This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A074334 #23 Sep 13 2024 10:55:56
%S A074334 0,1,20,234,2144,16750,117432,761460,4654848,27173718,152867000,
%T A074334 834212236,4438175040,23108423884,118111709744,594059985000,
%U A074334 2946077521920,14429322555750,69892354873080,335194270938780,1593211647720000,7511501237722020,35153884344493200
%N A074334 a(n) = Sum_{r=1..n} r^4*binomial(n,r)^2.
%D A074334 H. W. Gould, Combinatorial Identities, 1972. (See formulas 3.77, 3.78, and 3.79 on page 31.)
%H A074334 Andrew Howroyd, <a href="/A074334/b074334.txt">Table of n, a(n) for n = 0..200</a>
%H A074334 Nikita Gogin and Mika Hirvensalo, <a href="https://pca-pdmi.ru/2020/files/10/GoHi2020ExtAbstract.pdf">On the Moments of Squared Binomial Coefficients</a>, (2020).
%F A074334 For n>1 a(n) = n^2*(n^3+n^2-3*n-1)*C(n-2). Here C(n-2) = binomial(2*n-4, n-2)/(n-1) is a Catalan number.
%F A074334 From _G. C. Greubel_, Jun 23 2022: (Start)
%F A074334 a(n) = (n^2*(n^3 + n^2 - 3*n -1)/(2*(2*n-3)))*binomial(2*n-2, n-1).
%F A074334 G.f.: x*(1 + 2*x + 32*x^3 - 128*x^4 + 144*x^5)/(1-4*x)^(9/2).
%F A074334 E.g.f.: x*exp(2*x)*( (1+2*x)*(1 +6*x +4*x^2)*BesselI(0, 2*x) + 2*x*(2 + 7*x + 4*x^2)*BesselI(1, 2*x) ). (End)
%F A074334 D-finite with recurrence (n-1)*(39*n-106)*a(n) +4*(-38*n^2+n+290)*a(n-1) +4*(100*n^2-784*n+1145)*a(n-2) -64*(13*n+4)*(2*n-9)*a(n-3)=0. - _R. J. Mathar_, Sep 13 2024
%t A074334 Total/@Table[r^4 Binomial[n,r]^2,{n,0,20},{r,n}] (* _Harvey P. Dale_, Dec 04 2017 *)
%t A074334 Table[n^2*(n^3+n^2-3*n-1)*CatalanNumber[n-2] -Boole[n==1], {n,0,30}] (* _G. C. Greubel_, Jun 23 2022 *)
%o A074334 (PARI) vector(30, n, n--; sum(k=1, n, k^4*binomial(n,k)^2)) \\ _Michel Marcus_, Aug 19 2015
%o A074334 (Magma) [n le 1 select n else n^2*(n^3+n^2-3*n-1)*Catalan(n-2): n in [0..30]]; // _G. C. Greubel_, Jun 23 2022
%o A074334 (SageMath) [n^2*(n^3+n^2-3*n-1)*catalan_number(n-2) for n in (0..30)] # _G. C. Greubel_, Jun 23 2022
%Y A074334 Cf. A000108 (Catalan numbers).
%Y A074334 Cf. A000984, A002457, A037966, A037972, A329444, A329913.
%K A074334 easy,nonn
%O A074334 0,3
%A A074334 _Paul Boddington_, Mar 05 2003
%E A074334 Terms a(18) and beyond from _Andrew Howroyd_, Jan 16 2020