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A302645
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Values of unimodal polynomial analogous to A302612 and A302644 arising from a partition size <= 4 restriction.
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5
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0, 1, 2, 6, 20, 70, 252, 924, 3432, 12705, 45430, 152438, 472836, 1352078, 3578680, 8827080, 20439984, 44745513, 93185994, 185640070, 355452020, 656846190, 1175604980, 2044130980, 3462303000, 5725877625, 9264588606, 14692562262, 22874204836, 35009334470
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OFFSET
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0,3
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COMMENTS
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Consider the unimodal polynomial from O'Hara's proof of unimodality of q-binomials after making the restriction to partitions of size <=4. See G_4(n,k) from arXiv:1711.11252. If we make the simplification k=n and take the limit as q->1^-, we obtain the listed polynomial.
As the size restriction s increases, G_s->G_infinity=G: the q-binomials. Then substituting k=n and q=1 yields the central binomial coefficients: A000984.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).
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FORMULA
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a(n) = n*(n+2)*(n+1)*(n^8-32*n^7+462*n^6-3836*n^5+20013*n^4-66836*n^3+140804*n^2-171216*n+100800)/120960.
G.f.: x*(1 - 10*x + 48*x^2 - 140*x^3 + 281*x^4 - 390*x^5 + 430*x^6 - 220*x^7 + 330*x^8) / (1 - x)^12.
a(n) = 12*a(n-1) - 66*a(n-2) + 220*a(n-3) - 495*a(n-4) + 792*a(n-5) - 924*a(n-6) + 792*a(n-7) - 495*a(n-8) + 220*a(n-9) - 66*a(n-10) + 12*a(n-11) - a(n-12) for n>11.
(End)
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EXAMPLE
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For n=6, G_4(6,6)=q^36+q^35+2*q^34+3*q^33+5*q^32+7*q^31+11*q^30+13*q^29+18*q^28+22*q^27+28*q^26+32*q^25+39*q^24+42*q^23+48*q^22+51*q^21+55*q^20+55*q^19+58*q^18+55*q^17+55*q^16+51*q^15+48*q^14+42*q^13+39*q^12+32*q^11+28*q^10+22*q^9+18*q^8+13*q^7+11*q^6+7*q^5+5*q^4+3*q^3+2*q^2+q+1 (using the formula in the referenced paper). Then substituting q=1 yields 924.
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PROG
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(PARI) concat(0, Vec(x*(1 - 10*x + 48*x^2 - 140*x^3 + 281*x^4 - 390*x^5 + 430*x^6 - 220*x^7 + 330*x^8) / (1 - x)^12 + O(x^40))) \\ Colin Barker, Apr 19 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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0 prepended to the sequence and formulas adjusted accordingly by Colin Barker, Apr 19 2018
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STATUS
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approved
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