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%I #19 Dec 21 2023 11:27:01
%S 10,56,165,364,680,1140,1771,2600,3654,4960,6545,8436,10660,13244,
%T 16215,19600,23426,27720,32509,37820,43680,50116,57155,64824,73150,
%U 82160,91881,102340,113564,125580,138415,152096,166650,182104,198485,215820,234136,253460
%N a(n) = binomial(3*n + 2, 3).
%H Vincenzo Librandi, <a href="/A228888/b228888.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4, -6, 4, -1).
%F a(n) = binomial(3*n + 2, 3) = 1/6*(3*n)*(3*n + 1)*(3*n + 2).
%F a(-n) = - A006566(n).
%F a(n) = 1/6*A228889(n).
%F G.f.: (10*x + 16*x^2 + x^3)/(1 - x)^4 = 10*x + 56*x^2 + 165*x^3 + ....
%F Sum {n >= 1} 1/a(n) = 9/2 - 3/2*log(3) - 1/2*sqrt(3)*Pi.
%F Sum {n >= 1} (-1)^n/a(n) = 9/2 - 4*log(2) - 1/3*sqrt(3)*Pi.
%e From _Bruno Berselli_, Jun 26 2018: (Start)
%e Including 0, row sums of the triangle:
%e | 0| .................................................................. 0
%e | 1| 2 3 4 ..................................................... 10
%e | 5| 6 7 8 9 10 11 ......................................... 56
%e |12| 13 14 15 16 17 18 19 20 21 ............................ 165
%e |22| 23 24 25 26 27 28 29 30 31 32 33 34 ................ 364
%e |35| 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 .... 680
%e ...
%e in the first column of which we have the pentagonal numbers (A000326).
%e (End)
%p seq(binomial(3*n+2,3), n = 1..38);
%t Table[(Binomial[3 n + 2, 3]), {n, 1, 40}] (* _Vincenzo Librandi_, Sep 09 2013 *)
%o (Magma) [Binomial(3*n + 2, 3): n in [1..40]]; // _Vincenzo Librandi_, Sep 09 2013
%Y Cf. A006566 (binomial(3*n,3)) and A228887 (binomial(3*n + 1,3)).
%Y Cf. A228889.
%Y Similar sequences are listed in A316224.
%K nonn,easy
%O 1,1
%A _Peter Bala_, Sep 09 2013
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