,0,
%T A124796 0,25,0,0,0,10,0,0,0,0,0,0,0,15,0,0,0,0,0,15,0,5,0,0,0,30,0,0,0,1,0,0,
%U A124796 0,0,0,0,0,65,0,0,0,0,0,0,0,20,1,0,0,7,0,0,0,1,0,11,0,0,0,0,0,21,0,0,0,4,0
%N A124796 Coefficients in expansion of powers of the operator "multiplication by f(x) followed by differentiation", in the prime factorization order.
%C A124796 Let d o f(x) be an operator of multiplication by f(x) followed by differentiation. (d o f)^m = Sum a([k0,k1,...])*((d^0 f)^k0*(d^1 f)^k1*...)*d^(m-k1-2*k2-...) where the sum is taken over all nonnegative integer vectors [k0,k1,...] such that k0+k1+...=m and k1+2*k2+...<=m.
%C A124796 For all k >= 0 it holds that a(2^k) = a(3^k) = 1 and also a(p) = 0 for all primes p > 3. - _Alexander Adamchuk_, Dec 03 2006 and _Antti Karttunen_, Feb 28 2023
%H A124796 Antti Karttunen, <a href="/A124796/b124796.txt">Table of n, a(n) for n = 1..16384</a>
%H A124796 Antti Karttunen, <a href="/A124796/a124796.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F A124796 For n=p0^k0*p1^k1*... where 2=p0<p1<... are the sequence of all primes, a(n) = a([k0,k1,...]) satisfy the recurrence a([k0,k1,...]) = a([k0-1,k1,...]) + (k0+1)*a([k0,k1-1,...]) + Sum_{i=2..oo} (k(i-1)+1)*a([k0-1,k1,...,k(i-2),k(i-1)+1,ki-1,k(i+1),...]) with a([0,0,...])=1 and a([k0,k1,...])=0 as soon as some ki<0.
%F A124796 a([k0,k1,0,0,...]) = S(k0+k1+1,k0+1), Stirling number of the 2nd kind, see A008277.
%e A124796 From _Antti Karttunen_, Feb 28 2023: (Start)
%e A124796 For n=6, a(6) = a(2^1 * 3^1) = a([1,1,0,0,0,...]) = a([0,1,0,0,...]) + (1+1)*a([1,0,0,0,...]) + 0 = a(3) + 2*a(1) = 3.
%e A124796 For n=10, a(10) = a(2^1 * 5^1) = a([1,0,1,0,0,0...]) = a([0,0,1,0,0,0,...]) + 2*0 + 1*a([0,1,0,0,0,...]) = a(5) + 0 + 1*a(3) = 1.
%e A124796 For n=20, a(20) = a(2^2 * 5^1) = a([2,0,1,0,0,0...]) = a([1,0,1,0,0,0,...]) + 3*0 + 1*a([1,1,0,0,0,...]) = a(10) + 0 + 1*a(6) = 1+3 = 4.
%e A124796 (End)
%o A124796 (PARI) A124796(n) = if(1==n,1,my(u=primepi(vecmax(factor(n)[, 1]))); if(n%3,0,((1+valuation(n,2)) * A124796(n/3))) + if(n%2,0,(A124796(n/2) + sum(i=3,u,if(n%prime(i),0,(valuation(n,prime(i-1))+1)*A124796((n/2)*prime(i-1)/prime(i))))))); \\ _Antti Karttunen_, Feb 28 2023
%Y A124796 Cf. A139605, A145271.
%K A124796 nonn
%O A124796 1,6
%A A124796 _Max Alekseyev_, Nov 29 2006