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%I #20 May 10 2023 06:01:22
%S 1,2,8,32,141,672,3451,18962,110882,686866,4489422,30853656,222276063,
%T 1674067342,13149209956,107481488424,912490408782,8031867965568,
%U 73181346933680,689194657064660,6699707386510583,67143409071264516,692926011957479445,7356058078964945382
%N Number of partitions of {1...n} containing one string of 3 consecutive integers, counted within a block.
%H Augustine O. Munagi, <a href="https://doi.org/10.1155/IJMMS.2005.215">Set Partitions with Successions and Separations</a>, Int. J. Math and Math. Sc. 2005:3 (2005), 451-463.
%F a(n) = Sum_{k=1..n} c(n, k, 1), where c(n, k, 1) is the case r=1 of c(n, k, r) given by c(n, k, r)=c(n-1, k-1, r)+(k-1)c(n-1, k, r)+c(n-2, k-1, r)+(k-1)c(n-2, k, r)+c(n-1, k, r-1)-c(n-2, k-1, r-1)-(k-1)c(n-2, k, r-1), r=0, 1, .., n-k-1, k=1, 2, .., n-2r, c(n, k, 0) = Sum_{0..floor(n/2)} binomial(n-j, j)*S2(n-j-1, k-1).
%e a(5) = 8 because the partitions of {1,2,3,4,5} with one 3-string of consecutive integers are 1235/4, 1345/2, 15/234, 123/45, 12/345, 123/4/5, 1/234/5, 1/2/345.
%p c := proc(n,k,r) option remember ; local j ; if r =0 then add(binomial(n-j,j)*combinat[stirling2](n-j-1,k-1),j=0..floor(n/2)) ; else if r <0 or r > n-k-1 then RETURN(0) fi ; if n <1 then RETURN(0) fi ; if k <1 then RETURN(0) fi ; RETURN( c(n-1,k-1,r)+(k-1)*c(n-1,k,r)+c(n-2,k-1,r)+(k-1)*c(n-2,k,r) +c(n-1,k,r-1)-c(n-2,k-1,r-1)-(k-1)*c(n-2,k,r-1) ) ; fi ; end: A105483 := proc(n) local k ; add(c(n,k,1),k=1..n) ; end: for n from 3 to 26 do printf("%d, ",A105483(n)) ; od ; # _R. J. Mathar_, Feb 20 2007
%t S2[_, -1] = 0;
%t S2[n_, k_] = StirlingS2[n, k];
%t c [n_, k_, r_] := c[n, k, r] = Which[r == 0, Sum[Binomial[n - j, j]*S2[n - j - 1, k - 1], {j, 0, Floor[n/2]}], r < 0 || r > n - k - 1, 0, n < 1, 0, k < 1, 0, True, c[n - 1, k - 1, r] + (k - 1)*c[n - 1, k, r] + c[n - 2, k - 1, r] + (k - 1)*c[n - 2, k, r] + c[n - 1, k, r - 1] - c[n - 2, k - 1, r - 1] - (k - 1)*c[n - 2, k, r - 1]];
%t A105483[n_] := Sum[c[n, k, 1], {k, 1, n}];
%t Table[A105483[n], {n, 3, 26}] (* _Jean-François Alcover_, May 10 2023, after _R. J. Mathar_ *)
%Y Cf. A105484, A105489, A105493.
%K nonn
%O 3,2
%A _Augustine O. Munagi_, Apr 10 2005
%E More terms from _R. J. Mathar_, Feb 20 2007
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