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a(n) = binomial(3*n + 2, 3).
6

%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