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A090802 Triangle read by rows: a(n,k) = number of k-length walks in the Hasse diagram of a Boolean algebra of order n. 16
1, 2, 1, 4, 4, 2, 8, 12, 12, 6, 16, 32, 48, 48, 24, 32, 80, 160, 240, 240, 120, 64, 192, 480, 960, 1440, 1440, 720, 128, 448, 1344, 3360, 6720, 10080, 10080, 5040, 256, 1024, 3584, 10752, 26880, 53760, 80640, 80640, 40320 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Row sums = A010842(n); Row sums from column 1 on = A066534(n) = n*A010842(n-1) = A010842(n) - 2^n.

a(n,k) = n! = k! = A000142(n) for n = k; a(n,n-1) = 2*n! = A052849(n) for n > 1; a(n,n-2) = 2*n! = A052849(n) for n > 2; a(n,n-3) = (4/3)*n! = A082569(n) for n > 3; a(n,n-1)/a(2,1) = n!/2! = A001710(n) for n > 1; a(n,n-2)/ a(3,1) = n!/3! = A001715(n) for n > 2; a(n,n-3)/a(4,1) = n!/4! = A001720(n) for n > 3.

a(2k, k) = A052714(k+1). a(2k-1, k) = A034910(k).

a(n,0) = A000079(n); a(n,1) = A001787(n) = row sums of A003506; a(n,2) = A001815(n) = 2!*A001788(n-1); a(n,3) = A052771(n) = 3!*A001789(n); a(n,4) = A052796(n) = 4!*A003472(n); ceiling[a(n,1) / 2] = A057711(n); a(n,5) = 5!*A054849(n).

In a class of n students, the number of committees (of any size) that contain an ordered k-sized subcommittee is a(n,k). - Ross La Haye, Apr 17 2006

Antidiagonal sums [1,2,5,12,30,76,198,528,1448,4080...] appear to be binomial transform of A000522 interleaved with itself, i.e. 1,1,2,2,5,5,16,16,65,65... - Ross La Haye, Sep 09 2006

Let P(A) be the power set of an n-element set A. Then a(n,k) = the number of ways to add k elements of A to each element x of P(A) where the k elements are not elements of x and order of addition is important. - Ross La Haye, Nov 19 2007

The derivatives of x^n evaluated at x=2. - T. D. Noe, Apr 21 2011

LINKS

T. D. Noe, Rows n = 0..100, flattened

Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.

Eric Weisstein, Walk

Eric Weisstein, Boolean Algebra

Eric Weisstein, Hasse Diagram

FORMULA

a(n, k) = 0 for n < k. a(n, k) = k!*C(n, k)*2^(n-k) = P(n, k)*2^(n-k) = (2n)!!/((n-k)!*2^k) = k!*A038207(n, k) = A068424*2^(n-k) = Sum[C(n, m)*P(n-m, k), {m, 0, n-k}] = Sum[C(n, n-m)*P(n-m, k), {m, 0, n-k}] = n!*Sum[1/(m!*(n-m-k)!), {m, 0, n-k}] = k!*Sum[C(n, m)*C(n-m, k), {m, 0, n-k}] = k!*Sum[C(n, n-m)*C(n-m, k), {m, 0, n-k}] = k!*C(n, k)*Sum[C(n-k, n-m-k), {m, 0, n-k}] = k!*C(n, k)*Sum[C(n-k, m), {m, 0, n-k}] for n >= k.

a(n, k) = 0 for n < k. a(n, k) = n*a(n-1, k-1) for n >= k >= 1.

E.g.f. (by columns): exp(2x)*x^k.

EXAMPLE

{1};

{2, 1};

{4, 4, 2};

{8, 12, 12, 6};

{16, 32, 48, 48, 24};

{32, 80, 160, 240, 240, 120};

{64, 192, 480, 960, 1440, 1440, 720};

{128, 448, 1344, 3360, 6720, 10080, 10080, 5040};

{256, 1024, 3584, 10752, 26880, 53760, 80640, 80640, 40320}

a(5,3) = 240 because P(5,3) = 60, 2^(5-3) = 4 and 60 * 4 = 240.

MATHEMATICA

Flatten[Table[n!/(n-k)! * 2^(n-k), {n, 0, 8}, {k, 0, n}]] (* Ross La Haye, Feb 10 2004 *)

CROSSREFS

Cf. A000142, A001710, A001715, A001720, A001787, A001788, A001789, A001815, A003472, A010842, A052771, A052796, A052849, A054849, A057711, A066534, A082569.

Cf. A038207, A007318.

Sequence in context: A192017 A180566 A051289 * A129159 A095830 A193915

Adjacent sequences:  A090799 A090800 A090801 * A090803 A090804 A090805

KEYWORD

easy,nonn,tabl

AUTHOR

Ross La Haye, Feb 10 2004

EXTENSIONS

More terms from Ray Chandler, Feb 26 2004

Entry revised by Ross La Haye, Aug 18 2006

STATUS

approved

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Last modified September 22 00:25 EDT 2017. Contains 292326 sequences.