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A151627
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Number of permutations of 2 indistinguishable copies of 1..n with exactly 5 adjacent element pairs in decreasing order.
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2
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0, 0, 0, 72, 27664, 2864328, 163809288, 6727188848, 225167210712, 6590156148912, 175992170793456, 4407169187423736, 105396936343707456, 2437638848729751736, 55010494951127561400, 1219075824289276443744, 26652917330108137129544, 576864003740129587504224
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OFFSET
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1,4
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (126, -7245, 252750, -5998905, 102950910, -1325335665, 13104159030, -101064788055, 614053303570, -2956203883287, 11303748373482, -34308033838515, 82354980144330, -155288585585115, 227560437228978, -255076375732668, 213622511296680, -128922602864400, 52813189404000, -13115963880000, 1488034800000).
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FORMULA
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a(n) = Sum_{j=0..5} (-1)^j*binomial(2*n+1, j)*binomial(7-j, 2)^n.
G.f.: 8*x^4*(9 + 2324*x - 12462*x^2 - 1858545*x^3 + 34890010*x^4 - 134744022*x^5 - 1875623070*x^6 + 22965673068*x^7 - 95590873845*x^8 + 93562460910*x^9 + 576877450068*x^10 - 2203266593259*x^11 + 2865061552194*x^12 - 347005909980*x^13 - 2472141497400*x^14 + 1471264884000*x^15 + 318864600000*x^16)/((1-x)^6*(1-3*x)^5*(1-6*x)^4*(1-10*x)^3*(1-15*x)^2*(1-21*x)).
E.g.f.: exp(21*x) - (1 + 30*x)*exp(15*x) + 10*x*(3 + 20*x)*exp(10*x) - 6*x*(1 + 24*x + 48*x^2)*exp(6*x) + (9*x^2/2)*(5 + 20*x + 12*x^2)*exp(3*x) - (x^2/30)*(15 + 90*x + 60*x^2 + 8*x^3)*exp(x). (End)
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MATHEMATICA
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With[{B = Binomial}, Table[Sum[(-1)^j*B[2n+1, j]*B[7-j, 2]^n, {j, 0, 5}], {n, 30}]] (* G. C. Greubel, Sep 07 2022 *)
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PROG
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(Magma) [(&+[(-1)^j*Binomial(2*n+1, j)*Binomial(7-j, 2)^n: j in [0..5]]): n in [1..30]]; // G. C. Greubel, Sep 07 2022
(SageMath)
@CachedFunction
def A151627(n): return sum((-1)^j*binomial(2*n+1, j)*binomial(7-j, 2)^n for j in (0..5))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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