A151636
Number of permutations of 3 indistinguishable copies of 1..n with exactly 6 adjacent element pairs in decreasing order.
Original entry on oeis.org
0, 0, 1, 49682, 58571184, 21475242671, 4476844162434, 678770257169016, 84698452637705746, 9324662905839457490, 944619860914428706035, 90435965482528402360106, 8327298182652856026223632, 746238093776109096993716949, 65611401726068220422014371676
Offset: 1
- Andrew Howroyd, Table of n, a(n) for n = 1..200
- Index entries for linear recurrences with constant coefficients, signature (462, -97119, 12368586, -1071791874, 67276115172, -3179430045126, 116078176526940, -3333091664566125, 76240546809223870, -1401969472955910939, 20859439219374986298, -252205532159847743136, 2484342723967019291664, -19958746288798848738096, 130732178656572589908768, -697028928252901175309184, 3016166101164375614922240, -10546444216517128719718400, 29623887798829604653056000, -66331952042317220782080000, 117232249430274689433600000, -161447240088380473344000000, 170296114651151892480000000, -134298682034837913600000000, 76357985182875648000000000, -29486276845240320000000000, 6908379398144000000000000, -740183506944000000000000).
-
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}];
Table[T[n, 6], {n, 30}] (* G. C. Greubel, Mar 26 2022 *)
-
a(n) = sum(j=0, 8, (-1)^j*binomial(3*n+1, 8-j)*(binomial(j+1, 3))^n); \\ Michel Marcus, Mar 27 2022
-
@CachedFunction
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) )
[T(n, 6) for n in (1..30)] # G. C. Greubel, Mar 26 2022
A151637
Number of permutations of 3 indistinguishable copies of 1..n with exactly 7 adjacent element pairs in decreasing order.
Original entry on oeis.org
0, 0, 0, 6750, 40241088, 40396577931, 18096792917796, 5183615502649800, 1129236431002624116, 205937718403143468690, 33309411205799991188160, 4957409194925592040479126, 695659299332984273417824080, 93590807522941640152432361025, 12213007949715545409829962783732
Offset: 1
- Andrew Howroyd, Table of n, a(n) for n = 1..200
- Index entries for linear recurrences with constant coefficients, signature (792, -290862, 65984412, -10391950167, 1210552073172, -108511757620112, 7687874707991352, -438801842634100047, 20463952984838053792, -788123497343025648150, 25270669669098512733228, -678837532745427806095113, 15349821535045369264190388, -293201738441368171406215308, 4743013718033546118227289728, -65086051105034316579789479088, 758320289765381459651144067648, -7502862242647817789019372638528, 63008516937463808482656194692608, -448617173455769595833933418138624, 2703212640048390870750946882682880, -13751269076145632994683610138624000, 58868414966953079922480694640640000, -211239879261162169366157386547200000, 632288694113916466698811382169600000, -1569443030321576212767530483712000000, 3207541662867661775437679820800000000, -5350838506010097270908387328000000000, 7207961277223719420234301440000000000, -7733433557377208506751385600000000000, 6489065439953990845464576000000000000, -4151465243031120893706240000000000000, 1949961396576407938662400000000000000, -632475674323703562240000000000000000, 126348400886324133888000000000000000, -11698926007992975360000000000000000).
-
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}];
Table[T[n, 7], {n, 30}] (* G. C. Greubel, Mar 26 2022 *)
-
@CachedFunction
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) )
[T(n, 7) for n in (1..30)] # G. C. Greubel, Mar 26 2022
A151638
Number of permutations of 3 indistinguishable copies of 1..n with exactly 8 adjacent element pairs in decreasing order.
Original entry on oeis.org
0, 0, 0, 243, 12750255, 40396577931, 41106807537048, 22745757394235250, 8699569720553953791, 2617057246555282014495, 668634213456480163469985, 152325974081288304581227794, 31960987230978975148286275260, 6315174416665212479526100114476
Offset: 1
- Andrew Howroyd, Table of n, a(n) for n = 1..200
- Index entries for linear recurrences with constant coefficients, 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).
-
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}];
Table[T[n, 8], {n, 30}] (* G. C. Greubel, Mar 26 2022 *)
-
a(n) = sum(j=0, 10, (-1)^j*binomial(3*n+1, 10-j)*(binomial(j+1, 3))^n); \\ Michel Marcus, Mar 27 2022
-
@CachedFunction
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) )
[T(n, 8) for n in (1..30)] # G. C. Greubel, Mar 26 2022
A151639
Number of permutations of 3 indistinguishable copies of 1..n with exactly 9 adjacent element pairs in decreasing order.
Original entry on oeis.org
0, 0, 0, 1, 1722320, 21475242671, 53885342499340, 59751188387945950, 40765121565728774056, 20317575263352346466495, 8176401371550779497913310, 2829198944755856389638829950, 877500019282729363773560673680, 251039391334595768636642931100892
Offset: 1
- Andrew Howroyd, Table of n, a(n) for n = 1..200
- Index entries for linear recurrences with constant coefficients, 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).
-
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}];
Table[T[n, 9], {n, 30}] (* G. C. Greubel, Mar 26 2022 *)
-
@CachedFunction
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) )
[T(n, 9) for n in (1..30)] # G. C. Greubel, Mar 26 2022
A289254
a(n) = 4^n - 3*n - 1.
Original entry on oeis.org
0, 9, 54, 243, 1008, 4077, 16362, 65511, 262116, 1048545, 4194270, 16777179, 67108824, 268435413, 1073741778, 4294967247, 17179869132, 68719476681, 274877906886, 1099511627715, 4398046511040, 17592186044349, 70368744177594, 281474976710583, 1125899906842548
Offset: 1
-
[4^n -(3*n+1): n in [1..30]]; // G. C. Greubel, Mar 26 2022
-
Table[4^n - 3 n - 1, {n, 20}]
LinearRecurrence[{6, -9, 4}, {0, 9, 54}, 20]
CoefficientList[Series[-((9 x)/((-1 + x)^2 (-1 + 4 x))), {x, 0, 20}], x]
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concat(0, Vec(9*x^2 / ((1 - x)^2*(1 - 4*x)) + O(x^30))) \\ Colin Barker, Jun 30 2017
-
[4^n -(3*n+1) for n in (1..30)] # G. C. Greubel, Mar 26 2022
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