A135228 Triangle A000012(signed) * A007318 * A103451, read by rows.
1, 1, 1, 3, 1, 1, 5, 2, 2, 1, 11, 2, 4, 3, 1, 21, 3, 6, 7, 4, 1, 43, 3, 9, 13, 11, 5, 1, 85, 4, 12, 22, 24, 16, 6, 1, 171, 4, 16, 34, 46, 40, 22, 7, 1, 341, 5, 20, 50, 80, 86, 62, 29, 8, 1, 683, 5, 25, 70, 130, 166, 148, 91, 37, 9, 1, 1365, 6, 30, 95, 200, 296, 314, 239, 128, 46, 10, 1
Offset: 0
Examples
First few rows of the triangle are:
1;
1, 1;
3, 1, 1;
5, 2, 2, 1;
11, 2, 4, 3, 1;
21, 3, 6, 7, 4, 1;
43, 3, 9, 13, 11, 5, 1;
85, 4, 12, 22, 24, 16, 6, 1;
171, 4, 16, 34, 46, 40, 22, 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 0 then return (2^(n+1) +(-1)^n)/3; else return (&+[Binomial(n-2*j-1, k-1): j in [0..Floor((n-1)/2)]]); 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=0 then (2^(n+1) +(-1)^n)/3 else add(binomial(n-2*j-1, k-1), j=0..floor((n-1)/2)) 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==0, (2^(n+1) +(-1)^n)/3, 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 *) -
PARI
T(n,k) = if(k==0, (2^(n+1) +(-1)^n)/3, sum(j=0, (n-1)\2, binomial( n-2*j-1, k-1)) ); \\ G. C. Greubel, Nov 20 2019
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Sage
@CachedFunction def T(n, k): if (k==0): return (2^(n+1) +(-1)^n)/3 else: return sum(binomial(n-2*j-1, k-1) for j in (0..floor((n-1)/2))) [[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 20 2019
Formula
T(n,k) = A000012(signed) * A007318 * A103451 as infinite lower triangular matrices. A000012(signed) = (1; -1,1; 1,-1,1; ...).
T(n,k) = Sum_{j=0..floor((n-1)/2)} binomial(n-2*j-1, k-1), with T(n,0) =
(2^(n+1) - (-1)^(n+1))/3 (Jacobsthal_{n+1}).- G. C. Greubel, Nov 20 2019
Extensions
Offset changed by G. C. Greubel, Nov 20 2019
Comments