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%I #14 Nov 23 2020 09:28:39
%S 1,96,14763,957936,33177456,718506720,10837963166,122793273216,
%T 1103391397593,8187061491760,51724720525317,284976371277888,
%U 1395347280436638,6165194801711616,24889894891691712,92768491235726640,321987367305139071,1048378447871747424,3222195250935497833,9398840830661453088
%N Number of 6 by 6 magic squares with line sum n.
%H Guoce Xin, <a href="http://arxiv.org/abs/1208.6074">A Euclid style algorithm for MacMahon's partition analysis</a>, arxiv 1208.6074
%H G. Xin, <a href="https://doi.org/10.1016/j.jcta.2014.11.006">A Euclid style algorithm for MacMahon's partition analysis</a>, J. Comb. Theory A 131 (2015) 32 sect. 5.3
%F G.f.: (x^138+99*x^137+15057*x^136+1002806*x^135+36140317*x^134+823860011*x^133+13197261179*x^132+159778881431*x^131+1540197926928*x^130+12283604989433*x^129+83443844586997*x^128
%F +493826644119635*x^127+2591895971809073*x^126+12239625173465375*x^125+52618101897021930*x^124
%F +207948182505922572*x^123+761697282842373791*x^122+2603936594202983265*x^121
%F +8357520624415623570*x^120+25313244131813040492*x^119+72673216612249799707*x^118
%F +198540029295827265030*x^117+517913155627899876744*x^116+1293950334879519037064*x^115
%F +3104565556800370034675*x^114+7170548645642540233444*x^113+15977552472766155842750*x^112
%F +34412717940513453504180*x^111+71769782821380635837621*x^110+145167679454737704278880*x^109
%F +285189004474854548554157*x^108+544883332503752228347324*x^107
%F +1013692519414068545966383*x^106+1838319814003865364502115*x^105
%F +3253035784774708879439262*x^104+5622314253334154424175766*x^103
%F +9498907763273239021574685*x^102+15700357961071728256043309*x^101
%F +25406320589195514110356366*x^100+40277791473075750762252075*x^99
%F +62597197699253178187339298*x^98+95425280193517651890574674*x^97
%F +142766762407648666487568356*x^96+209732150155458679271033099*x^95
%F +302678001784712603830421513*x^94+429303207319389562327707454*x^93
%F +598674963030494000816618195*x^92+821156092631443052249172731*x^91
%F +1108206045308608891199410839*x^90+1472032087920610932242371227*x^89
%F +1925075439230166802560415829*x^88+2479329488091630543216144069*x^87
%F +3145503368703854928491254853*x^86+3932062984462037001968113054*x^85
%F +4844201407852058337442332388*x^84+5882809249486653844574028923*x^83
%F +7043530583232146694988816214*x^82+8315998814445857390844541404*x^81
%F +9683347293907738803126233896*x^80+11122080015097990434647761713*x^79
%F +12602367905141556425711508726*x^78+14088806780184052230859053795*x^77
%F +15541636034748392591830628113*x^76+16918375811338196658691711642*x^75
%F +18175798884655835561351408187*x^74+19272116367842845200134757907*x^73
%F +20169228060755970451363952559*x^72+20834872558688610557869003806*x^71
%F +21244511627696474156825956913*x^70+21382798694422310755770332936*x^69
%F +21244511627696474156825956913*x^68+20834872558688610557869003806*x^67
%F +20169228060755970451363952559*x^66+19272116367842845200134757907*x^65
%F +18175798884655835561351408187*x^64+16918375811338196658691711642*x^63
%F +15541636034748392591830628113*x^62+14088806780184052230859053795*x^61
%F +12602367905141556425711508726*x^60+11122080015097990434647761713*x^59
%F +9683347293907738803126233896*x^58+8315998814445857390844541404*x^57
%F +7043530583232146694988816214*x^56+5882809249486653844574028923*x^55
%F +4844201407852058337442332388*x^54+3932062984462037001968113054*x^53
%F +3145503368703854928491254853*x^52+2479329488091630543216144069*x^51
%F +1925075439230166802560415829*x^50+1472032087920610932242371227*x^49
%F +1108206045308608891199410839*x^48+821156092631443052249172731*x^47
%F +598674963030494000816618195*x^46+429303207319389562327707454*x^45
%F +302678001784712603830421513*x^44+209732150155458679271033099*x^43
%F +142766762407648666487568356*x^42+95425280193517651890574674*x^41
%F +62597197699253178187339298*x^40+40277791473075750762252075*x^39
%F +25406320589195514110356366*x^38+15700357961071728256043309*x^37
%F +9498907763273239021574685*x^36+5622314253334154424175766*x^35
%F +3253035784774708879439262*x^34+1838319814003865364502115*x^33
%F +1013692519414068545966383*x^32+544883332503752228347324*x^31
%F +285189004474854548554157*x^30+145167679454737704278880*x^29
%F +71769782821380635837621*x^28+34412717940513453504180*x^27
%F +15977552472766155842750*x^26+7170548645642540233444*x^25
%F +3104565556800370034675*x^24+1293950334879519037064*x^23
%F +517913155627899876744*x^22+198540029295827265030*x^21
%F +72673216612249799707*x^20+25313244131813040492*x^19+8357520624415623570*x^18
%F +2603936594202983265*x^17+761697282842373791*x^16+207948182505922572*x^15
%F +52618101897021930*x^14+12239625173465375*x^13+2591895971809073*x^12
%F +493826644119635*x^11+83443844586997*x^10+12283604989433*x^9+1540197926928*x^8
%F +159778881431*x^7+13197261179*x^6+823860011*x^5
%F +36140317*x^4+1002806*x^3+15057*x^2+99*x+1)*(x-1)^3/((x^4-1)^5*(x^8-1)^2*(x^3-1)^5*(x^9-1)*(x^5-1)^4*(x^6-1)^6*(x^7-1)^3*(x^10-1)) [typos corrected by _Georg Fischer_, Apr 17 2020]
%e For n = 1, there are a(1) = 96 order 6 permutation matrices with exactly one 1 in each of the two diagonals.
%Y Cf. A111158.
%K nonn
%O 0,2
%A _Guoce Xin_, Aug 30 2012
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