A374452 - OEIS

%I #18 Sep 30 2024 12:41:56

%S 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,

%T 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,

%U 45,120,195,226,195,120,45,10,1

%N Iterated rascal triangle R3: T(n,k) = Sum_{m=0..3} binomial(n-k,m)*binomial(k,m).

%C Triangle T(n,k) is the third triangle R3 among the rascal-family triangles; A077028 is triangle R1, A374378 is triangle R2.

%C Triangle T(n,k) equals Pascal's triangle A007318 through row 2i+1, i=2 (i.e., row 7).

%C Triangle T(n,k) equals Pascal's triangle A007318 through column i, i=2 (i.e., column 3).

%H Jena Gregory, Brandt Kronholm and Jacob White, <a href="https://doi.org/10.1007/s00010-023-00987-6">Iterated rascal triangles</a>, Aequationes mathematicae, 2023.

%H Jena Gregory, <a href="https://scholarworks.utrgv.edu/etd/1050/">Iterated rascal triangles</a>, Theses and Dissertations. 1050., The University of Texas Rio Grande Valley, 2022.

%H Amelia Gibbs and Brian K. Miceli, <a href="https://arxiv.org/abs/2405.11045">Two Combinatorial Interpretations of Rascal Numbers</a>, arXiv:2405.11045 [math.CO], 2024.

%H Philip K. Hotchkiss, <a href="https://arxiv.org/abs/1907.07749">Student Inquiry and the Rascal Triangle</a>, arXiv:1907.07749 [math.HO], 2019.

%H Philip K. Hotchkiss, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Hotchkiss/hotchkiss4.html">Generalized Rascal Triangles</a>, Journal of Integer Sequences, Vol. 23, 2020.

%H Petro Kolosov, <a href="https://kolosovpetro.github.io/pdf/IdentitiesInRascalTriangle.pdf">Identities in Iterated Rascal Triangles</a>, 2024.

%F 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).

%F Row sums give A008860(n).

%F Diagonal T(n+1, n) gives A000027(n).

%F Diagonal T(n+2, n) gives A000217(n).

%F Diagonal T(n+3, n) gives A000292(n).

%F Diagonal T(n+4, n) gives A005894(n).

%F Diagonal T(n+6, n) gives A247608(n).

%F Column k=4 difference binomial(n+8, 4) - T(n+8, 4) gives C(n+4,4)=A007318(n+4,4).

%F Column k=5 difference binomial(n+9, 5) - T(n+9, 5) gives sixth column of (1,5)-Pascal triangle A096943.

%F 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

%e Triangle begins:

%e --------------------------------------------------

%e k=     0   1   2   3    4    5    6   7   8   9 10

%e --------------------------------------------------

%e n=0:   1

%e n=1:   1   1

%e n=2:   1   2   1

%e n=3:   1   3   3   1

%e n=4:   1   4   6   4    1

%e n=5:   1   5  10  10    5    1

%e n=6:   1   6  15  20   15    6    1

%e n=7:   1   7  21  35   35   21    7   1

%e n=8:   1   8  28  56   69   56   28   8   1

%e n=9:   1   9  36  84  121  121   84  36   9   1

%e n=10:  1  10  45 120  195  226  195  120  45  10  1

%t 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]

%Y Cf. A077028, A000027, A000217, A000292, A005894, A247608, A008860, A374378, A007318, A096943, A096940

%K nonn,tabl

%O 0,5

%A _Kolosov Petro_, Jul 08 2024