A151636 - OEIS

%I #15 Jun 11 2023 11:57:13

%S 0,0,1,49682,58571184,21475242671,4476844162434,678770257169016,

%T 84698452637705746,9324662905839457490,944619860914428706035,

%U 90435965482528402360106,8327298182652856026223632,746238093776109096993716949,65611401726068220422014371676

%N Number of permutations of 3 indistinguishable copies of 1..n with exactly 6 adjacent element pairs in decreasing order.

%H Andrew Howroyd, <a href="/A151636/b151636.txt">Table of n, a(n) for n = 1..200</a>

%H <a href="/index/Rec#order_28">Index entries for linear recurrences with constant coefficients</a>, 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).

%F a(n) = Sum_{j=0..8} (-1)^j*binomial(3*n+1, 8-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, 6], {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, 6) for n in (1..30)] # _G. C. Greubel_, Mar 26 2022

%o (PARI) a(n) = sum(j=0, 8, (-1)^j*binomial(3*n+1, 8-j)*(binomial(j+1, 3))^n); \\ _Michel Marcus_, Mar 27 2022

%Y Column k=6 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