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A151685
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a(n) = Sum_{k >= 0} bin2(wt(n+k),k+1), where bin2(i,j) = A013609(i,j), wt(i) = A000120(i).
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11
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3, 7, 5, 7, 17, 17, 7, 7, 17, 17, 19, 41, 51, 31, 9, 7, 17, 17, 19, 41, 51, 31, 21, 41, 51, 55, 101, 143, 113, 49, 11, 7, 17, 17, 19, 41, 51, 31, 21, 41, 51, 55, 101, 143, 113, 49, 23, 41, 51, 55, 101, 143, 113, 73, 103, 143, 161, 257, 387, 369, 211, 71, 13, 7, 17, 17, 19, 41, 51
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OFFSET
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0,1
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COMMENTS
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Or, a(n) = Sum_{k >= 0} 2^wt(k) * binomial(wt(n+k),k).
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LINKS
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FORMULA
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G.f.: Product_{ k >= 0 } (1 + 2*x^(2^k-1) + x^(2^k)).
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EXAMPLE
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Triangle begins:
.3;
.7,5;
.7,17,17,7;
.7,17,17,19,41,51,31,9;
.7,17,17,19,41,51,31,21,41,51,55,101,143,113,49,11;
.7,17,17,19,41,51,31,21,41,51,55,101,143,113,49,23,41,51,55,101,143,113,...
(End)
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MAPLE
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bin2:=proc(n, k) option remember; if k<0 or k>n then 0
elif k=0 then 1 else 2*bin2(n-1, k-1)+bin2(n-1, k); fi; end;
wt := proc(n) local w, m, i;
w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end:
f:=n->add( bin2(wt(n+k), k), k=0..120 );
# or:
f := n->add( 2^k*binomial(wt(n+k), k), k=0..20 );
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MATHEMATICA
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max = 70; (* number of terms *)
CoefficientList[Product[1 + 2*x^(2^k-1) + x^(2^k), {k, 0, Log2[max+1] // Ceiling}] + O[x]^max, x] (* Jean-François Alcover, Aug 03 2022 *)
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CROSSREFS
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For generating functions of the form Product_{k>=c} (1+a*x^(2^k-1)+b*x^2^k)) for the following values of (a,b,c) see: (1,1,0) A160573, (1,1,1) A151552, (1,1,2) A151692, (2,1,0) A151685, (2,1,1) A151691, (1,2,0) A151688 and A152980, (1,2,1) A151550, (2,2,0) A151693, (2,2,1) A151694.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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