The On-Line Encyclopedia of Integer Sequences (OEIS)

A075271 revision #19

A075271

a(0)=1 and, for n>=1, (BM)a(n)=2a(n-1), where BM is the BinomialMean transform. BM is defined by (BM)a(n)=(M^n)a(0) where (M)a(n) is the mean (a(n)+a(n+1))/2, or, alternatively, by (BM)a(n)=Sum[C(n,k)a(k),k=0..n]/(2^n).

1, 3, 17, 211, 5793, 339491, 41326513, 10282961907, 5181436229441, 5258784071302723, 10717167529963833681, 43779339268428732008723, 358114286723184561034838497, 5862685570087914880854259126371, 192026370558313054275618817346778353

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OFFSET

0,2

COMMENTS

The BinomialMean transform of this sequence is given in A075272.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..50

Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

EXAMPLE

Given that a(0)=1 and a(1)=3. Then (BM)a(2)=(1+2*3+a(2))/4=2a(1)=6, hence a(2)=17.

MAPLE

iBM:= proc(p) proc(n) option remember; add(2^(k)*p(k)*(-1)^(n-k) *binomial(n, k), k=0..n) end end: a:= iBM(aa): aa:= n-> `if`(n=0, 1, 2*a(n-1)): seq(a(n), n=0..16); # Alois P. Heinz, Sep 09 2008

MATHEMATICA

iBM[p_] := Module[{proc}, proc[n_] := proc[n] = Sum[2^k*p[k]*(-1)^(n-k) * Binomial[n, k], {k, 0, n}]; proc]; a = iBM[aa]; aa[n_] := If[n == 0, 1, 2*a[n-1]]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Nov 08 2015, after Alois P. Heinz *)

Table[Sum[QFactorial[k, 2] Binomial[n + 1, k]/2, {k, 0, n + 1}], {n, 0, 15}] (* Vladimir Reshetnikov, Oct 16 2016 *)

CROSSREFS

Sequence in context: A210898 A009494 A267659 * A194925 A072350 A181032

Adjacent sequences: A075268 A075269 A075270 * A075272 A075273 A075274

KEYWORD

eigen,nonn

AUTHOR

John W. Layman, Sep 11 2002

EXTENSIONS

More terms from Alois P. Heinz, Sep 09 2008

STATUS

approved