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A212393
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Expansion of (1-4*x+7*x^2-5*x^3+4*x^4-6*x^5+21*x^6+18*x^7-5*x^8)/(1-x)^5.
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3
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1, 1, 2, 5, 14, 30, 72, 195, 485, 1059, 2065, 3682, 6120, 9620, 14454, 20925, 29367, 40145, 53655, 70324, 90610, 115002, 144020, 178215, 218169, 264495, 317837, 378870, 448300, 526864, 615330, 714497, 825195, 948285, 1084659, 1235240, 1400982, 1582870, 1781920
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OFFSET
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0,3
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COMMENTS
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In the paper of Kitaev, Remmel and Tiefenbruck (see the Links section), Q_(132)^(0,k,0,0)(x,0) represents a generating function depending on k and x.
For successive values of k we have:
k=2, the g.f. of A028310: (1-x+x^2)/(1-x)^2;
k=3, the g.f. (1-2*x+2*x^2+x^3-x^4)/(1-x)^3, whose coefficients (except the first two) are given by A000096 (for n>0);
k=4, the g.f. (1-3*x+4*x^2-x^3+3*x^4-5*x^5+2*x^6)/(1-x)^4, whose coefficients (except the first three) are given by A005586 (for n>0).
This sequence corresponds to the case k=5.
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LINKS
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FORMULA
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G.f.: (1-4*x+7*x^2-5*x^3+4*x^4-6*x^5+21*x^6+18*x^7-5*x^8)/(1-x)^5.
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) for n>8, a(0)=a(1)=1, a(2)=2, a(3)=5, a(4)=14, a(5)=30, a(6)=72, a(7)=195, a(8)=485.
a(n) = (n-3)*(31*n^3-369*n^2+1454*n-1560)/24 for n>3, a(0)=a(1)=1, a(2)=2, a(3)=5.
G.f.: 1+x+2*x^2+5*x^3 + 14*x^4*G(0), where G(k)= 1 + x*(k+1)*(124*k^3+192*k^2+89*k+180)/( (2*k+1)*(62*k^3+3*k^2-5*k+84) - x*(2*k+1)*(62*k^3+3*k^2-5*k+84)*(2*k+3)*(62*k^3+189*k^2+187*k+144)/(x*(2*k+3)*(62*k^3+189*k^2+187*k+144) + (k+1)*(124*k^3+192*k^2+89*k+180)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 06 2013
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MATHEMATICA
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CoefficientList[Series[(1 - 4 x + 7 x^2 - 5 x^3 + 4 x^4 - 6 x^5 + 21 x^6 + 18 x^7 - 5 x^8)/(1 - x)^5, {x, 0, 38}], x]
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PROG
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(PARI) Vec((1-4*x+7*x^2-5*x^3+4*x^4-6*x^5+21*x^6+18*x^7-5*x^8)/(1-x)^5+O(x^39))
(Magma) m:=39; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-4*x+7*x^2-5*x^3+4*x^4-6*x^5+21*x^6+18*x^7-5*x^8)/(1-x)^5));
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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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STATUS
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approved
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