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A105253 a(n) = binomial(n+6,n)*binomial(n+10,n).

%I A105253 #31 Mar 04 2025 07:36:18
%S A105253 1,77,1848,24024,210210,1387386,7399392,33372768,131405274,462351890,
%T A105253 1479526048,4365213216,12004336344,31040798712,76018282560,
%U A105253 177375992640,396324483555,851617661895,1766318113560,3547314771000,6917263803450,13128684361650,24304341297600
%N A105253 a(n) = binomial(n+6,n)*binomial(n+10,n).
%H A105253 T. D. Noe, <a href="/A105253/b105253.txt">Table of n, a(n) for n = 0..1000</a>
%H A105253 <a href="/index/Rec#order_17">Index entries for linear recurrences with constant coefficients</a>, signature (17,-136,680,-2380,6188,-12376,19448,-24310,24310,-19448,12376,-6188,2380,-680,136,-17,1).
%F A105253 G.f.: (1 + 60*x + 675*x^2 + 2400*x^3 + 3150*x^4 + 1513*x^5 + 210*x^6)/(1-x)^17. - _Colin Barker_, Jan 21 2013
%F A105253 From _Amiram Eldar_, Sep 01 2022: (Start)
%F A105253 Sum_{n>=0} 1/a(n) = 20020*Pi^2 - 1493768807/7560.
%F A105253 Sum_{n>=0} (-1)^n/a(n) = 131072*log(2)/21 - 100*Pi^2 - 88332653/26460. (End)
%e A105253 a(0): C(0+6,0)*C(0+10,0) = C(6,0)*C(10,0) = 1*1 = 1;
%e A105253 a(10): C(10+6,10)*C(10+10,10) = C(16,10)*(20,10) = 8008*184756 = 1479526048.
%t A105253 f[n_] := Binomial[n + 6, n]Binomial[n + 10, n]; Table[ f[n], {n, 0, 20}] (* _Robert G. Wilson v_, Apr 20 2005 *)
%o A105253 (Magma) [Binomial(n+6,n)*Binomial(n+10,n): n in [0..30]]; // _Vincenzo Librandi_, Jul 31 2015
%o A105253 (Python)
%o A105253 A105253_list, m = [], [8008, -22022, 23023, -11297, 2563, -209] + [1]*11
%o A105253 for _ in range(10**2):
%o A105253     A105253_list.append(m[-1])
%o A105253     for i in range(16):
%o A105253         m[i+1] += m[i] # _Chai Wah Wu_, Jan 24 2016
%o A105253 (SageMath)
%o A105253 def A105253(n): return binomial(n+6,n)*binomial(n+10,n)
%o A105253 print([A105253(n) for n in range(31)]) # _G. C. Greubel_, Mar 04 2025
%Y A105253 Cf. A062264.
%K A105253 easy,nonn
%O A105253 0,2
%A A105253 _Zerinvary Lajos_, Apr 14 2005
%E A105253 More terms from _Robert G. Wilson v_, Apr 20 2005