A151638 - OEIS

%I #13 Jun 11 2023 12:00:25

%S 0,0,0,243,12750255,40396577931,41106807537048,22745757394235250,

%T 8699569720553953791,2617057246555282014495,668634213456480163469985,

%U 152325974081288304581227794,31960987230978975148286275260,6315174416665212479526100114476

%N Number of permutations of 3 indistinguishable copies of 1..n with exactly 8 adjacent element pairs in decreasing order.

%H Andrew Howroyd, <a href="/A151638/b151638.txt">Table of n, a(n) for n = 1..200</a>

%H <a href="/index/Rec#order_45">Index entries for linear recurrences with constant coefficients</a>, signature (1287, -778635, 295087143, -78757785579, 15778863379215, -2469906458492215, 310433833720029195, -31951771730535427065, 2733179350270223477785, -196528389527687790031593, 11985141042154490418785781, -624319353245531324368589785, 27937742235090714610172185269, -1078918430758501220128475037837, 36090971895951595049632092872905, -1048807183921997729421431750802210, 26538626394588804244411344362289180, -585742712867636473889820333895325960, 11290946875479197719981055898705842640, -190242978590642131810466071594577045856, 2803028334513544005305538345639354644032, -36116842200479350538673233701931574526080, 406839176866275971760705035377939565311488, -4003890839024013630299437079281080555536384, 34391681475642311442636394036792617457582080, -257483345852835701281401936856484593110220800, 1677354698611307266147622604716869996838912000, -9487912103085101966952941170538539017830400000, 46483514384733904314355601898864606157209600000, -196665838715803401733927598517905248747520000000, 716085486868940810143492111107549922590720000000, -2234951903777824995459451037159105809612800000000, 5951411268951166700745578197262375321600000000000, -13448380314552698433108601331259958886400000000000, 25625033743682596345813503616255996723200000000000, -40863903217821737298129553070185512960000000000000, 54045744994528210207993057131469209600000000000000, -58623407588610300485897706188832768000000000000000, 51413975240876145030443038385635328000000000000000, -35776951355628605313894780040642560000000000000000, 19242984319988159850768781384089600000000000000000, -7696862042758387149551324626944000000000000000000, 2151367384135060324918650470400000000000000000000, -374436012994639297259765760000000000000000000000, 30509601058822461258203136000000000000000000000).

%F a(n) = Sum_{j=0..10} (-1)^j*binomial(3*n+1, 10-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, 8], {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, 8) for n in (1..30)] # _G. C. Greubel_, Mar 26 2022

%o (PARI) a(n) = sum(j=0, 10, (-1)^j*binomial(3*n+1, 10-j)*(binomial(j+1, 3))^n); \\ _Michel Marcus_, Mar 27 2022

%Y Column k=8 of A174266.

%K nonn

%O 1,4

%A _R. H. Hardin_, May 29 2009

%E Terms a(9) and beyond from _Andrew Howroyd_, May 06 2020