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A157898
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Triangle read by rows: inverse binomial transform of A059576
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3
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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, 12, 40, 40, 96, 32, 64, 1, 4, 20, 40, 120, 96, 224, 64, 128, 0, 5, 20, 80, 120, 336, 224, 512, 128, 256
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,6
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COMMENTS
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The inverse binomial transform of the triangle A059576 is given by multiplying the triangle with A130595 from the left.
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LINKS
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FORMULA
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Sum_{k=0..n} T(n, k) = A097076(n+1).
T(n, k) = Sum_{j=k..n} (-1)^(n-j)*binomial(n,j)*A059576(j,k).
T(n, n-1) = A011782(n) - [n==0]. (End)
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EXAMPLE
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First few rows of the triangle =
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, 12, 40, 40, 96, 32, 64;
1, 4, 20, 40, 120, 96, 224, 64, 128;
0, 5, 20, 80, 120, 336, 224, 512, 128, 256;
...
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MAPLE
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if n = 0 then
return 1;
end if;
if k <= n and k >= 0 then
add((-1)^j*2^(n-j-1)*binomial(k, j)*binomial(n-j, k), j = 0 .. min(k, n-k))
else
0 ;
end if
end proc:
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MATHEMATICA
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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 *)
A157898[n_, k_]:= A157898[n, k]= Sum[(-1)^(n-j)*Binomial[n, j]*t[j, k], {j, k, n}];
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PROG
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(Magma)
A011782:= func< n | n eq 0 select 1 else 2^(n-1) >;
if k eq 0 or k eq n then return A011782(n);
else return 2*t(n-1, k-1) + 2*t(n-1, k) - (2 - 0^(n-2))*t(n-2, k-1);
end if; return t;
end function;
A157898:= func< n, k | (&+[(-1)^(n-j)*Binomial(n, j)*t(j, k): j in [k..n]]) >;
(SageMath)
@CachedFunction
if (k==0 or k==n): return bool(n==0)/2 + 2^(n-1) # A011782
else: return 2*t(n-1, k-1) + 2*t(n-1, k) - (2 - 0^(n-2))*t(n-2, k-1)
def A157898(n, k): return sum((-1)^(n-j)*binomial(n, j)*t(j, k) for j in (k..n))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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