A151629 Number of permutations of 2 indistinguishable copies of 1..n with exactly 7 adjacent element pairs in decreasing order.

0, 0, 0, 0, 232, 450048, 163809288, 27306899520, 2898916824320, 230479103253264, 15045786224718576, 853790829031070016, 43726349865720132216, 2073954076439134340896, 92786533933117718314680, 3968339124661862533557120, 163867469809854783921566544

Offset

1,5

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Index entries for linear recurrences with constant coefficients, 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).

Formula

From G. C. Greubel, Sep 07 2022: (Start)

a(n) = Sum_{j=0..7} (-1)^j*binomial(2*n+1, j)*binomial(9-j, 2)^n.

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) ).

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)

Mathematica

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 *)

Prog

(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
(SageMath)
def A151629(n): return sum((-1)^j*binomial(2*n+1, j)*binomial(9-j, 2)^n for j in (0..7))
[A151629(n) for n in (1..30)] # G. C. Greubel, Sep 08 2022

Crossrefs

Column k=7 of A154283.

Sequence in context: A125850 A139673 A252003 * A301828 A132917 A139652

Adjacent sequences: A151626 A151627 A151628 * A151630 A151631 A151632

Keyword

nonn

Author

R. H. Hardin, May 29 2009

Extensions

Terms a(11) and beyond from Andrew Howroyd, May 06 2020

Status

approved