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%I #99 Nov 03 2023 06:39:36
%S 1,1,7,37,231,1451,9331,60691,398567,2636263,17538157,117224317,
%T 786588243,5295520699,35751527189,241958082737,1641010879207,
%U 11150608945863,75894449584849,517331384963959,3531097638576781,24131083600660801,165090433568378523
%N Central heptanomial coefficients: largest coefficient of (1+x+...+x^6)^n.
%C Apparently: number of n-step 1-D walks ending at the origin with steps of size 0, 1, 2 or 3. - _David Scambler_, Apr 09 2012
%C Generally, largest coefficient of (1+x+...+x^k)^n is asymptotic to (k+1)^n * sqrt(6/(k*(k+2)*Pi*n)). - _Vaclav Kotesovec_, Aug 09 2013
%D Rudolph-Lilith, Michelle, and Lyle E. Muller. "On a link between Dirichlet kernels and central multinomial coefficients." Discrete Mathematics 338.9 (2015): 1567-1572.
%H Alois P. Heinz, <a href="/A025012/b025012.txt">Table of n, a(n) for n = 0..1000</a> (first 201 terms from T. D. Noe)
%H M. Rudolph-Lilith and L. E. Muller, <a href="http://arxiv.org/abs/1403.5942">On an explicit representation of central (2k+1)-nomial coefficients</a>, arXiv preprint arXiv:1403.5942 [math.CO], 2014.
%H Doron Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/tokhniot/EKHAD">EKHAD</a>; <a href="/A025012/a025012.txt">Local copy</a>.
%F D-finite with recurrence: 3*n*(3*n-1)*(3*n-2)*a(n) +(41*n^3-723*n^2+1132*n-456)*a(n-1) +7*(-383*n^3+2607*n^2-5992*n+4608)*a(n-2) +49*(-83*n^3+1317*n^2-6706*n +10512)*a(n-3) +343*(199*n^3-2487*n^2+10394*n-14496)*a(n-4) +2401*(n-4)*(43*n^2-437*n+1116)*a(n-5) -16807*(n-4)*(n-5)*(5*n-24)*a(n-6) -117649*(n-5)*(n-6)*(n-4)*a(n-7) = 0. - _R. J. Mathar_, Feb 21 2010
%F I checked that Mathar's conjectured formula fits the 200 terms of the b-file. Note that the coefficients are powers of 7: 49=7^2, 343=7^3, 2401=7^4, 16807=7^5, 117649=7^6. - _T. D. Noe_, Nov 15 2010
%F Comment from Doron Zeilberger, Apr 12 2011: The recurrence can be rigorously proved (and also discovered at the same time, no need for a guessing program) automatically via the Almkvist-Zeilberger algorithm (procedure AZd in the EKHAD link). In fact the program finds a 4th-order recurrence, not a 7th-order one: 343/3*(4*n+15)*(n+3)*(n+2)*(n+1)/(3*n+7)/(4*n+7)/(11+3*n)/(n+4)*a(n)+98/3*(4*n+15)*(n+3)*(n+2)*(2*n+7) /(11+3*n)/(3*n+10)/(4*n+11)/(n+4)*a(n+1) -14/3*(n+3)*(92*n^3+713*n^2+1747*n+1372)/(3*n+7)/(4*n+7)/(11+3*n)/(n+4)*a(n+2) -2/3*(4*n+15)*(2*n+7)*(37*n^2 +222*n+332)/(11+3*n)/(3*n+10)/(4*n+1)/(n+4)*a(n+3) +a(n+4) = 0.
%F G.f.: sqrt((-4*x*(7*x+6)*(1+2*x)*g^3-4*x*(77*x^2+117*x+43)*g^2-4*x*(119*x^2+188*x+72)*g-307*x^2-217*x^3+27-67*x)/((7*x-1)*(49*x^2-14*x-27)))/(1+x) where (2*g+1)*(g^3+5*g^2+6*g+1)*x^2+g*(g^3+3+11*g+7*g^2)*x-g = 0. - _Mark van Hoeij_, May 06 2013
%F a(n) ~ 7^n / sqrt(8*Pi*n). - _Vaclav Kotesovec_, Aug 09 2013
%p g := RootOf((2*g+1)*(g^3+5*g^2+6*g+1)*x^2+g*(g^3+3+11*g+7*g^2)*x-g, g);
%p ogf := sqrt((-4*x*(7*x+6)*(1+2*x)*g^3-4*x*(77*x^2+117*x+43)*g^2-4*x*(119*x^2+188*x+72)*g-307*x^2-217*x^3+27-67*x)/((7*x-1)*(49*x^2-14*x-27)))/(1+x);
%p series(ogf,x=0,30); # _Mark van Hoeij_, May 06 2013
%t With[{poly=Plus@@(x^Range[0,6])},Table[Max[CoefficientList[Expand[poly^i],x]], {i,0,20}]] (* _Harvey P. Dale_, Mar 09 2011 *)
%Y Cf. A001405, A002426, A005190, A005191, A018901, A025013, A025014.
%K easy,nonn
%O 0,3
%A _David W. Wilson_
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