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 A374378 #57 Sep 30 2024 12:41:48
%S A374378 1,1,1,1,2,1,1,3,3,1,1,4,6,4,1,1,5,10,10,5,1,1,6,15,19,15,6,1,1,7,21,
%T A374378 31,31,21,7,1,1,8,28,46,53,46,28,8,1,1,9,36,64,81,81,64,36,9,1,1,10,
%U A374378 45,85,115,126,115,85,45,10,1,1,11,55,109,155,181,181,155,109,55,11,1
%N A374378 Iterated rascal triangle R2: T(n,k) = Sum_{m=0..2} binomial(n-k,m)*binomial(k,m).
%C A374378 Triangle T(n,k) is the second triangle R2 among the rascal-family triangles; A374452 is triangle R3; A077028 is triangle R1.
%C A374378 Triangle T(n,k) equals Pascal's triangle A007318 through row 2i+1, i=2 (i.e., row 5).
%C A374378 Triangle T(n,k) equals Pascal's triangle A007318 through column i, i=2 (i.e., column 2).
%H A374378 Kolosov Petro, <a href="/A374378/b374378.txt">Table of n, a(n) for n = 0..1325</a>
%H A374378 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 A374378 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 A374378 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 A374378 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 A374378 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 A374378 Petro Kolosov, <a href="https://kolosovpetro.github.io/pdf/IdentitiesInRascalTriangle.pdf">Identities in Iterated Rascal Triangles</a>, 2024.
%F A374378 T(n,k) = 1 + k*(n-k) + (1/4)*(k-1)*k*(n-k-1)*(n-k).
%F A374378 Row sums give A006261(n).
%F A374378 Diagonal T(n+1, n) gives A000027(n).
%F A374378 Diagonal T(n+2, n) gives A000217(n).
%F A374378 Diagonal T(n+3, n) gives A005448(n).
%F A374378 Diagonal T(n+4, n) gives A056108(n).
%F A374378 Diagonal T(n+5, n) gives A212656(n).
%F A374378 Column k=3 difference binomial(n+6, 3) - T(n+6, 3) gives C(n+3,3)=A007318(n+3,3).
%F A374378 Column k=4 difference binomial(n+7, 4) - T(n+7, 4) gives fifth column of (1,4)-Pascal triangle A095667.
%F A374378 G.f.: (1 + 3*x^4*y^2 - (2*x + 3*x^3*y)*(1 + y) + x^2*(1 + 5*y + y^2))/((1 - x)^3*(1 - x*y)^3). - _Stefano Spezia_, Jul 09 2024
%e A374378 Triangle begins:
%e A374378 --------------------------------------------------
%e A374378 k= 0 1 2 3 4 5 6 7 8 9 10
%e A374378 --------------------------------------------------
%e A374378 n=0: 1
%e A374378 n=1: 1 1
%e A374378 n=2: 1 2 1
%e A374378 n=3: 1 3 3 1
%e A374378 n=4: 1 4 6 4 1
%e A374378 n=5: 1 5 10 10 5 1
%e A374378 n=6: 1 6 15 19 15 6 1
%e A374378 n=7: 1 7 21 31 31 21 7 1
%e A374378 n=8: 1 8 28 46 53 46 28 8 1
%e A374378 n=9: 1 9 36 64 81 81 64 36 9 1
%e A374378 n=10: 1 10 45 85 115 126 115 85 45 10 1
%t A374378 t[n_, k_]:=Sum[Binomial[n - k, m]*Binomial[k, m], {m, 0, 2}]; Column[Table[t[n, k], {n, 0, 12}, {k, 0, n}], Center]
%Y A374378 Cf. A077028, A007318, A000027, A000217, A005448, A056108, A212656, A000292, A095667, A095666, A006261.
%K A374378 nonn,tabl
%O A374378 0,5
%A A374378 _Kolosov Petro_, Jul 06 2024