|
| |
|
|
A103210
|
|
(1/n) * Sum[i=0..n-1, C(n,i)*C(n,i-1)*2^i*3^(n-i) ], a(0)=1.
|
|
14
| |
|
|
1, 3, 15, 93, 645, 4791, 37275, 299865, 2474025, 20819307, 178003815, 1541918901, 13503125805, 119352115551, 1063366539315, 9539785668657, 86104685123025, 781343125570515, 7124072211203775
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| The Hankel transform of this sequence is 6^C(n+1,2). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 28 2007
The Hankel transform of the sequence starting 1, 1, 3, 15, ... is A081955. [From Paul Barry (pbarry(AT)wit.ie), Dec 09 2008]
Number of Schroeder paths from (0,0) to (0,2n) allowing two colors for the down steps (or alternatively for the rise steps). [From Paul Barry (pbarry(AT)wit.ie), Feb 01 2009]
Essentially reversion of x(1-2x)/(1+x). [From Paul Barry (pbarry(AT)wit.ie), Apr 28 2009]
|
|
|
LINKS
| E. Ackerman, G. Barequet, R. Y. Pinter and D. Romik, The number of guillotine partitions in d dimensions
|
|
|
FORMULA
| G.f.: [1-z-(z^2-10z+1)^(1/2)]/(4z).
a(n)=sum{k=0..n, C(n+k, 2k)2^k*C(k)}, C(n) given by A000108. - Paul Barry (pbarry(AT)wit.ie), May 21 2005
a(n)=Sum_{k, 0<=k<=n}A060693(n,k)*2^(n-k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 02 2007
a(0)=1, a(n)=a(n-1)+2*Sum_{k, 0<=k<=n-1}a(k)*a(n-1-k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 23 2007
a(n)=(3/2)*A107841(n) for n>0 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 28 2007
G.f.: 1/(1-x-2x/(1-x-2x/(1-x-2x/(1-.... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Feb 01 2009]
G.f.: 1/(1-3x-6x^2/(1-5x-6x^2/(1-5x-6x^2/(1-... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Apr 28 2009]
G.f.: 1/(1-3x/(1-2x/(1-3x/(1-2x/(1-3x/(1-... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), May 14 2009]
a(n) = Hypergeometric2F1(-n,n+1,2,-2) = sum{k=0..n, C(n+k,k) * C(n,k) * 2^k/ (k+1)}. [Paul Barry, Feb 8 2011]
G.f.: A(x)=(1-x-(x^2-10*x+1)^(1/2))/(4*x)= 1/(G(0)-x); G(k)= 1 + x - 3*x/G(k+1); (continued fraction, 1-step ). - Sergei N. Gladkovskii, Jan 05 2012
|
|
|
CROSSREFS
| Third column of array A103209.
Sequence in context: A192296 A002893 A074539 * A203014 A060066 A206177
Adjacent sequences: A103207 A103208 A103209 * A103211 A103212 A103213
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Ralf Stephan, Jan 27 2005
|
|
|
EXTENSIONS
| Spelling/notation corrections by Charles R Greathouse IV (charles.greathouse(AT)case.edu), Mar 18 2010
|
| |
|
|
