1,2

Paul Curtz, Comments on this sequence

G.f.: x*(1-x)^3/((1-2*x)*(1-3*x+3*x^2)).

a(1) = 1, a(2) = 2, a(3) = 4, a(4) = 7; for n > 4, a(n) = 5*a(n-1) - 9*a(n-2) + 6*a(n-3).

Binomial transform of A088911. - Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jun 17 2007

a(n+1) = A057083(n)/3+2^(n-1), n>1. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 22 2009

First seven rows of T are

[ 1 ]

[ 1, 2 ]

[ 1, 2, 4 ]

[ 0, 1, 3, 7 ]

[ 0, 0, 1, 4, 11 ]

[ 0, 0, 0, 1, 5, 16 ]

[ 1, 1, 1, 1, 2, 7, 23 ].

(PARI) {m=33; v=concat([1, 2, 4, 7], vector(m-4)); for(n=5, m, v[n]=5*v[n-1]-9*v[n-2]+6*v[n-3]); v} /* Klaus Brockhaus, Jun 10 2007 */

(MAGMA) m:=33; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do if (j-1) mod 6 lt 3 then M[j, 1]:=1; end if; end for; for k:=2 to m do for j:=k to m do M[j, k]:=M[j-1, k-1]+M[j, k-1]; end for; end for; [ M[n, n]: n in [1..m] ]; /* Klaus Brockhaus, Jun 10 2007 */

(MAGMA) m:=33; S:=[ [1, 1, 1, 0, 0, 0][(n-1) mod 6 + 1]: n in [1..m] ]; [ &+[ Binomial(i-1, k-1)*S[k]: k in [1..i] ]: i in [1..m] ]; - Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jun 17 2007

Cf. A038504, A131022 (T read by rows), A131023 (first subdiagonal of T), A131024 (row sums of T), A131025 (antidiagonal sums of T). First through sixth column of T are in A088911, A131026, A131027, A131028, A131029, A131030 resp.

Sequence in context: A062433 A065095 A005253 this_sequence A011912 A063676 A099385

Adjacent sequences: A129336 A129337 A129338 this_sequence A129340 A129341 A129342

nonn

Paul Curtz (bpcrtz(AT)free.fr), May 28 2007

Edited and extended by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jun 10 2007

G.f multiplied by x to match the offset - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 22 2009