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A301972 a(n) = n*(n^2 - 2*n + 4)*binomial(2*n,n)/((n + 1)*(n + 2)).

%I A301972 #7 Jan 30 2020 21:29:18
%S A301972 0,1,4,21,112,570,2772,13013,59488,266526,1175720,5123426,22108704,
%T A301972 94645460,402503220,1702300725,7165821120,30043474230,125523450360,
%U A301972 522857438070,2172127120800,9002522512620,37233403401480,153704429299746,633442159732032,2606543487445100,10710790748646352,43957192722175908
%N A301972 a(n) = n*(n^2 - 2*n + 4)*binomial(2*n,n)/((n + 1)*(n + 2)).
%C A301972 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).
%H A301972 <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%F A301972 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)).
%F A301972 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).
%F A301972 a(n) = [x^n] x*(1 - 3*x + n*x)/(1 - x)^(n+3).
%F A301972 a(n) ~ 4^n*sqrt(n)/sqrt(Pi).
%F A301972 D-finite with recurrence: -(n+2)*(961*n-3215)*a(n) +4*(2081*n^2-4414*n-4668)*a(n-1) -28*(320*n-389)*(2*n-3)*a(n-2)=0. - _R. J. Mathar_, Jan 27 2020
%e A301972 For n = 5 we have:
%e A301972 ----------------------------
%e A301972 0   1    2    3     4    [5]
%e A301972 ----------------------------
%e A301972 0,  1,   5,  12,   22,   35,  ... A000326 (pentagonal numbers)
%e A301972 0,  1,   6,  18,   40,   75,  ... A002411 (pentagonal pyramidal numbers)
%e A301972 0,  1,   7,  25,   65,  140,  ... A001296 (4-dimensional pyramidal numbers)
%e A301972 0,  1,   8,  33,   98,  238,  ... A051836 (partial sums of A001296)
%e A301972 0,  1,   9,  42,  140,  378,  ... A051923 (partial sums of A051836)
%e A301972 0,  1,  10,  52,  192, [570], ... A050494 (partial sums of A051923)
%e A301972 ----------------------------
%e A301972 therefore a(5) = 570.
%t A301972 Table[n (n^2 - 2 n + 4) Binomial[2 n, n]/((n + 1) (n + 2)), {n, 0, 27}]
%t A301972 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]
%t A301972 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]!
%t A301972 Table[SeriesCoefficient[x (1 - 3 x + n x)/(1 - x)^(n + 3), {x, 0, n}], {n, 0, 27}]
%Y A301972 Cf. A000984, A002457, A006484, A057145, A060354, A080851, A080852, A100119, A180266, A275490, A292551.
%K A301972 nonn
%O A301972 0,3
%A A301972 _Ilya Gutkovskiy_, Mar 29 2018