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A096957 Fourth column (m=3) of (1,6)-Pascal triangle A096956.

%I A096957 #23 May 02 2025 13:44:54
%S A096957 6,19,40,70,110,161,224,300,390,495,616,754,910,1085,1280,1496,1734,
%T A096957 1995,2280,2590,2926,3289,3680,4100,4550,5031,5544,6090,6670,7285,
%U A096957 7936,8624,9350,10115,10920,11766,12654,13585,14560,15580,16646,17759,18920
%N A096957 Fourth column (m=3) of (1,6)-Pascal triangle A096956.
%C A096957 If Y is a 6-subset of an n-set X then, for n>=8, a(n-8) is the number of 3-subsets of X having at most one element in common with Y. - _Milan Janjic_, Dec 16 2007
%H A096957 Vincenzo Librandi, <a href="/A096957/b096957.txt">Table of n, a(n) for n = 0..3000</a>
%H A096957 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A096957 a(n) = A096956(n+3, 3) = 6*b(n) - 5*b(n-1) = (n+18)*binomial(n+2, 2)/3, with b(n) = A000292(n) = binomial(n+3, 3).
%F A096957 G.f.: (6-5*x)/(1-x)^4.
%F A096957 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. - _Vincenzo Librandi_, Apr 19 2017
%F A096957 E.g.f.: exp(x)*(36 + 78*x + 24*x^2 + x^3)/6. - _Stefano Spezia_, May 02 2025
%t A096957 CoefficientList[Series[(6 - 5*x)/(1 - x)^4, {x, 0, 40}], x] (* _Wesley Ivan Hurt_, Apr 18 2017 *)
%t A096957 LinearRecurrence[{4, -6, 4, -1}, {6, 19, 40, 70}, 50] (* _Vincenzo Librandi_, Apr 19 2017 *)
%o A096957 (Magma) I:=[6,19,40,70]; [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..50]]; // _Vincenzo Librandi_, Apr 19 2017
%Y A096957 Cf. A000292, A096956.
%Y A096957 Cf. other columns: A096958 (m = 4), A096959 (m = 5), A097297 (m = 6), A097298 (m = 7), A097299 (m = 8), A097300 (m = 9).
%K A096957 nonn,easy
%O A096957 0,1
%A A096957 _Wolfdieter Lang_, Aug 13 2004