A099266 Partial sums of A056273.

1, 3, 8, 23, 75, 278, 1154, 5265, 25913, 135212, 736704, 4139831, 23767895, 138468210, 814675838, 4824766301, 28699128501, 171207852152, 1023332115836, 6124430348355, 36684624841811, 219860794899518, 1318179574171578

Offset: 1

Author

Nelma Moreira, Oct 10 2004

Keywords

• easy
• nonn

Comments

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.

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*).

Links

• Nelma Moreira and Rogerio Reis, On the density of languages representing finite set partitions, Technical Report DCC-2004-07, August 2004, DCC-FC & LIACC, Universidade do Porto.
• Nelma Moreira and Rogerio Reis, On the Density of Languages Representing Finite Set Partitions, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.
• Index entries for linear recurrences with constant coefficients, signature (17,-111,355,-584,468,-144).

Crossrefs

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

Programs

Maple

with (combinat):seq(sum(sum(stirling2(k, j),j=1..6), k=1..n), n=1..23); # Zerinvary Lajos, Dec 04 2007

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

PARI

a(n) = sum(m=1, n, sum(i=1, 6, stirling(m, i, 2))) \\ Petros Hadjicostas, Mar 09 2021

Formula

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.

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

Extensions

Shorter name by Joerg Arndt, Oct 28 2014

Comments edited by Petros Hadjicostas, Mar 09 2021