A055278 Number of rooted trees with n nodes and 3 leaves.
1, 3, 8, 18, 35, 62, 103, 161, 241, 348, 487, 664, 886, 1159, 1491, 1890, 2364, 2922, 3574, 4329, 5198, 6192, 7322, 8600, 10039, 11651, 13450, 15450, 17665, 20110, 22801, 25753, 28983, 32508, 36345, 40512, 45028, 49911, 55181, 60858, 66962, 73514, 80536
Offset: 4
Examples
G.f. = x^4 + 3*x^5 + 8*x^6 + 18*x^7 + 35*x^8 + 62*x^9 + 103*x^10 + ...
Links
- Georg Fischer, Table of n, a(n) for n = 4..200
- Index entries for sequences related to rooted trees
- Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,0,1,2,-3,1).
Programs
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Mathematica
a[ n_] := Quotient[ 3 n^4 - 20 n^3 + 54 n^2 - 60 n + 32, 144];
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PARI
{a(n) = (3*n^4 - 20*n^3 + 54*n^2 - 60*n + 32) \ 144}; /* Michael Somos, Jun 29 2015 */ -
PARI
{a(n) = if( n<0, n = -1-n; polcoeff( (1 + x + x^3) / ((1 - x)^3 * (1 - x^2) * (1 - x^3)) + x * O(x^n), n), polcoeff( x^4 * (1 + x^2 + x^3) / ((1 - x)^3 * (1 - x^2) * (1 - x^3)) + x * O(x^n), n))}; /* Michael Somos, Jun 29 2015 */
Formula
G.f.: x^4*(x^3+x^2+1)/((1-x^2)*(1-x^3)*(1-x)^3) (conjectured). - Ralf Stephan, Mar 07 2004
a(n) = A055364(4 - n) for all n in Z. - Michael Somos, Jun 29 2015