A323211 Level 1 of Pascal's pyramid. T(n, k) triangle read by rows for n >= 0 and 0 <= k <= n.
1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 2, 4, 4, 2, 1, 1, 2, 5, 7, 5, 2, 1, 1, 2, 6, 11, 11, 6, 2, 1, 1, 2, 7, 16, 21, 16, 7, 2, 1, 1, 2, 8, 22, 36, 36, 22, 8, 2, 1, 1, 2, 9, 29, 57, 71, 57, 29, 9, 2, 1, 1, 2, 10, 37, 85, 127, 127, 85, 37, 10, 2, 1
Offset: 0
Examples
Triangle starts:
1
1, 1
1, 2, 1
1, 2, 2, 1
1, 2, 3, 2, 1
1, 2, 4, 4, 2, 1
1, 2, 5, 7, 5, 2, 1
1, 2, 6, 11, 11, 6, 2, 1
1, 2, 7, 16, 21, 16, 7, 2, 1
1, 2, 8, 22, 36, 36, 22, 8, 2, 1
1, 2, 9, 29, 57, 71, 57, 29, 9, 2, 1
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Crossrefs
Programs
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Magma
A323211:= func< n,k | n le 1 select 1 else 1 + Binomial(n-2,k-1) >; [A323211(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Sep 26 2024
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Maple
T := (n, k) -> `if`(n=1, 1, binomial(n-2, k-1) + 1): seq(seq(T(n, k), k=0..n), n=0..10); # Alternative: T := proc(n, k) option remember; if k = n then return 1 fi; if k < 2 then return k+1 fi; T(n-1, k-1) + T(n-1, k) - 1 end: seq(seq(T(n, k), k=0..n), n=0..10);
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Mathematica
A323211[n_, k_]:= If[n<2, 1, Binomial[n-2, k-1] +1]; Table[A323211[n,k], {n,0,13}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 26 2024 *)
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SageMath
def A323211(n,k): return 1 if (n<2) else 1 + binomial(n-2,k-1) flatten([[A323211(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Sep 26 2024
Formula
T(n, k) = binomial(n-2, k-1) + 1 if n != 1 else 1.
G.f.: (1 + 3*y + y^2 + x^4*y^2*(1 + y)^2 + x^2*y*(2 + 5*y + 2*y^2) - x^3*y*(1 + 4*y + 4*y^2 + y^3) - x*(1 + 5*y + 5*y^2 + y^3)/((1 - x)*(1 + y)^2*(1 - x*y)*(1 - x - x*y)). - Stefano Spezia, Sep 26 2024
From G. C. Greubel, Sep 26 2024: (Start)
T(n, n-k) = T(n, k) (symmetry).
T(2*n, n) = A323230(n).
Sum_{k=0..n} (-1)^k*T(n, k) = (n+1 mod 2) - [n=2].
Sum_{k=0..floor(n/2)} T(n-k, k) = Fibonacci(n-2) + (1/4)*(2*n + 3 + (-1)^n) +[n=0] - [n=1]. (End)
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