3,2

a(n) = sum((-1)^(m-p-1)*sum(2^(n-q-1)*binomial(n-q-1,m-p-1)*A094665(n-1,q)* A156919(q,p),q=1..n-m+p), p=0..m-1) with m = 3.

restart; nmax:=15; mmax:=nmax: imax := nmax: i:=0: T1(0, x):=1: T1(0, x+1):=1: for i from 1 to imax do T1(i, x):= expand((2*x+1)*(x+1)*T1(i-1, x+1)-2*x^2*T1(i-1, x)): dx:=degree(T1(i, x)): for k from 0 to dx do c(k):=coeff(T1(i, x), x, k) od: T1(i, x+1):=sum(c(j)*(x+1)^(j), j=0..dx): od: for i from 0 to imax do for j from 0 to i do A083061(i, j):=coeff(T1(i, x), x, j) od: od: for n from 0 to nmax do for k from 0 to n do A094665(n+1, k+1) := A083061(n, k) od: od: A094665(0, 0):=1: for n from 1 to nmax do A094665(n, 0):=0 od: for m from 1 to mmax do A156919(0, m):= 0 end do: for n from 0 to nmax do A156919(n, 0):=2^n end do: for n from 1 to nmax do for m from 1 to mmax do A156919(n, m):=(2*m+2)*A156919(n-1, m)+(2*n-2*m+1)* A156919(n-1, m-1) end do end do: m:=3; for n from m to nmax do a(n, m):= sum((-1)^(m-p-1)*sum(2^(n-q-1)*binomial(n-q-1, m-p-1)*A094665(n-1, q)*A156919(q, p), q=1..n-m+p), p=0..m-1) od: seq(a(n, m), n=m..nmax);

Third left hand column of the EG1 triangle A162005.

Other left hand columns are A000182 and A162006.

Related to A094665, A083061 and A156919.

A000079, A036289 and A100381 appear in the a(n, 3) formula.

A001789, A003472, A054849, A002409, A054851, A140325 and A140354 (scaled by 2^(m-1)) appear by one by one in the a(n, m) formulae for m= 4 and higher .

Sequence in context: A109025 A028535 A108094 this_sequence A104844 A086003 A048295

Adjacent sequences: A162004 A162005 A162006 this_sequence A162008 A162009 A162010

easy,nonn

Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 27 2009