A105494 -id:A105494 - cp's OEIS frontend

A105481 Number of partitions of {1...n} containing 4 pairs of consecutive integers, where each pair is counted within a block and a string of more than 2 consecutive integers are counted two at a time.

1, 5, 30, 175, 1050, 6552, 42630, 289410, 2049300, 15120105, 116090975, 926248050, 7668746540, 65793760060, 584151925320, 5360347320420, 50776288702215, 495946245776940, 4989391837053085, 51648932225779735, 549620905409062872

Offset: 5

Augustine O. Munagi, Apr 10 2005

			a(6) = 5 because the partitions of {1,2,3,4,5,6} with 4 pairs of consecutive integers are 12345/6,1234/56,123/456,12/3456,1/23456.

A. O. Munagi, Set Partitions with Successions and Separations, Int. J. Math and Math. Sc. 2005, no. 3 (2005), 451-463.

A. O. Munagi, Set Partitions with Successions and Separations, IJMMS 2005:3 (2005), 451-463.

Cf. A105480, A105482, A105486, A105491, A105494.

seq(binomial(n-1,4)*combinat[bell](n-5),n=5..25);

a(n) = binomial(n-1, 4)*Bell(n-5), the case r = 4 in the general case of r pairs: c(n, r) = binomial(n-1, r)*B(n-r-1).
Let A be the upper Hessenberg matrix of order n defined by: A[i,i-1]=-1, A[i,j]=binomial(j-1,i-1), (i<=j), and A[i,j]=0 otherwise. Then, for n>=4, a(n+1)=(-1)^(n-4)*coeff(charpoly(A,x),x^4). [Milan Janjic, Jul 08 2010]
E.g.f.: (1/4!) * Integral (x^4 * exp(exp(x) - 1)) dx. - Ilya Gutkovskiy, Jul 10 2020

A105486 Number of partitions of {1...n} containing 4 strings of 3 consecutive integers, where each string is counted within a block and a string of more than 3 consecutive integers are counted three at a time.

Original entry on oeis.org

1, 2, 11, 50, 255, 1362, 7746, 46556, 294965, 1963865, 13703812, 99974851, 760824922, 6027441398, 49616033975, 423649629415, 3746306203604, 34259548971914, 323564415957687, 3152120868598090, 31638011553779137, 326825518800658174, 3471291152755614386

Offset: 6

Augustine O. Munagi, Apr 10 2005

nonn

			a(7) = 2 because the partitions of {1,...,7} with 4 strings of 3 consecutive integers are 123456/7, 1/234567.

Augustine O. Munagi, Set Partitions with Successions and Separations, Int. J. Math and Math. Sc., 2005:3 (2005), 451-463.

Cf. A105485, A105487, A105490, A105494.

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: A105486 := proc(n) local k ; add(c(n,k,4),k=1..n) ; end: for n from 6 to 29 do printf("%d, ",A105486(n)) ; od ; # R. J. Mathar, Feb 20 2007

S2[_, -1] = 0;
S2[n_, k_] = StirlingS2[n, k];
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]];
A105486[n_] := Sum[c[n, k, 4], {k, 1, n}];
Table[A105486[n], {n, 6, 29}] (* Jean-François Alcover, May 10 2023, after R. J. Mathar *)

a(n) = Sum_{k=1..n} (c(n, k, 4), ), where c(n, k, 4) is the case r=4 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_{j= 0..floor(n/2)} binomial(n-j, j)*S2(n-j-1, k-1).

More terms from R. J. Mathar, Feb 20 2007

cp's OEIS Frontend

A105481 Number of partitions of {1...n} containing 4 pairs of consecutive integers, where each pair is counted within a block and a string of more than 2 consecutive integers are counted two at a time.

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