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Partial sums of A056273.
1

%I #42 Mar 10 2021 08:52:35

%S 1,3,8,23,75,278,1154,5265,25913,135212,736704,4139831,23767895,

%T 138468210,814675838,4824766301,28699128501,171207852152,

%U 1023332115836,6124430348355,36684624841811,219860794899518,1318179574171578

%N Partial sums of A056273.

%C Some previous names were a(6,n) := (1/600)*6^n + (1/36)*4^n + (1/12)*3^n + (3/8)*2^n + (11/30)*n - (439/900) = Sum_{m=1..n} Sum_{i=1..6} S(m,i), where S(n,i) = A008277(n,i) are the Stirling numbers of the second kind.

%C Density of the regular language L{0}* over {0, 1, 2, 3, 4, 5, 6} (i.e., the number of strings of length n), where L is described by regular expression with c = 6: Sum_{i=1..c} Prod_{j=1..i} (j(1+...+j)*), where "Sum" stands for union and "Product" for concatenation. I.e., L = L((11* + ... + 11*2(1 + 2)*3(1 + 2 + 3)*4(1 + 2 + 3 + 4)*5(1 + 2 + 3 + 4 + 5)*6(1 + 2 + 3 + 4 + 5 + 6)*)0*).

%H Nelma Moreira and Rogerio Reis, <a href="https://web.archive.org/web/20170810170007/https://www.dcc.fc.up.pt/dcc/Pubs/TReports/TR04/dcc-2004-07.pdf">On the density of languages representing finite set partitions</a>, Technical Report DCC-2004-07, August 2004, DCC-FC & LIACC, Universidade do Porto.

%H Nelma Moreira and Rogerio Reis, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Moreira/moreira8.html">On the Density of Languages Representing Finite Set Partitions</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (17,-111,355,-584,468,-144).

%F For c = 6, a(c, n) = g(1, c)*n + Sum_{k=2..c} g(k, c)*k*(k^n - 1)/(k - 1), where g(1, 1) = 1, g(1, c) = g(1, c-1) + (-1)^(c-1)/(c-1)! for c > 1, and g(k, c) = g(k-1, c-1)/k for c > 1 and 2 <= k <= c.

%F G.f.: x*(91*x^4 - 135*x^3 + 68*x^2 - 14*x + 1) / ((x - 1)^2*(2*x - 1)*(3*x - 1)*(4*x - 1)*(6*x - 1)). - _Colin Barker_, Oct 28 2014

%p with (combinat):seq(sum(sum(stirling2(k, j),j=1..6), k=1..n), n=1..23); # _Zerinvary Lajos_, Dec 04 2007

%o (PARI) Vec(x*(91*x^4-135*x^3+68*x^2-14*x+1)/((x-1)^2*(2*x-1)*(3*x-1)*(4*x-1)*(6*x-1)) + O(x^100)) \\ _Colin Barker_, Oct 28 2014

%o (PARI) a(n) = sum(m=1, n, sum(i=1, 6, stirling(m, i, 2))) \\ _Petros Hadjicostas_, Mar 09 2021

%Y Cf. A047926, A056273, A008277, A099264, A099265.

%K easy,nonn

%O 1,2

%A _Nelma Moreira_, Oct 10 2004

%E Shorter name by _Joerg Arndt_, Oct 28 2014

%E Comments edited by _Petros Hadjicostas_, Mar 09 2021