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A187004 Expansion of A(x) = (1 + 2*x^2 + 6*x^3 + 9*x^4 + 8*x^5 + 5*x^6) / (1 - x - 2*x^2 - 3*x^3 - 3*x^4 - 2*x^5 - x^6). 1

%I #32 Feb 27 2019 01:20:51

%S 1,5,16,41,101,254,638,1609,4048,10185,25631,64502,162319,408476,

%T 1027931,2586793,6509675,16381622,41224415,103741401,261065639,

%U 656972695,1653274340,4160471302,10469842201,26347398589,66303330946

%N Expansion of A(x) = (1 + 2*x^2 + 6*x^3 + 9*x^4 + 8*x^5 + 5*x^6) / (1 - x - 2*x^2 - 3*x^3 - 3*x^4 - 2*x^5 - x^6).

%C The value of a(0) is undefined. The A(x) leads to a(0)=1, the a(n) formula to a(0)=0 and the recurrence relation to a(0)=6 as pointed out by _Bruno Berselli_. - _Johannes W. Meijer_, Jul 10 2011

%H Ivan Panchenko, <a href="/A187004/b187004.txt">Table of n, a(n) for n = 1..200</a>

%H Kruchinin Vladimir Victorovich, <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1, 2, 3, 3, 2, 1).

%F a(n) = n*Sum_{m=1..n} Sum_{k=m..n} binomial(m, k-m)*Sum_{j=0..k} binomial(k,j) * binomial(j, n-3*k+2*j)))/m).

%F a(n) = a(n-1) + 2*a(n-2) + 3*a(n-3) + 3*a(n-4) + 2*a(n-5) + a(n-6); a(1)=1, a(2)=5, a(3)=16, a(4)=41, a(5)=101, a(6)=254. - _Harvey P. Dale_, Jun 14 2011

%p A187004 := proc(n) local m,k,j: (n*add(add(binomial(m,k-m)*add(binomial(k,j)*binomial(j,n-3*k+2*j),j=0..k), k=m..n)/m, m= 1..n)) end: seq(A187004(n),n=1..27); # _Johannes W. Meijer_, Jul 10 2011

%t LinearRecurrence[{1,2,3,3,2,1},{1,5,16,41,101,254},30] (* or *) Rest[ CoefficientList[Series[(1+2x^2+6x^3+9x^4+8x^5+5x^6)/(1-x-2x^2- 3x^3-3x^4-2x^5-x^6), {x,1,30}],x]] (* _Harvey P. Dale_, Jun 14 2011 *)

%o (Maxima) a(n):=n*sum(sum(binomial(m,k-m)*sum(binomial(k,j)*binomial(j,n-3*k+2*j),j,0,k),k,m,n)/m,m,1,n);

%K nonn,easy

%O 1,2

%A _Vladimir Kruchinin_, Mar 01 2011

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Last modified March 28 13:42 EDT 2024. Contains 371254 sequences. (Running on oeis4.)