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A361474 a(n) = 1*binomial(n,2) + 3*binomial(n,3) + 6*binomial(n,4) + 10*binomial(n,5).

%I A361474 #24 Apr 16 2023 13:40:41
%S A361474 0,0,1,6,24,80,225,546,1176,2304,4185,7150,11616,18096,27209,39690,
%T A361474 56400,78336,106641,142614,187720,243600,312081,395186,495144,614400,
%U A361474 755625,921726,1115856,1341424,1602105,1901850,2244896,2635776,3079329,3580710,4145400,4779216,5488321,6279234
%N A361474 a(n) = 1*binomial(n,2) + 3*binomial(n,3) + 6*binomial(n,4) + 10*binomial(n,5).
%C A361474 a(n) is the number of ordered set partitions of an n-set into 2 sets such that the first set has either 3, 2, 1 or no elements, the second set has no restrictions, and two elements are selected from the second set.
%C A361474 Note the coefficients 1,3,6,10 in a(n) are triangular numbers (in accordance with the selection of two elements from the second set).
%H A361474 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A361474 E.g.f.: (1 + x + x^2/2 + x^3/6)*x^2/2*exp(x).
%F A361474 O.g.f.: x^2*(1 + 3*x^2 + 6*x^3)/(1 - x)^6. - _Stefano Spezia_, Mar 16 2023
%e A361474 The 546 set partitions for n=7 are the following (where the 2 elements selected from the second set are in parentheses):
%e A361474    { }, {(1),(2),3,4,5,6,7}  (21 of these);
%e A361474    {1}, {(2),(3),4,5,6,7}    (105 of these);
%e A361474    {1,2}, {(3),(4),5,6,7}    (210 of these);
%e A361474    {1,2,3}, {(4),(5),6,7}    (210 of these).
%t A361474 a[n_] := Total[Binomial[n, Range[2, 5]]*{1, 3, 6, 10}]; Array[a, 40, 0] (* _Amiram Eldar_, Mar 28 2023 *)
%o A361474 (PARI) a(n) = binomial(n,2) + 3*binomial(n,3) + 6*binomial(n,4) + 10*binomial(n,5); \\ _Michel Marcus_, Mar 16 2023
%o A361474 (Python)
%o A361474 def A361474(n): return n*(n*(n*(n*(n - 7) + 23) - 29) + 12)//12 # _Chai Wah Wu_, Apr 16 2023
%Y A361474 Cf. A006000, A361099.
%K A361474 nonn,easy
%O A361474 0,4
%A A361474 _Enrique Navarrete_, Mar 13 2023