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A331987 a(n) = ((n + 1) - 9*(n + 1)^2 + 8*(n + 1)^3)/6.

%I A331987 #17 Apr 11 2024 16:04:35
%S A331987 0,5,23,62,130,235,385,588,852,1185,1595,2090,2678,3367,4165,5080,
%T A331987 6120,7293,8607,10070,11690,13475,15433,17572,19900,22425,25155,28098,
%U A331987 31262,34655,38285,42160,46288,50677,55335,60270,65490,71003,76817,82940,89380,96145
%N A331987 a(n) = ((n + 1) - 9*(n + 1)^2 + 8*(n + 1)^3)/6.
%C A331987 The start values of the partial rows on the main diagonal of A332662 in the representation in the example section.
%C A331987 Apparently the sum of the hook lengths over the partitions of 2*n + 1 with exactly 2 parts (cf. A180681).
%H A331987 G. C. Greubel, <a href="/A331987/b331987.txt">Table of n, a(n) for n = 0..1000</a>
%H A331987 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A331987 a(n) = [x^n] (x*(5 + 3*x)/(1 - x)^4).
%F A331987 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
%F A331987 a(n) = binomial(n+2, 3) + binomial(n+1, 3) + 2*(n+1)*binomial(n+1, 2).
%F A331987 From _G. C. Greubel_, Apr 19 2023: (Start)
%F A331987 a(n) = 3*binomial(n+1,1) - 11*binomial(n+2,2) + 8*binomial(n+3,3).
%F A331987 a(n) = n*binomial(8*n+8,2)/24.
%F A331987 a(n) = n*(n+1)*(8*n+7)/6.
%F A331987 E.g.f.: (1/6)*x*(30 + 39*x + 8*x^2)*exp(x). (End)
%p A331987 a := n -> ((n+1) - 9*(n+1)^2 + 8*(n+1)^3)/6: seq(a(n), n=0..41);
%p A331987 gf := (x*(3*x + 5))/(x - 1)^4: ser := series(gf, x, 44):
%p A331987 seq(coeff(ser, x, n), n=0..41);
%t A331987 LinearRecurrence[{4,-6,4,-1}, {0,5,23,62}, 42]
%t A331987 Table[(n-9n^2+8n^3)/6,{n,50}] (* _Harvey P. Dale_, Apr 11 2024 *)
%o A331987 (Magma) [n*(n+1)*(8*n+7)/6: n in [0..50]]; // _G. C. Greubel_, Apr 19 2023
%o A331987 (SageMath)
%o A331987 def A331987(n): return n*(n+1)*(8*n+7)/6
%o A331987 [A331987(n) for n in range(51)] # _G. C. Greubel_, Apr 19 2023
%Y A331987 Apparently a bisection of A049779 and of A024862.
%Y A331987 Cf. A180681, A332662.
%K A331987 nonn
%O A331987 0,2
%A A331987 _Peter Luschny_, Feb 19 2020