This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A374452 #18 Sep 30 2024 12:41:56
%S A374452 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 A374452 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 A374452 45,120,195,226,195,120,45,10,1
%N A374452 Iterated rascal triangle R3: T(n,k) = Sum_{m=0..3} binomial(n-k,m)*binomial(k,m).
%C A374452 Triangle T(n,k) is the third triangle R3 among the rascal-family triangles; A077028 is triangle R1, A374378 is triangle R2.
%C A374452 Triangle T(n,k) equals Pascal's triangle A007318 through row 2i+1, i=2 (i.e., row 7).
%C A374452 Triangle T(n,k) equals Pascal's triangle A007318 through column i, i=2 (i.e., column 3).
%H A374452 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 A374452 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 A374452 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 A374452 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 A374452 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 A374452 Petro Kolosov, <a href="https://kolosovpetro.github.io/pdf/IdentitiesInRascalTriangle.pdf">Identities in Iterated Rascal Triangles</a>, 2024.
%F A374452 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 A374452 Row sums give A008860(n).
%F A374452 Diagonal T(n+1, n) gives A000027(n).
%F A374452 Diagonal T(n+2, n) gives A000217(n).
%F A374452 Diagonal T(n+3, n) gives A000292(n).
%F A374452 Diagonal T(n+4, n) gives A005894(n).
%F A374452 Diagonal T(n+6, n) gives A247608(n).
%F A374452 Column k=4 difference binomial(n+8, 4) - T(n+8, 4) gives C(n+4,4)=A007318(n+4,4).
%F A374452 Column k=5 difference binomial(n+9, 5) - T(n+9, 5) gives sixth column of (1,5)-Pascal triangle A096943.
%F A374452 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 A374452 Triangle begins:
%e A374452 --------------------------------------------------
%e A374452 k= 0 1 2 3 4 5 6 7 8 9 10
%e A374452 --------------------------------------------------
%e A374452 n=0: 1
%e A374452 n=1: 1 1
%e A374452 n=2: 1 2 1
%e A374452 n=3: 1 3 3 1
%e A374452 n=4: 1 4 6 4 1
%e A374452 n=5: 1 5 10 10 5 1
%e A374452 n=6: 1 6 15 20 15 6 1
%e A374452 n=7: 1 7 21 35 35 21 7 1
%e A374452 n=8: 1 8 28 56 69 56 28 8 1
%e A374452 n=9: 1 9 36 84 121 121 84 36 9 1
%e A374452 n=10: 1 10 45 120 195 226 195 120 45 10 1
%t A374452 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 A374452 Cf. A077028, A000027, A000217, A000292, A005894, A247608, A008860, A374378, A007318, A096943, A096940
%K A374452 nonn,tabl
%O A374452 0,5
%A A374452 _Kolosov Petro_, Jul 08 2024