A129880 -id:A129880 - cp's OEIS frontend

A129872 Sequence M_n arising in enumeration of arrays of directed blocks (see 2007 Quaintance reference for precise definition). [The next term is not an integer.].

1, 4, 16, 72, 364, 1916, 10581, 59681, 343903, 2010089

Offset: 1

N. J. A. Sloane, May 26 2007

nonn

Warning: as defined the terms are not integral in general: 1, 4, 16, 72, 364, 1916, 10581, 59681, 343903, 2010089, 23798969/2, ... - Jocelyn Quaintance, Mar 31 2013.

Jocelyn Quaintance, Letter Representations of m x n x p Proper Arrays (2004), arXiv:math/0412244. See Table 1.
Jocelyn Quaintance, Combinatoric Enumeration of Two-Dimensional Proper Arrays, Discrete Math., 307 (2007), 1844-1864. See Table 1.

Cf. A129873-A129886.

listrn(m) = {R = t*O(t); for (n= 1, m, R = (2*t^11 + t^10 + 3*t^9 + 8*t^8 + 12*t^7 + 16*t^6 + 26*t^5 + 20*t^4 + 16*t^3 + 7*t^2 + t - (t^14 - t^12 - 8*t^10 - 11*t^8 - 5*t^6 + t^4)*R^3 - (3*t^13 + t^12 - 5*t^11 - 2*t^10 - 34*t^9 - 14*t^8 - 39*t^7 - 16*t^6 - 11*t^5 - 2*t^4 + 6*t^3 + 2*t^2)*R^2)/(t^12 + t^11 - 7*t^10 - 7*t^9 - 42*t^8 - 35*t^7 - 51*t^6 - 41*t^5 - 14*t^4 - 4*t^3 + 8*t^2 + 6*t + 1);); return(vector(m, i , polcoeff(R, i, t)));}
listbn(m) = {B = t*O(t); for (n= 1, m, B = (1 + 2*t*B^2 - t^2*B^3 );); return(vector(m, i , polcoeff(B, i, t)));}
listMn(m) = {b = listbn(m); /* see also A006013 */ s = listsn(m); /* see A129873 */ d = listdn(m); /* see A129880 */ r = listrn(m); for (i=1, m, v = (b[i] + s[i] - d[i] - r[i])/2; print1(v, ", "););}
\\ Michel Marcus, Mar 30 2013

See Quaintance reference for generating functions that produce A129872-A129886.

Edited by N. J. A. Sloane, Nov 29 2016

A124837 Numerators of third-order harmonic numbers (defined by Conway and Guy, 1996).

Original entry on oeis.org

1, 7, 47, 57, 459, 341, 3349, 3601, 42131, 44441, 605453, 631193, 655217, 1355479, 23763863, 24444543, 476698557, 162779395, 166474515, 34000335, 265842403, 812400067, 20666950267, 21010170067, 192066102203, 194934439103

Offset: 1

Jonathan Vos Post, Nov 10 2006

Denominators are A124838. All fractions reduced. Thanks to Jonathan Sondow for verifying these calculations. He suggests that the equivalent definition in terms of first order harmonic numbers may be computationally simpler. We are happy with the description of A027612 Numerator of 1/n + 2/(n-1) + 3/(n-2) + ... + (n-1)/2 + n, but baffled by the description of A027611.
From Alexander Adamchuk, Nov 11 2006: (Start)
a(n) is the numerator of H(n, (3)) = Sum_{m=1..n} Sum_{k=1..m} HarmonicNumber(k).
Denominators are listed in A124838.
p divides a(p-5) for prime p > 5.
Primes are listed in A129880.
Numbers k such that a(k) is prime are listed in A129881. (End)

			a(1) = 1 = numerator of 1/1.
a(2) = 7 = numerator of 1/1 + 5/2 = 7/2.
a(3) = 47 = numerator of 7/2 + 13/3 = 47/6.
a(4) = 57 = numerator of 47/6 + 77/12 = 57/4.
a(5) = 549 = numerator of 57/4 + 87/10 = 549/20.
a(6) = 341 = numerator of 549/20 + 223/20 = 341/10
a(7) = 3349 = numerator of 341/10 + 481/35 = 3349/70.
a(8) = 88327 = numerator of 3349/70 + 4609/280 = 88327/1260.
a(9) = 3844 = numerator of 88327/1260 + 4861/252 = 3844/45.
a(10) = 54251 = numerator of 3844/45 + 55991/2520 = 54251/504, or, untelescoping:
a(10) = 54251 = numerator of 1/1 + 5/2 + 13/3 + 77/12 + 87/10 + 223/20 + 481/35 + 4609/252 + 4861/252 + 55991/2520 = 54251/504.

J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, pp. 143 and 258-259, 1996.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Harmonic Number. See equation for third order harmonic numbers.

Cf. A027611, A027612, A124838.

Cf. A001008, A002805, A067657, A056903, A124878, A124879, A124837, A129880, A129881.

a124837 n = a213998 (n + 2) (n - 1) -- Reinhard Zumkeller, Jul 03 2012

Table[Numerator[(n+2)!/2!/n!*Sum[1/k,{k,3,n+2}]],{n,1,30}] (* Alexander Adamchuk, Nov 11 2006 *)

A124837(n)/A124838(n) = Sum{i=1..n} A027612(n)/A027611(n+1).
From Alexander Adamchuk, Nov 11 2006: (Start)
a(n) = numerator(Sum_{m=1..n} Sum_{l=1..m} Sum_{k=1..l} 1/k).
a(n) = numerator(((n+2)!/(2!*n!)) * Sum_{k=3..n+2} 1/k).
a(n) = numerator(((n+2)*(n+1)/2) * Sum_{k=3..n+2} 1/k). (End)
a(n) = numerator(Sum_{k=0..n-1} (-1)^k*binomial(-3,k)/(n-k)). - Gary Detlefs, Jul 18 2011
a(n) = A213998(n+2,n-1). - Reinhard Zumkeller, Jul 03 2012

Corrected and extended by Alexander Adamchuk, Nov 11 2006

cp's OEIS Frontend

A129872 Sequence M_n arising in enumeration of arrays of directed blocks (see 2007 Quaintance reference for precise definition). [The next term is not an integer.].

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