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A081181 Staircase on Pascal's triangle. 4

%I #14 Jan 14 2024 02:33:52

%S 1,3,6,20,35,126,210,792,1287,5005,8008,31824,50388,203490,319770,

%T 1307504,2042975,8436285,13123110,54627300,84672315,354817320,

%U 548354040,2310789600,3562467300,15084504396,23206929840,98672427616,151532656696

%N Staircase on Pascal's triangle.

%C Arrange Pascal's triangle as a square array. a(n) is then a diagonal staircase on the square array. A companion staircase is given by A065942.

%H Vincenzo Librandi, <a href="/A081181/b081181.txt">Table of n, a(n) for n = 0..300</a>

%F a(n) = binomial(floor((n + 1)/2) + (n + 1), n).

%F Conjecture: 8*n*(n+3)*(1845*n-2882)*a(n) +4*(-5097*n^3+11143*n^2 +42110*n-27416)*a(n-1) +6*(-16605*n^3-7272*n^2-16701*n+9490)*a(n-2) +3*(3*n-5)*(5097*n-949)*(3*n-4)*a(n-3)=0. - _R. J. Mathar_, Oct 29 2014

%t Table[Binomial[Floor[(n + 1) / 2] + (n + 1), n], {n, 0, 30}] (* _Vincenzo Librandi_, Aug 06 2013 *)

%o (Magma) [Binomial(Floor((n+1)/2)+(n+1), n): n in [0..30]]; // _Vincenzo Librandi_, Aug 06 2013

%o (SageMath) [binomial(((n+1)//2)+(n+1), n) for n in range(41)] # _G. C. Greubel_, Jan 14 2024

%Y Cf. A065942.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Mar 11 2003

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Last modified June 22 14:17 EDT 2024. Contains 373587 sequences. (Running on oeis4.)