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URL: https://oeis.org/A072913

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A072913
Numerators of (1/4!)*(H(n,1)^4+6*H(n,1)^2*H(n,2)+8*H(n,1)*H(n,3)+3*H(n,2)^2+6*H(n,4)), where H(n,m) = Sum_{i=1..n} 1/i^m are generalized harmonic numbers.
4
1, 31, 3661, 76111, 58067611, 68165041, 187059457981, 3355156783231, 300222042894631, 327873266234371, 5194481903600608411, 5578681466128739761, 170044702211669500782121, 180514164422163370751221
OFFSET
1,2
COMMENTS
a(n) is also the numerator of binomial transform of (-1)^n/(n+1)^5.
LINKS
Jerry Metzger and Thomas Richards, A Prisoner Problem Variation, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.
Javier Sesma, The Roman harmonic numbers revisited, Journal of Number Theory Vol. 180, Nov. 2017, pp. 544-565; arXiv:1702.03718v2 [math.NT], 2017.
FORMULA
Numerators of 1/4!*((gamma + Psi(n+1))^4 + 6*(gamma+Psi(n+1))^2*(1/6*Pi^2 - Psi(1, n+1)) + 8*(gamma + Psi(n+1))*(Zeta(3) + 1/2*Psi(2, n+1)) + 3*(1/6*Pi^2 - Psi(1, n+1))^2 + 6*(1/90*Pi^4 - 1/6*Psi(3, n+1))).
For n >= 1, H(n,1)^4 + 6*H(n,1)^2*H(n,2) + 8*H(n,1)*H(n,3) + 3*H(n,2)^2 + 6*H(n,4) = Integral_{x = 0..1} x^(n-1)*(log(1-x))^4 dx.
From Peter Bala, Dec 05 2025: (Start)
a(n) = numerator(c_4(n)), where c_1(n) = Sum_{k = 1..n} 1/k (the harmonic numbers) and c_(i+1)(n) = Sum_{k = 1..n} c_i(k)/k for i >= 1. See Sesma.
The o.g.f. Sum_{n >= 1} c_4(n)*x^n = 1/(1 - x) * Sum_{n >= 1} (-1)^(n+1)/n^4 * (x/(1 - x))^n.
The e.g.f. Sum_{n >= 1} c_4(n)*x^n/n! = exp(x) * Sum_{n >= 1} (-1)^(n+1)/n^4 * x^n/n!.
a(n) = numerator( Sum_{k = 1..n} (-1)^(k+1)/k^4 * binomial(n, k) ). (End)
MAPLE
seq( numer(add((-1)^(k+1)/k^4 * binomial(n, k), k = 1..n)), n = 1..20 ); # Peter Bala, Dec 04 2025
PROG
(PARI)
x(n)=sum(k=1, n, 1/k);
y(n)=sum(k=1, n, 1/k^2);
z(n)=sum(k=1, n, 1/k^3);
w(n)=sum(k=1, n, 1/k^4);
a(n)=numerator(1/4!*(x(n)^4+6*x(n)^2*y(n)+8*x(n)*z(n)+3*y(n)^2+6*w(n)))
CROSSREFS
KEYWORD
easy,nonn,frac
AUTHOR
Vladeta Jovovic, Aug 10 2002
EXTENSIONS
More terms from Benoit Cloitre, Aug 13 2002
STATUS
approved