A374452 Iterated rascal triangle R3: T(n,k) = Sum_{m=0..3} binomial(n-k,m)*binomial(k,m).
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 15, 6, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 8, 28, 56, 69, 56, 28, 8, 1, 1, 9, 36, 84, 121, 121, 84, 36, 9, 1, 1, 10, 45, 120, 195, 226, 195, 120, 45, 10, 1
Offset: 0
Examples
Triangle begins: -------------------------------------------------- k= 0 1 2 3 4 5 6 7 8 9 10 -------------------------------------------------- n=0: 1 n=1: 1 1 n=2: 1 2 1 n=3: 1 3 3 1 n=4: 1 4 6 4 1 n=5: 1 5 10 10 5 1 n=6: 1 6 15 20 15 6 1 n=7: 1 7 21 35 35 21 7 1 n=8: 1 8 28 56 69 56 28 8 1 n=9: 1 9 36 84 121 121 84 36 9 1 n=10: 1 10 45 120 195 226 195 120 45 10 1
Links
- Jena Gregory, Brandt Kronholm and Jacob White, Iterated rascal triangles, Aequationes mathematicae, 2023.
- Jena Gregory, Iterated rascal triangles, Theses and Dissertations. 1050., The University of Texas Rio Grande Valley, 2022.
- Amelia Gibbs and Brian K. Miceli, Two Combinatorial Interpretations of Rascal Numbers, arXiv:2405.11045 [math.CO], 2024.
- Philip K. Hotchkiss, Student Inquiry and the Rascal Triangle, arXiv:1907.07749 [math.HO], 2019.
- Philip K. Hotchkiss, Generalized Rascal Triangles, Journal of Integer Sequences, Vol. 23, 2020.
- Petro Kolosov, Identities in Iterated Rascal Triangles, 2024.
Crossrefs
Programs
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Mathematica
t[n_, k_] := Sum[Binomial[n - k, m]*Binomial[k, m], {m, 0, 3}]; Column[Table[t[n, k], {n, 0, 12}, {k, 0, n}], Left]
Formula
T(n,k) = 1 + k*(n-k) + 1/4*(k-1)*k*(n-k-1)*(n-k) + 1/36*(k-2)*(k-1)*k*(n-k-2)*(n-k-1)*(n-k).
Row sums give A008860(n).
Diagonal T(n+1, n) gives A000027(n).
Diagonal T(n+2, n) gives A000217(n).
Diagonal T(n+3, n) gives A000292(n).
Diagonal T(n+4, n) gives A005894(n).
Diagonal T(n+6, n) gives A247608(n).
Column k=4 difference binomial(n+8, 4) - T(n+8, 4) gives C(n+4,4)=A007318(n+4,4).
Column k=5 difference binomial(n+9, 5) - T(n+9, 5) gives sixth column of (1,5)-Pascal triangle A096943.
G.f.: (1 + 4*x^6*y^3 - 3*x*(1 + y) - 6*x^5*y^2*(1 + y) + 2*x^4*y*(2 + 7*y+ 2*y^2) + x^2*(3 + 10*y + 3*y^2) - x^3*(1 + 11*y + 11*y^2 + y^3))/((1 - x)^4*(1 - x*y)^4). - Stefano Spezia, Jul 09 2024
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