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%I #12 Jun 11 2023 12:01:43
%S 0,0,0,1,1722320,21475242671,53885342499340,59751188387945950,
%T 40765121565728774056,20317575263352346466495,
%U 8176401371550779497913310,2829198944755856389638829950,877500019282729363773560673680,251039391334595768636642931100892
%N Number of permutations of 3 indistinguishable copies of 1..n with exactly 9 adjacent element pairs in decreasing order.
%H Andrew Howroyd, <a href="/A151639/b151639.txt">Table of n, a(n) for n = 1..200</a>
%H <a href="/index/Rec#order_55">Index entries for linear recurrences with constant coefficients</a>, signature (2002, -1903473, 1145540604, -490697246100, 159504703878456, -40962938199359892, 8543917891847861088, -1476612487288484536734, 214675942140068854920940, -26565076527531007381806478, 2824272739283822276672151576, -259926043826785157432428807284, 20836537288650855023543220951672, -1462352927921603694734632985134980, 90234742223036589432125233952549568, -4912790794256403282637163015492980521, 236700371529443998052889391662516495954, -10117044203868226837490381498627715891337, 384392825539032417065324935501966842465100, -13004256775286258922566221096216296355899696, 392250739476898367357151508534457623894980672, -10559947991607095603218681080920151843846838368, 253930915878445088171984501266369006919853028224, -5457008736007770393930005757333651419545424209920, 104835678793820064850370954223248750262209382590976, -1800591441813144805434585972840310807932181304723712, 27645160835526261254255724891832138131239988591582208, -379297401494603160308262696677824295041524194821218304, 4648046415897612983578549543440415906139180127325224960, -50836811152257276881232477862297786085497310732407603200, 495797528206249652557511784489086218654098684461285376000, -4306864041233327994708273688892836293229391593828515840000, 33278954943219381604637582548198385534549365246289510400000, -228378313437880516350730344435042790180832441478414336000000, 1389439890170722317506967476479999670196450188486246400000000, -7478904815075104227138008886230783209082840352922009600000000, 35534015681764477478511600769987832363545712272605184000000000, -148635503292844144648387165778592391627025675735531520000000000, 545746861188001821470613819055564655367778498038988800000000000, -1753092258313596312335129565355450174371464849915904000000000000, 4908287237887375627713223935002134384231418670612480000000000000, -11926567015623705908482713680801772960036536136499200000000000000, 25029412170469626086237828839303721592269064634368000000000000000, -45113800504590260658677604425025133931371450859520000000000000000, 69385718142932409879033870377080183947002694860800000000000000000, -90364789184554934855367459384763800482020851712000000000000000000, 98736713417057627145562379386668809285460295680000000000000000000, -89483278154843571438748621153510164400806297600000000000000000000, 66291975848611425632217745091071357489250304000000000000000000000, -39379593895079017458958787895990101779415040000000000000000000000, 18264099584981881548449365838653458638438400000000000000000000000, -6359618920578418521780902682541745504256000000000000000000000000, 1561358972594287955292211036133366169600000000000000000000000000, -240686894019238180512616839527792640000000000000000000000000000, 17504501383217322219099406511112192000000000000000000000000000).
%F a(n) = Sum_{j=0..11} (-1)^(j+1)*binomial(3*n+1, 11-j)*(binomial(j+1, 3))^n. - _G. C. Greubel_, Mar 26 2022
%t T[n_, k_]:= T[n, k]= Sum[(-1)^(k-j)*Binomial[3*n+1, k-j+2]*(Binomial[j+1,3])^n, {j, 0, k+2}];
%t Table[T[n, 9], {n, 30}] (* _G. C. Greubel_, Mar 26 2022 *)
%o (Sage)
%o @CachedFunction
%o def T(n, k): return sum( (-1)^(k-j)*binomial(3*n+1, k-j+2)*(binomial(j+1,3))^n for j in (0..k+2) )
%o [T(n, 9) for n in (1..30)] # _G. C. Greubel_, Mar 26 2022
%Y Column k=9 of A174266.
%K nonn
%O 1,5
%A _R. H. Hardin_, May 29 2009
%E Terms a(9) and beyond from _Andrew Howroyd_, May 06 2020
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