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%I #10 Feb 17 2024 11:53:33
%S 1,120,56628,41983200,38244074820,39137678949600,43169977801676880,
%T 50180219346847075200,60633191914827463116900,
%U 75481112829367580702796000,96214948596107910313766029008,125026188575803676432586848856960,165076420520740156599642652986224784
%N a(n) = Product_{k=1..L} hypergeom([-n, -n], [1], k) with L = 4.
%H A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, <a href="http://arxiv.org/abs/1507.03227">Diagonals of rational functions and selected differential Galois groups</a>, arXiv preprint arXiv:1507.03227 [math-ph], 2015.
%H Jacques-Arthur Weil, <a href="http://www.unilim.fr/pages_perso/jacques-arthur.weil/diagonals/">Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"</a>
%F Recurrence: (n-3)^2*(n-2)^2*(n-1)^2*n^4*(2*n-13)*(2*n-11)*(2*n-9)*(58844*n^6 - 1412256*n^5 + 13592380*n^4 - 66837440*n^3 + 176157576*n^2 - 234562624*n + 122345655)*a(n)=120*(n-3)^2*(n-2)^2*(n-1)^2*(2*n-13)*(2*n-11)*(2*n-9)*(2*n-1)^2*(235376*n^8 - 5884400*n^7 + 60077388*n^6 - 323129466*n^5 + 985542429*n^4 - 1709581035*n^3 + 1603577752*n^2 - 723821269*n + 122325360)*a(n-1) - 16*(n-3)^2*(n-2)^2*(2*n-13)*(2*n-11)*(2*n-3)^2*(2*n-1)*(546307696*n^10 - 16935538576*n^9 + 225057696392*n^8 - 1678347886040*n^7 + 7722946671584*n^6 - 22718876464232*n^5 + 42836897175604*n^4 - 50512102808434*n^3 + 35149893258801*n^2 - 12841591780296*n + 1871682205320)*a(n-2) + 1920*(n-3)^2*(2*n-13)*(2*n-11)*(2*n-5)^2*(2*n-3)*(2*n-1)^2*(123101648*n^10 - 4062354384*n^9 + 58223209796*n^8 - 475879945898*n^7 + 2447982317405*n^6 - 8247410195747*n^5 + 18335033385744*n^4 - 26375282324477*n^3 + 23256044918502*n^2 - 11156176523664*n + 2136088451160)*a(n-3) - 256*(2*n-13)*(2*n-7)^2*(2*n-5)*(2*n-3)*(2*n-1)^2*(37315922600*n^12 - 1567268749200*n^11 + 29452637855264*n^10 - 326877445085240*n^9 + 2381202054498224*n^8 - 11964823606010912*n^7 + 42393024452273562*n^6 - 106320184974511802*n^5 + 186423157681832475*n^4 - 221423574944289564*n^3 + 167504632422689313*n^2 - 71361665488438164*n + 12581492486282280)*a(n-4) + 276480*(2*n-13)*(2*n-9)^2*(2*n-7)*(2*n-5)*(2*n-3)^2*(2*n-1)^2*(123101648*n^10 - 4554760976*n^9 + 73734017444*n^8 - 685490500982*n^7 + 4036636234787*n^6 - 15651262473677*n^5 + 40189058083536*n^4 - 66839237050563*n^3 + 67905381181320*n^2 - 37111550536602*n + 7867011090105)*a(n-5) - 331776*(2*n-11)^2*(2*n-9)*(2*n-7)*(2*n-5)*(2*n-3)^2*(2*n-1)^2*(546307696*n^10 - 21306000144*n^9 + 362727235784*n^8 - 3536617831208*n^7 + 21770125653896*n^6 - 87904346623040*n^5 + 234073849141972*n^4 - 401839017403046*n^3 + 419321125129479*n^2 - 234120782139840*n + 50374929826575)*a(n-6) + 358318080*(2*n-13)^2*(2*n-11)*(2*n-9)*(2*n-7)*(2*n-5)^2*(2*n-3)^2*(2*n-1)^2*(235376*n^8 - 7296656*n^7 + 94677660*n^6 - 666175438*n^5 + 2750313619*n^4 - 6721504509*n^3 + 9292612455*n^2 - 6418417212*n + 1579020075)*a(n-7) - 429981696*(n-7)^2*(2*n-15)*(2*n-13)*(2*n-11)*(2*n-9)*(2*n-7)*(2*n-5)^2*(2*n-3)^2*(2*n-1)^2*(58844*n^6 - 1059192*n^5 + 7413760*n^4 - 25413600*n^3 + 43959636*n^2 - 35098488*n + 9342135)*a(n-8). - _Vaclav Kotesovec_, Feb 17 2024
%p a := n -> mul(hypergeom([-n, -n], [1], k), k=1..4):
%p seq(simplify(a(k)), k=0..11);
%t a[n_] := Product[Hypergeometric2F1[-n, -n, 1, k], {k, 1, 4}];
%t Table[a[n], {n, 0, 12}] (* _Jean-François Alcover_, Mar 20 2018 *)
%Y With the parameter L in the name: A000012 (L=0), A000984 (L=1), A268555 (L=2), A301392 (L=3), this seq. (L=4).
%K nonn
%O 0,2
%A _Peter Luschny_, Mar 20 2018
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