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 A179095 Rectified 5-cell numbers: the coefficient of x^{2n-2} in (1+x+x^2+ ... + x^{n-1})^5. 5
 0, 1, 10, 45, 135, 320, 651, 1190, 2010, 3195, 4840, 7051, 9945, 13650, 18305, 24060, 31076, 39525, 49590, 61465, 75355, 91476, 110055, 131330, 155550, 182975, 213876, 248535, 287245, 330310, 378045, 430776, 488840, 552585, 622370, 698565, 781551 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS J. Conrad, Table of n, a(n) for n = 0..260 FORMULA Conjectures: a(n) = n*(11*n^3+6*n^2+n+6)/24. G.f.: x*(1+5*x+5*x^2)/(1-x)^5. - Colin Barker, Jan 09 2012 Comment from Doron Zeilberger, Feb 18 2016 (Start): The conjectures in A179095-A170099 are true. Proof: The geometric series 1+x+x^2+..+x^(n-1) = (1-x^n)/(1-x). Hence for a fixed k (in the above cases k=5..9, but the argument holds in general) the coefficient of  x^(2*n-2) in (1+x+...+x^(n-1))^k = coefficient of  x^(2*n-2) in (1-x^n)^k*(1-x)^(-k) = coefficient of  x^(2*n-2) in (1-k*x^n + Sum of powers higher than x^(2*n-2)..) = coefficient of x^(2*n-2) in (1-x)^(-k) -k*(the coefficient of x^(n-2) in (1-x)^(-k)) = (-1)^(2*n-2)*binomial(-k,2*n-2)- k* (-1)^(n-2)*binomial(-k,n-2)= Using  (-1)^m *binomial(-m,k)= binomial(m+k-1,k-1) this is binomial(k+2*n-3,k-1) - k *binomial(k+n-3,k-1) and this agrees with the conjectures for k=5..9 (End) MATHEMATICA f[n_] := CoefficientList[ Series[ Sum[x^k, {k, 0, n - 1}]^5, {x, 0, 2 n + 3}], x][[2 n - 1]]; Array[f, 36] (* Robert G. Wilson v, Jul 30 2010 *) PROG (PARI) a(n) = polcoeff(((x^n-1)/(x-1))^5, 2*n-2); \\ Michel Marcus, Feb 17 2016 (PARI) A179095(n)=n*(11*n^3+6*n^2+n+6)\24 \\ M. F. Hasler, Feb 19 2016 CROSSREFS Cf. A179096, A179097, A179098, A179099. Sequence in context: A226254 A022605 A211032 * A213188 A037270 A027800 Adjacent sequences:  A179092 A179093 A179094 * A179096 A179097 A179098 KEYWORD nonn AUTHOR Michael A. Jackson, Jun 29 2010 EXTENSIONS More terms from Robert G. Wilson v, Jul 30 2010 STATUS approved

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Last modified December 13 22:07 EST 2018. Contains 318087 sequences. (Running on oeis4.)