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a(n) = Sum_{r=1..n} r^4*binomial(n,r)^2.
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%I #23 Sep 13 2024 10:55:56

%S 0,1,20,234,2144,16750,117432,761460,4654848,27173718,152867000,

%T 834212236,4438175040,23108423884,118111709744,594059985000,

%U 2946077521920,14429322555750,69892354873080,335194270938780,1593211647720000,7511501237722020,35153884344493200

%N a(n) = Sum_{r=1..n} r^4*binomial(n,r)^2.

%D H. W. Gould, Combinatorial Identities, 1972. (See formulas 3.77, 3.78, and 3.79 on page 31.)

%H Andrew Howroyd, <a href="/A074334/b074334.txt">Table of n, a(n) for n = 0..200</a>

%H 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 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 From _G. C. Greubel_, Jun 23 2022: (Start)

%F a(n) = (n^2*(n^3 + n^2 - 3*n -1)/(2*(2*n-3)))*binomial(2*n-2, n-1).

%F G.f.: x*(1 + 2*x + 32*x^3 - 128*x^4 + 144*x^5)/(1-4*x)^(9/2).

%F 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 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 Total/@Table[r^4 Binomial[n,r]^2,{n,0,20},{r,n}] (* _Harvey P. Dale_, Dec 04 2017 *)

%t 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 (PARI) vector(30, n, n--; sum(k=1, n, k^4*binomial(n,k)^2)) \\ _Michel Marcus_, Aug 19 2015

%o (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 (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 Cf. A000108 (Catalan numbers).

%Y Cf. A000984, A002457, A037966, A037972, A329444, A329913.

%K easy,nonn

%O 0,3

%A _Paul Boddington_, Mar 05 2003

%E Terms a(18) and beyond from _Andrew Howroyd_, Jan 16 2020