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A062136 Twelfth column of Losanitsch's triangle A034851 (formatted as lower triangular matrix). 3

%I

%S 1,6,42,182,693,2184,6216,15912,37854,83980,176484,352716,676270,

%T 1248072,2229096,3863080,6519591,10737090,17299646,27313650,42337659,

%U 64512240,96770544,143048880,208616044

%N Twelfth column of Losanitsch's triangle A034851 (formatted as lower triangular matrix).

%C Also seventh column (m=6) of triangle A062135.

%C Number of homeomorphically irreducible (or series-reduced) trees (no vertices of degree 2) with n+9 leaves which become tree P(7) (path on 7 nodes (vertices) or 6 edges (links) when all leaves are omitted. A leave is an edge together with a node of degree 1 at one end. Proof by Polya enumeration. See illustration for A034851.

%H G. C. Greubel, <a href="/A062136/b062136.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>

%F G.f.: Pe(6, x^2)/((1-x)^(2*6)*(1+x)^6), with Pe(6, x^2) := sum(A034839(6, m)*x^(2*m), m=0..3)= 1+15*x^2+15*x^4+x^6.

%F a(n) = A034851(n+11,11).

%F a(2n+1) = A001288(2n+12)/2; a(2n) = (A001288(2n+11)+A000389(n+5))/2. [Gary W. Adamson, Dec 15 2010]

%F a(n) = (1/(2*11!))*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7)*(n+8)*(n+9)*(n+10)*(n+11) + (1/15)*(1/2^9)*(n+2)*(n+4)*(n+6)*(n+8)*(n+10)*(1/2)*(1+(-1)^n). - _Yosu Yurramendi_, Jun 24 2013

%t Table[(1/(2*11!))*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(n + 5)*(n + 6)*(n + 7)*(n + 8)*(n + 9)*(n + 10)*(n + 11) + (1/15)*(1/2^9)*(n + 2)*(n + 4)*(n + 6)*(n + 8)*(n + 10)*(1/2)*(1 + (-1)^n), {n, 0, 50}] (* _G. C. Greubel_, Nov 24 2017 *)

%o (PARI) for(n=0,50, print1((1/(2*11!))*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(n + 5)*(n + 6)*(n + 7)*(n + 8)*(n + 9)*(n + 10)*(n + 11) + (1/15)*(1/2^9)*(n + 2)*(n + 4)*(n + 6)*(n + 8)*(n + 10)*(1/2)*(1 + (-1)^n), ", ")) \\ _G. C. Greubel_, Nov 24 2017

%o (MAGMA) [(1/(2*Factorial(11)))*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(n + 5)*(n + 6)*(n + 7)*(n + 8)*(n + 9)*(n + 10)*(n + 11) + (1/15)*(1/2^9)*(n + 2)*(n + 4)*(n + 6)*(n + 8)*(n + 10)*(1/2)*(1 + (-1)^n): n in [0..30]]; // _G. C. Greubel_, Nov 24 2017

%Y Cf. A018213.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Jun 19 2001

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Last modified September 20 15:27 EDT 2020. Contains 337265 sequences. (Running on oeis4.)