A151637 - OEIS

%I #12 Jun 11 2023 11:58:17

%S 0,0,0,6750,40241088,40396577931,18096792917796,5183615502649800,

%T 1129236431002624116,205937718403143468690,33309411205799991188160,

%U 4957409194925592040479126,695659299332984273417824080,93590807522941640152432361025,12213007949715545409829962783732

%N Number of permutations of 3 indistinguishable copies of 1..n with exactly 7 adjacent element pairs in decreasing order.

%H Andrew Howroyd, <a href="/A151637/b151637.txt">Table of n, a(n) for n = 1..200</a>

%H <a href="/index/Rec#order_36">Index entries for linear recurrences with constant coefficients</a>, 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).

%F a(n) = Sum_{j=0..9} (-1)^(j+1)*binomial(3*n+1, 9-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, 7], {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, 7) for n in (1..30)] # _G. C. Greubel_, Mar 26 2022

%Y Column k=7 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