cp's OEIS Frontend

A259525 First differences of A007318, when Pascal's triangle is seen as flattened list.

%I A259525 #20 Apr 25 2024 13:45:11
%S A259525 0,0,0,1,-1,0,2,0,-2,0,3,2,-2,-3,0,4,5,0,-5,-4,0,5,9,5,-5,-9,-5,0,6,
%T A259525 14,14,0,-14,-14,-6,0,7,20,28,14,-14,-28,-20,-7,0,8,27,48,42,0,-42,
%U A259525 -48,-27,-8,0,9,35,75,90,42,-42,-90,-75,-35,-9,0,10,44,110
%N A259525 First differences of A007318, when Pascal's triangle is seen as flattened list.
%C A259525 A214292 gives first differences per row in Pascal's triangle.
%H A259525 Reinhard Zumkeller, <a href="/A259525/b259525.txt">Table of n, a(n) for n = 0..10000</a>
%H A259525 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A259525 From _G. C. Greubel_, Apr 25 2024: (Start)
%F A259525 If viewed as a triangle then:
%F A259525 T(n, k) = binomial(n, k+1) - binomial(n, k), with T(n, n) = 0.
%F A259525 T(n, n-k) = - T(n, k), for 0 <= k < n.
%F A259525 T(2*n, n) = [n=0] - A000108(n).
%F A259525 Sum_{k=0..n} T(n, k) = 0 (row sums).
%F A259525 Sum_{k=0..floor(n/2)} T(n, k) = A047171(n).
%F A259525 Sum_{k=0..n} (-1)^k*T(n, k) = A021499(n).
%F A259525 Sum_{k=0..floor(n/2)} T(n-k, k) = A074331(n-1). (End)
%t A259525 Table[If[k==n, 0, ((n-2*k-1)/(n-k))*Binomial[n,k+1]], {n,0,12}, {k,0, n}]//Flatten (* _G. C. Greubel_, Apr 25 2024 *)
%o A259525 (Haskell)
%o A259525 a259525 n = a259525_list !! n
%o A259525 a259525_list = zipWith (-) (tail pascal) pascal
%o A259525                            where pascal = concat a007318_tabl
%o A259525 (Magma)
%o A259525 [k eq n select 0 else (n-2*k-1)*Binomial(n,k+1)/(n-k): k in [0..n], n in [0..14]]; // _G. C. Greubel_, Apr 25 2024
%o A259525 (SageMath)
%o A259525 flatten([[binomial(n,k+1) -binomial(n,k) +int(k==n) for k in range(n+1)] for n in range(15)]) # _G. C. Greubel_, Apr 25 2024
%Y A259525 Cf. A007318, A014473, A163866, A214292.
%Y A259525 Cf. A000108, A021499, A047171, A074331.
%K A259525 sign
%O A259525 0,7
%A A259525 _Reinhard Zumkeller_, Jul 18 2015