0,2

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 (rlahaye(AT)new.rr.com), 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 (rlahaye(AT)new.rr.com), 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 (rlahaye(AT)new.rr.com), Nov 19 2007

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. [From Ross La Haye (rlahaye(AT)new.rr.com), Feb 22 2009]

Eric Weisstein, Walk

Eric Weisstein, Boolean Algebra

Eric Weisstein, Hasse Diagram

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.

{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.

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

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

Cf. A038207, A007318.

Sequence in context: A113421 A135366 A051289 this_sequence A129159 A095830 A101621

Adjacent sequences: A090799 A090800 A090801 this_sequence A090803 A090804 A090805

easy,nonn,tabl

Ross La Haye (rlahaye(AT)new.rr.com), Feb 10 2004

More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net), Feb 26 2004

Entry revised by Ross La Haye (rlahaye(AT)new.rr.com), Aug 18 2006