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 A301972 a(n) = n*(n^2 - 2*n + 4)*binomial(2*n,n)/((n + 1)*(n + 2)). 0
 0, 1, 4, 21, 112, 570, 2772, 13013, 59488, 266526, 1175720, 5123426, 22108704, 94645460, 402503220, 1702300725, 7165821120, 30043474230, 125523450360, 522857438070, 2172127120800, 9002522512620, 37233403401480, 153704429299746, 633442159732032, 2606543487445100, 10710790748646352, 43957192722175908 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS For n > 2, a(n) is the n-th term of the main diagonal of iterated partial sums array of n-gonal numbers (in other words, a(n) is the n-th (n+2)-dimensional n-gonal number, see also example). LINKS FORMULA O.g.f.: (-4 + 31*x - 66*x^2 + 28*x^3 + (4 - 7*x)*(1 - 4*x)^(3/2))/(2*x^2*(1 - 4*x)^(3/2)). E.g.f.: exp(2*x)*(4 - x + 2*x^2)*BesselI(1,2*x)/x - 2*exp(2*x)*(2 - x)*BesselI(0,2*x). a(n) = [x^n] x*(1 - 3*x + n*x)/(1 - x)^(n+3). a(n) ~ 4^n*sqrt(n)/sqrt(Pi). EXAMPLE For n = 5 we have: ---------------------------- 0   1    2    3     4    [5] ---------------------------- 0,  1,   5,  12,   22,   35,  ... A000326 (pentagonal numbers) 0,  1,   6,  18,   40,   75,  ... A002411 (pentagonal pyramidal numbers) 0,  1,   7,  25,   65,  140,  ... A001296 (4-dimensional pyramidal numbers) 0,  1,   8,  33,   98,  238,  ... A051836 (partial sums of A001296) 0,  1,   9,  42,  140,  378,  ... A051923 (partial sums of A051836) 0,  1,  10,  52,  192, [570], ... A050494 (partial sums of A051923) ---------------------------- therefore a(5) = 570. MATHEMATICA Table[n (n^2 - 2 n + 4) Binomial[2 n, n]/((n + 1) (n + 2)), {n, 0, 27}] nmax = 27; CoefficientList[Series[(-4 + 31 x - 66 x^2 + 28 x^3 + (4 - 7 x) (1 - 4 x)^(3/2))/(2 x^2 (1 - 4 x)^(3/2)), {x, 0, nmax}], x] nmax = 27; CoefficientList[Series[Exp[2 x] (4 - x + 2 x^2) BesselI[1, 2 x]/x - 2 Exp[2 x] (2 - x) BesselI[0, 2 x], {x, 0, nmax}], x] Range[0, nmax]! Table[SeriesCoefficient[x (1 - 3 x + n x)/(1 - x)^(n + 3), {x, 0, n}], {n, 0, 27}] CROSSREFS Cf. A000984, A002457, A006484, A057145, A060354, A080851, A080852, A100119, A180266, A275490, A292551. Sequence in context: A117381 A010908 A136786 * A026335 A027909 A127111 Adjacent sequences:  A301969 A301970 A301971 * A301973 A301974 A301975 KEYWORD nonn AUTHOR Ilya Gutkovskiy, Mar 29 2018 STATUS approved

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Last modified March 22 10:07 EDT 2019. Contains 321421 sequences. (Running on oeis4.)