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%I #15 Jun 11 2023 11:47:23
%S 0,0,0,0,232,450048,163809288,27306899520,2898916824320,
%T 230479103253264,15045786224718576,853790829031070016,
%U 43726349865720132216,2073954076439134340896,92786533933117718314680,3968339124661862533557120,163867469809854783921566544
%N Number of permutations of 2 indistinguishable copies of 1..n with exactly 7 adjacent element pairs in decreasing order.
%H Andrew Howroyd, <a href="/A151629/b151629.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 (330, -51513, 5066402, -352805739, 18532822542, -764011192951, 25389927552654, -693360910567062, 15782229920923084, -302672805175992858, 4931504254236896916, -68703084605396486102, 822462579151947202524, -8492510968865324313198, 75847875003485374953052, -587051007000593357006397, 3942243867122711876178882, -22980348438736769272554525, 116263149690925646738764650, -510093882106104569146940943, 1937990696541806422436512950, -6362481040931909732744369259, 17998658175922136342871558966, -43712735312807208911181978972, 90728113301609002223916131208, -160019275941921567466090870848, 238141442344924194341467169088, -296416786921079653107637015680, 305161844579852353104555820800, -256124655172887234493061088000, 171916771042339597743180480000, -89854952889081931534972800000, 35162508668121498752928000000, -9674457989784467806080000000, 1666424129434910092800000000, -135019896025206528000000000).
%F From _G. C. Greubel_, Sep 07 2022: (Start)
%F a(n) = Sum_{j=0..7} (-1)^j*binomial(2*n+1, j)*binomial(9-j, 2)^n.
%F G.f.: 8*x^5*(29 +46686*x +3405558*x^2 -592781020*x^3 +15959334952*x^4 +631633031922*x^5 -49837254287872*x^6 +1300883760100354*x^7 -12994364551718898*x^8 -140176079949572802*x^9 +6513756576348329884*x^10 -101042319163019645166*x^11 +848633388017107293828*x^12 -2913665757033808948194*x^13 -19357175742148303993152*x^14 +332871592406004436180230*x^15 -2265050438781150240585891*x^16 +8844782645551069762176780*x^17 -16577175062101039893470178*x^18 -216035122652452146094327988*x^19 +244246494424905520901547660*x^20 -780226424729404888409973432*x^21 +1345511462530423731597208080*x^22 -1027054667766768116706056160*x^23 -747115159033132605830894400*x^24 +2731966566484322974432464000*x^25 -2760478881311463186555360000*x^26 +892027667079782450985600000*x^27 +450814927116061418400000000*x^28 -303214961231096241600000000*x^29 -30004421338934784000000000*x^30)/( Product_{j=1..8} (1-binomial(j+1,2)*x)^(9-j) ).
%F E.g.f.: exp(36*x) - (1 + 56*x)*exp(28*x) + 63*x*(1 + 14*x)*exp(21*x) - 15*x*(1 + 60*x + 300*x^2)*exp(15*x) + (250/3)*x^2*(3 + 40*x + 80*x^2)*exp(10*x) - (18/5)*x^2*(5 + 180*x + 720*x^2 + 576*x^3)*exp(6*x) + (9/10)*x^3*(35 + 210*x + 252*x^2 + 72*x^3)*exp(3*x) - (1/630)*x^3*(105 + 840*x + 840*x^2 + 224*x^3 + 16*x^4)*exp(x). (End)
%t With[{B=Binomial}, Table[Sum[(-1)^j*B[2n+1, j]*B[9-j, 2]^n, {j,0,7}], {n, 30}]] (* _G. C. Greubel_, Sep 08 2022 *)
%o (Magma) [(&+[(-1)^j*Binomial(2*n+1, j)*Binomial(9-j, 2)^n: j in [0..7]]): n in [1..30]]; // _G. C. Greubel_, Sep 08 2022
%o (SageMath)
%o def A151629(n): return sum((-1)^j*binomial(2*n+1, j)*binomial(9-j, 2)^n for j in (0..7))
%o [A151629(n) for n in (1..30)] # _G. C. Greubel_, Sep 08 2022
%Y Column k=7 of A154283.
%K nonn
%O 1,5
%A _R. H. Hardin_, May 29 2009
%E Terms a(11) and beyond from _Andrew Howroyd_, May 06 2020
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