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%I #48 Aug 01 2022 09:15:52
%S 1,14,126,924,6006,36036,204204,1108536,5819814,29745716,148728580,
%T 730122120,3528923580,16830250920,79342611480,370265520240,
%U 1712478031110,7857252142740,35794148650260
%N Expansion of 1/(1-4*x)^(7/2).
%C Also convolution of A000984 with A038845, also convolution of A000302 with A002802, also convolution of A002457 with A002697. - _Rui Duarte_, Oct 08 2011
%C 5*a(n) is the number of (n+3) X 2 Young tableaux with a four horizontal walls between the first and second column. If there is a wall between two cells, the entries may be decreasing; see [Banderier, Wallner 2021], A000984 for one horizontal wall, A002457 for two, and A002802 for three. - _Michael Wallner_, Mar 09 2022
%H Vincenzo Librandi, <a href="/A020918/b020918.txt">Table of n, a(n) for n = 0..200</a>
%H Cyril Banderier and Michael Wallner, <a href="https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2021/47.html">Young Tableaux with Periodic Walls: Counting with the Density Method</a>, Séminaire Lotharingien de Combinatoire, 85B (2021), Art. 47, 12 pp.
%F a(n) = binomial(n+3, 3)*A000984(n+3)/A000984(3), where A000984 are the central binomial coefficients. - _Wolfdieter Lang_
%F a(n) ~ 8/15*Pi^(-1/2)*n^(5/2)*2^(2*n)*{1 + 35/8*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 22 2001
%F a(n) = Sum_{a+b+c+d+e+f+g=n} f(a)*f(b)*f(c)*f(d)*f(e)*f(f)*f(g) with f(n)=A000984(n). - _Philippe Deléham_, Jan 22 2004
%F a(n) = A000292(n)*A000984(n+2)/20. - _Zerinvary Lajos_, May 05 2007
%F From _Rui Duarte_, Oct 08 2011: (Start)
%F a(n) = ((2n+5)(2n+3)(2n+1)/(5*3*1)) * binomial(2n, n).
%F a(n) = binomial(2n+6, 6) * binomial(2n, n) / binomial(n+3, 3).
%F a(n) = binomial(n+3, 3) * binomial(2n+6, n+3) / binomial(6, 3). (End)
%F a(n) = 4^n*hypergeom([-n,-5/2], [1], 1). - _Peter Luschny_, Apr 26 2016
%F Boas-Buck recurrence: a(n) = (14/n)*Sum_{k=0..n-1} 4^(n-k-1)*a(k), n >= 1, a(0) = 1. Proof from a(n) = A046521(n+3, 3). See a comment there. - _Wolfdieter Lang_, Aug 10 2017
%F From _Amiram Eldar_, Apr 07 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 10*sqrt(3)*Pi - 160/3.
%F Sum_{n>=0} (-1)^n/a(n) = 10*sqrt(5)*log(phi) - 320/3, where phi is the golden ratio (A001622). (End)
%F D-finite with recurrence n*a(n) +2*(-2*n-5)*a(n-1)=0. - _R. J. Mathar_, Aug 01 2022
%p seq(binomial(2*n,n)*binomial(n,(n-3))/20, n=2..21); # _Zerinvary Lajos_, May 05 2007
%p seq(simplify(4^n*hypergeom([-n,-5/2], [1], 1)),n=0..18); # _Peter Luschny_, Apr 26 2016
%t CoefficientList[Series[1/(1-4x)^(7/2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Jul 04 2013 *)
%o (PARI) vector(30, n, n--; binomial(2*n+6, n+3)*binomial(n+3, 3)/20 ) \\ _G. C. Greubel_, Jul 20 2019
%o (Magma) [Binomial(2*n+6, n+3)*Binomial(n+3, 3)/20: n in [0..30]]; // _G. C. Greubel_, Jul 20 2019
%o (Sage) [binomial(2*n+6, n+3)*binomial(n+3, 3)/20 for n in (0..30)] # _G. C. Greubel_, Jul 20 2019
%o (GAP) List([0..30], n-> Binomial(2*n+6, n+3)*Binomial(n+3, 3)/20); # _G. C. Greubel_, Jul 20 2019
%Y Cf. A000302, A000984, A001622, A002457, A002697, A002802, A000292, A038845, A046521 (fourth column).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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