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

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, 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, 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

Offset

1,6

Comments

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.

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Formula

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.

a([k0,k1,0,0,...]) = S(k0+k1+1,k0+1), Stirling number of the 2nd kind, see A008277.

Example

From Antti Karttunen, Feb 28 2023: (Start)
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.
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.
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.
(End)

Prog

(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

Crossrefs

Cf. A139605, A145271.

Sequence in context: A367452 A119612 A101949 * A343858 A065714 A110700

Adjacent sequences: A124793 A124794 A124795 * A124797 A124798 A124799

Keyword

nonn

Author

Max Alekseyev, Nov 29 2006

Status

approved