A075272 BinomialMean (BM) transform of A075271, which see for the definition of (BM).

1, 2, 6, 34, 422, 11586, 678982, 82653026, 20565923814, 10362872458882, 10517568142605446, 21434335059927667362, 87558678536857464017446, 716228573446369122069676994, 11725371140175829761708518252742

Offset: 0

Author

John W. Layman, Sep 11 2002

Keywords

• nonn
• easy
• changed

Comments

a(n) = 2*A075271(n-1), for n >= 1.

Binomial transform of A005329. - Vladimir Reshetnikov, Nov 20 2015

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, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.

Crossrefs

Cf. A075271, A005329.

Programs

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:='a': aa:= iBM(a): a:= n-> `if` (n=0, 1, 2*aa(n-1)): seq (a(n), n=0..16); # Alois P. Heinz, Sep 09 2008

Mathematica

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

Formula

G.f.: Sum_{n>=0} x^n*Product_{i=1..n}(2^i/(1+(2^i-1)*x)). - Vladeta Jovovic, Mar 10 2008

O.g.f. as a continued fraction of Stieltjes's type: 1/(1 - 2*x/(1 - x/(1 - 2^3*x/(1 - 3^2*x/(1 - 2^5*x/(1 - 7^2*x/(1 - 2^7*x/(1 - 15^2*x/(1 - 2^9*x/(1 - 31^2*x - ... )))))))))). Cf. A005329. - Peter Bala, Nov 10 2017

Extensions

More terms from Alois P. Heinz, Sep 09 2008