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%I #29 Sep 08 2022 08:46:09
%S 1,7,28,84,195,381,662,1058,1589,2275,3136,4192,5463,6969,8730,10766,
%T 13097,15743,18724,22060,25771,29877,34398,39354,44765,50651,57032,
%U 63928,71359,79345,87906,97062,106833,117239,128300,140036,152467,165613,179494
%N a(n) = Sum_{k=0..3} binomial(6,k)*binomial(n,k).
%H Vincenzo Librandi, <a href="/A247608/b247608.txt">Table of n, a(n) for n = 0..1000</a>
%H C. Krattenthaler, <a href="http://www.mat.univie.ac.at/~slc/wpapers/s42kratt.html">Advanced determinant calculus</a> Séminaire Lotharingien de Combinatoire, B42q (1999), 67 pp, (see p. 54).
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F G.f.: (1+3*x+6*x^2+10*x^3)/(1-x)^4.
%F a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4).
%F a(n) = (6+31*n-15*n^2+20*n^3)/6.
%F a(n) = 1+6*Binomial(n,1)+15*Binomial(n,2)+20*Binomial(n,3).
%t Table[(6 + 31 n - 15 n^2 + 20 n^3)/6, {n, 0, 50}] (* or *) CoefficientList[Series[(1 + 3 x + 6 x^2 + 10 x^3)/(1-x)^4,{x, 0, 50}], x]
%o (Magma) [(6+31*n-15*n^2+20*n^3)/6: n in [0..40]]; /* or */ [1+6*Binomial(n,1)+15*Binomial(n,2)+20*Binomial(n,3): n in [0..40]]; /* or */ I:=[1, 7, 28, 84]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]
%o (PARI) Vec((1+3*x+6*x^2+10*x^3)/(1-x)^4 + O (x^50)) \\ _Michel Marcus_, Sep 22 2014
%o (Sage) m=3; [sum((binomial(2*m,k)*binomial(n,k)) for k in (0..m)) for n in (0..40)] # _Bruno Berselli_, Sep 22 2014
%Y Cf. A005408, A056108.
%K nonn,easy
%O 0,2
%A _Vincenzo Librandi_, Sep 22 2014
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