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A157898 Triangle read by rows: inverse binomial transform of A059576 3

%I #10 Sep 03 2022 09:56:08

%S 1,0,1,1,1,2,0,2,2,4,1,2,6,4,8,0,3,6,16,8,16,1,3,12,16,40,16,32,0,4,

%T 12,40,40,96,32,64,1,4,20,40,120,96,224,64,128,0,5,20,80,120,336,224,

%U 512,128,256

%N Triangle read by rows: inverse binomial transform of A059576

%C The inverse binomial transform of the triangle A059576 is given by multiplying the triangle with A130595 from the left.

%H G. C. Greubel, <a href="/A157898/b157898.txt">Rows n = 0..50 of the triangle, flattened</a>

%F Sum_{k=0..n} T(n, k) = A097076(n+1).

%F From _G. C. Greubel_, Sep 03 2022: (Start)

%F T(n, k) = Sum_{j=k..n} (-1)^(n-j)*binomial(n,j)*A059576(j,k).

%F T(n, 0) = A059841(n).

%F T(n, 1) = A004526(n-1).

%F T(n, 2) = 2*A008805(n-2).

%F T(n, 3) = 4*A058187(n-3).

%F T(n, 4) = 8*A189976(n+4).

%F T(n, n) = A011782(n).

%F T(n, n-1) = A011782(n) - [n==0]. (End)

%e First few rows of the triangle =

%e 1;

%e 0, 1;

%e 1, 1, 2;

%e 0, 2, 2, 4;

%e 1, 2, 6, 4, 8;

%e 0, 3, 6, 16, 8, 16;

%e 1, 3, 12, 16, 40, 16, 32;

%e 0, 4, 12, 40, 40, 96, 32, 64;

%e 1, 4, 20, 40, 120, 96, 224, 64, 128;

%e 0, 5, 20, 80, 120, 336, 224, 512, 128, 256;

%e ...

%p A059576 := proc (n, k)

%p if n = 0 then

%p return 1;

%p end if;

%p if k <= n and k >= 0 then

%p add((-1)^j*2^(n-j-1)*binomial(k, j)*binomial(n-j, k), j = 0 .. min(k, n-k))

%p else

%p 0 ;

%p end if

%p end proc:

%p A157898 := proc(n,k)

%p add ( A130595(n,j)*A059576(j,k),j=k..n) ;

%p end proc: # _R. J. Mathar_, Feb 13 2013

%t t[n_, k_]:= t[n, k]= If[k==0 || k==n, 2^(n-1) +Boole[n==0]/2, 2*t[n-1, k-1] + 2*t[n-1, k] -(2 -Boole[n==2])*t[n-2, k-1]]; (* t= A059576 *)

%t A157898[n_, k_]:= A157898[n, k]= Sum[(-1)^(n-j)*Binomial[n,j]*t[j,k], {j,k,n}];

%t Table[A157898[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Sep 03 2022 *)

%o (Magma)

%o A011782:= func< n | n eq 0 select 1 else 2^(n-1) >;

%o function t(n, k) // t = A059576

%o if k eq 0 or k eq n then return A011782(n);

%o else return 2*t(n-1, k-1) + 2*t(n-1, k) - (2 - 0^(n-2))*t(n-2, k-1);

%o end if; return t;

%o end function;

%o A157898:= func< n,k | (&+[(-1)^(n-j)*Binomial(n,j)*t(j,k): j in [k..n]]) >;

%o [A157898(n, k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Sep 03 2022

%o (SageMath)

%o @CachedFunction

%o def t(n, k): # t = A059576

%o if (k==0 or k==n): return bool(n==0)/2 + 2^(n-1) # A011782

%o else: return 2*t(n-1, k-1) + 2*t(n-1, k) - (2 - 0^(n-2))*t(n-2, k-1)

%o def A157898(n,k): return sum((-1)^(n-j)*binomial(n,j)*t(j,k) for j in (k..n))

%o flatten([[A157898(n, k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Sep 03 2022

%Y Cf. A059576, A097076 (row sums), A130595.

%Y Cf. A004526, A008805, A011782, A058187, A059841, A189976.

%K nonn,tabl,easy

%O 0,6

%A _Gary W. Adamson_ and _Roger L. Bagula_, Mar 08 2009

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Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)