0,2

Construct the infinite array of polynomials

a(0,t) = 1

a(1,t) = 2

a(2,t) = 6 + 2 t

a(3,t) = 24 + 24 t

a(4,t) = 120 + 240 t + 24 t^2

a(5,t) = 720 + 2400 t + 720 t^2

a(6,t) =5040 + 25200 t + 15120 t^2 + 720 t^3

This array is the reciprocal array of the following array b(n,t) under the list partition transform and its associated operations described in A133314.

b(0,t) = 1, b(1,t) = -2, b(2,t) = -2*(t-1), b(n,t) = 0 for n>2 .

Then A000165(n) = a(n,1) .

Lower triangular matrix A110327 = binomial(n,k)*a(n-k,2) .

n! * A000129(n+1) = a(n,2) = A110327(n,0) .

A110330 = matrix inverse of binomial(n,k)*a(n-k,2) = binomial(n,k)*b(n-k,2) .

A000142(n+1) = a(n,0) .

exp[b(.,t)*x] = 1 - 2x - (t-1) * x^2; therefore exp[a(.,t)*x] = 1 / [ 1 - 2x - (t-1) * x^2 ] = (t-1) / { t - [ 1 + x*(t-1) ]^2 }.

Also, a(n,t) = (1 - t*u^2)^(n+1) (D_u)^n [ 1 / (1 - t*u^2) ] with eval. at u = 1/t . Compare A076743.

a(n,t) = n! sum(k=0,1,...) binomial(n+1,2k+1) * t^k = n! sum(k=0,1,...) A034867(n,k) * t^k .

E.g.f. = I_o[2*(u*x)^(1/2)] * exp[a(.,t)*x], analogous to A132382,

where I_o is the zeroth modified Bessel function of the first kind, i.e. I_o[2*(u*x)^(1/2)] = sum(j=0,1,...) u^j/j! * x^j/j! .

Additional relations are given by formulae in A133314.

a(n,t) = n! sum(k=0,...,n) binomial(n+1,2k+1) * t^k = n! sum(k=0,...,n) A034867(n,k) * t^k .

Sequence in context: A008556 A096485 A125032 this_sequence A076743 A141056 A027760

Adjacent sequences: A131977 A131978 A131979 this_sequence A131981 A131982 A131983

easy,nonn

Tom Copeland (tcjpn(AT)msn.com), Oct 30 2007, Nov 29 2007, Nov 30 2007