A135229 - OEIS

%I #11 Sep 08 2022 08:45:32

%S 1,1,1,1,1,1,1,2,2,1,1,2,4,3,1,1,3,6,7,4,1,1,3,9,13,11,5,1,1,4,12,22,

%T 24,16,6,1,1,4,16,34,46,40,22,7,1,1,5,20,50,80,86,62,29,8,1

%N Triangle A000012(signed) * A103451 * A007318, read by rows.

%C row sums = A005578 starting (1, 2, 3, 6, 11, 22, 43, 86, ...).

%H G. C. Greubel, <a href="/A135229/b135229.txt">Rows n = 0..100 of triangle, flattened</a>

%F T(n,k) = A000012(signed) * A103451 * A007318 as infinite lower triangular matrices, where A000012(signed) = (1; -1,1; 1,-1,1; ...).

%F T(n,k) = Sum_{j=0..floor((n-1)/2)} binomial(n-2*j-1, k-1), with T(n,0) = 1. - _G. C. Greubel_, Nov 20 2019

%e First few rows of the triangle are:

%e   1;

%e   1, 1;

%e   1, 1,  1;

%e   1, 2,  2,  1;

%e   1, 2,  4,  3,  1;

%e   1, 3,  6,  7,  4,  1;

%e   1, 3,  9, 13, 11,  5,  1;

%e   1, 4, 12, 22, 24, 16,  6, 1;

%e   1, 4, 16, 34, 46, 40, 22, 7, 1;

%e ...

%p T:= proc(n, k) option remember;

%p       if k=0 then 1

%p     else add(binomial(n-2*j-1, k-1), j=0..floor((n-1)/2))

%p       fi; end:

%p seq(seq(T(n, k), k=0..n), n=0..12); # _G. C. Greubel_, Nov 20 2019

%t T[n_, k_]:= T[n, k]= If[k==0, 1, Sum[Binomial[n-1-2*j, k-1], {j, 0, Floor[(n-1)/2]}]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 20 2019 *)

%o (PARI) T(n,k) = if(k==0, 1, sum(j=0, (n-1)\2, binomial( n-2*j-1, k-1)) ); \\ _G. C. Greubel_, Nov 20 2019

%o (Magma)

%o function T(n,k)

%o   if k eq 0 then return 1;

%o   else return (&+[Binomial(n-2*j-1, k-1): j in [0..Floor((n-1)/2)]]);

%o   end if; return T; end function;

%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 20 2019

%o (Sage)

%o @CachedFunction

%o def T(n, k):

%o     if (k==0): 1

%o     else: return sum(binomial(n-2*j-1, k-1) for j in (0..floor((n-1)/2)))

%o [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Nov 20 2019

%Y Cf. A005578, A007318, A103451.

%K nonn,tabl

%O 0,8

%A _Gary W. Adamson_, Nov 23 2007

%E Offset changed by _G. C. Greubel_, Nov 20 2019