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A135089 Triangle T(n,k) = 5*binomial(n,k) with T(0,0) = 1, read by rows.

%I A135089 #24 Mar 27 2022 18:59:46
%S A135089 1,5,5,5,10,5,5,15,15,5,5,20,30,20,5,5,25,50,50,25,5,5,30,75,100,75,
%T A135089 30,5,5,35,105,175,175,105,35,5,5,40,140,280,350,280,140,40,5,5,45,
%U A135089 180,420,630,630,420,180,45,5,5,50,225,600,1050,1260,1050,600,225,50,5
%N A135089 Triangle T(n,k) = 5*binomial(n,k) with T(0,0) = 1, read by rows.
%C A135089 Row sums = A020714 (except for the first term).
%C A135089 Triangle T(n,k), 0 <= k <= n, read by rows given by (5, -4, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (5, -4, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Nov 24 2013
%H A135089 G. C. Greubel, <a href="/A135089/b135089.txt">Table of n, a(n) for the first 50 rows</a>
%F A135089 T(n,k) = 5*binomial(n,k), n > 0, 0 <= k <= n.
%F A135089 Equals 2*A134059(n,k) - A007318(n,k).
%F A135089 G.f.: (1+4*x+4*x*y)/(1-x-x*y). - _Philippe Deléham_, Nov 24 2013
%F A135089 Sum_{k=0..n} T(n,k) = A020714(n) - 4*[n=0]. - _G. C. Greubel_, May 03 2021
%e A135089 First few rows of the triangle:
%e A135089   1;
%e A135089   5,  5;
%e A135089   5, 10,  5;
%e A135089   5, 15, 15,   5;
%e A135089   5, 20, 30   20,  5;
%e A135089   5, 25, 50,  50, 25,  5;
%e A135089   5, 30, 75, 100, 75, 30, 5.
%t A135089 Table[5*Binomial[n,k] -4*Boole[n==0], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Sep 22 2016; May 03 2021 *)
%o A135089 (Magma) [1] cat [5*Binomial(n,k): k in [0..n], n in [1..12]]; // _G. C. Greubel_, May 03 2021
%o A135089 (Sage)
%o A135089 def A135089(n,k): return 5*binomial(n,k) - 4*bool(n==0)
%o A135089 flatten([[A135089(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 03 2021
%Y A135089 Cf. A007318, A020714, A132200, A134058, A134059.
%K A135089 nonn,tabl
%O A135089 0,2
%A A135089 _Gary W. Adamson_, Nov 18 2007