A135233 Triangle A007318 * A193554, read by rows.
1, 2, 1, 5, 3, 1, 14, 7, 5, 1, 41, 15, 17, 7, 1, 122, 31, 49, 31, 9, 1, 365, 63, 129, 111, 49, 11, 1, 1094, 127, 321, 351, 209, 71, 13, 1, 3281, 255, 769, 1023, 769, 351, 97, 15, 1, 9842, 511, 1793, 2815, 2561, 1471, 545, 127, 17, 1
Offset: 0
Examples
First few rows of the triangle: 1; 2, 1; 5, 3, 1; 14, 7, 5, 1; 41, 15, 17, 7, 1; ...
Links
- G. C. Greubel, Rows n = 0..100 of triangle, flattened
Programs
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Magma
function T(n,k) if k eq n then return 1; elif k eq 0 then return (3^n+1)/2; else return (&+[(-1)^(n-k+j)*2^j*Binomial(n, j): j in [0..n-k]]); end if; return T; end function; [T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2019
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Maple
T:= proc(n, k) option remember; if k=n then 1 elif k=0 then (3^n_1)/2 else add((-1)^(n-k+j)*binomial(n, j)*2^j, j=0..n-k) fi; end: seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Nov 20 2019 -
Mathematica
T[n_, k_]:= T[n, k]= If[k==n, 1, If[k==0, (3^n+1)/2, Sum [(-1)^(n-k+i)* Binomial[n, i]*2^i, {i, 0, n-k}]]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 20 2019 *) -
PARI
T(n,k) = if(k==n, 1, if(k==0, (3^n+1)/2, sum(j=0, n-k, (-1)^(n-k+j)*binomial(n,j)*2^j) )); \\ G. C. Greubel, Nov 20 2019
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Sage
@CachedFunction def T(n, k): if (k==n): return 1 elif (k==0): return (3^n+1)/2 else: return sum((-1)^(n-k+j)*2^j*binomial(n, j) for j in (0..n-k)) [[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 20 2019
Formula
Binomial transform of A193554, as infinite lower triangular matrices.
T(n,k) = Sum_{j=0..n-k} (-1)^(n-k+j)*binomial(n,j)*2^j, with T(n,n) = 1, and T(n,0) = (3^n + 1)/2. - G. C. Greubel, Nov 20 2019
Extensions
Definition corrected by N. J. A. Sloane, Jul 30 2011
Comments