A099265 Partial sums of A056272.
1, 3, 8, 23, 75, 277, 1132, 4977, 22979, 109451, 531456, 2610931, 12917683, 64181625, 319695980, 1594859885, 7963472187, 39784944799, 198827606704, 993846943839, 4968361974491, 24839192686973, 124188113975628, 620917025694793, 3104514504312595, 15522360665856147, 77611167795714752
Offset: 1
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.
Programs
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Maple
with (combinat):seq(sum(sum(stirling2(k, j),j=1..5), k=1..n), n=1..23); # Zerinvary Lajos, Dec 04 2007
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PARI
a(n) = sum(m=1, n, sum(i=1, 5, stirling(m, i, 2))) \\ Petros Hadjicostas, Mar 10 2021
Formula
a(5,n) = (1/96)*5^n + (1/8)*3^n + (1/3)*2^n + (3/8)*n - 15/32.
a(n) = Sum_{m=1..n} Sum_{i=1..5} S(m,i), where S(m,i) = A008277(m,i) (i.e., partial sum of the sum of Stirling numbers of second kind S(n,i) for i = 1..5).
For c = 5, 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*(-1 + 19*x^3 - 24*x^2 + 9*x)/((3*x-1)*(2*x-1)*(5*x-1)*(x-1)^2). [Maksym Voznyy (voznyy(AT)mail.ru), Jul 28 2009]
Extensions
Name and Formula section edited by Petros Hadjicostas, Mar 10 2021
More terms from Michel Marcus, Jan 05 2025
Comments