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A075900 G.f.: Product_{n>0} 1/(1 - 2^(n-1)*x^n). 7
1, 1, 3, 7, 19, 43, 115, 259, 659, 1523, 3731, 8531, 20883, 47379, 113043, 259219, 609683, 1385363, 3245459, 7344531, 17028499, 38579603, 88585619, 199845267, 457864595, 1028904339, 2339763603, 5256820115, 11896157587, 26626389395 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..3180 (terms 0..1000 from Alois P. Heinz)

N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.

FORMULA

a(n) = Sum_{ partitions n = c_1 + ... + c_k } 2^(n-k). If p(n, m) = number of partitions of n into m parts, a(n) = sum_{m=1..n} p(n, m)*2^(n-m).

G.f.: Sum_{n>=0} (a(n)/2^n)*x^n = Product_{n>0} 1/(1-x^n/2). - Vladeta Jovovic, Feb 11 2003

a(n) = 1/n*Sum_{k=1..n} A080267(k)*a(n-k). - Vladeta Jovovic, Feb 11 2003

G.f.: exp( Sum_{n>=1} x^n / (n*(1 - 2^n*x^n)) ). - Paul D. Hanna, Jan 13 2013

a(n) = s(1,n), a(0)=1, where s(m,n)=sum(k=m..n/2, 2^(k-1)*s(k,n-k))+2^(n-1), s(n,n) = 2^(n-1), s(m,n)=0, m>. - Vladimir Kruchinin, Sep 06 2014

a(n) ~ 2^(n-2) * (Pi^2 - 6*log(2)^2)^(1/4) * exp(sqrt((Pi^2 - 6*log(2)^2)*n/3)) / (3^(1/4) * sqrt(Pi) * n^(3/4)). - Vaclav Kotesovec, Mar 09 2018

MAPLE

oo := 101; t1 := mul(1/(1-x^n/2), n=1..oo): t2 := series(t1, x, oo-1): t3 := seriestolist(t2): A075900 := n->2^n*t3[n+1];

with(combinat); A075900 := proc(n) local i, t1, t2, t3; t1 := partition(n); t2 := 0; for i from 1 to nops(t1) do t3 := t1[i]; t2 := t2+2^(n-nops(t3)); od: t2; end;

MATHEMATICA

b[n_] := b[n] = Sum[d*2^(n - n/d), {d, Divisors[n]}]; a[0] = 1; a[n_] := a[n] = 1/n*Sum[b[k]*a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 30}] (* Jean-Fran├žois Alcover, Mar 20 2014, after Vladeta Jovovic, fixed by Vaclav Kotesovec, Mar 08 2018 *)

PROG

(PARI) {a(n)=polcoeff(prod(k=1, n, 1/(1-2^(k-1)*x^k+x*O(x^n))), n)} \\ Paul D. Hanna, Jan 13 2013

(PARI) {a(n)=polcoeff(exp(sum(k=1, n+1, x^k/(k*(1-2^k*x^k)+x*O(x^n)))), n)} \\ Paul D. Hanna, Jan 13 2013

(Maxima)

s(m, n):=if n<m then 0 else if n=m then 2^(n-1) else sum(2^(k-1)*s(k, n-k), k, m, ceiling(n/2))+2^(n-1);

makelist(s(1, n), n, 1, 27); /* Vladimir Kruchinin, Sep 06 2014 */

CROSSREFS

Cf. A300579.

Sequence in context: A127990 A192301 A055622 * A176500 A136041 A146685

Adjacent sequences:  A075897 A075898 A075899 * A075901 A075902 A075903

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Oct 15 2002

EXTENSIONS

More terms from Vladeta Jovovic, Feb 11 2003

STATUS

approved

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Last modified February 20 12:57 EST 2019. Contains 320327 sequences. (Running on oeis4.)