A135227 Triangle A000012 * A135225, read by rows.
1, 2, 1, 3, 2, 1, 4, 3, 3, 1, 5, 4, 6, 4, 1, 6, 5, 10, 10, 5, 1, 7, 6, 15, 20, 15, 6, 1, 8, 7, 21, 35, 35, 21, 7, 1, 9, 8, 28, 56, 70, 56, 28, 8, 1, 10, 9, 36, 84, 126, 126, 84, 36, 9, 1, 11, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 12, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1
Offset: 0
Examples
First few rows of the triangle: 1; 2, 1; 3, 2, 1; 4, 3, 3, 1; 5, 4, 6, 4, 1; 6, 5, 10, 10, 5, 1; 7, 6, 15, 20, 15, 6, 1; ...
Links
- G. C. Greubel, Rows n = 0..100 of triangle, flattened
Programs
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GAP
T:= function(n,k) if k=0 then return 1; else return Binomial(n,k); fi; end; Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 20 2019 -
Magma
[k eq 0 select n+1 else Binomial(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2019
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Maple
seq(seq( `if`(k=0, n+1, binomial(n,k)), k=0..n), n=0..12); # G. C. Greubel, Nov 20 2019
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Mathematica
Table[If[k==0, n+1, Binomial[n, k]], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 20 2019 *) -
PARI
T(n,k) = if(k==0, n+1, binomial(n,k)); \\ G. C. Greubel, Nov 20 2019
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Sage
def T(n, k): if (k==0): return 1 else: return binomial(n, k) [[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 20 2019
Formula
A000012 * A135225 as infinite lower triangular matrices. Left border of 1's in Pascal's Triangle (A007318) is replaced with a column of (1,2,3,...).
T(n,k) = binomial(n,k), with T(n,0) = n+1. - G. C. Greubel, Nov 20 2019
Extensions
More terms added by G. C. Greubel, Nov 20 2019
Comments