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 A104978 #44 May 04 2025 08:02:06
%S A104978 1,1,1,2,5,3,5,21,28,12,14,84,180,165,55,42,330,990,1430,1001,273,132,
%T A104978 1287,5005,10010,10920,6188,1428,429,5005,24024,61880,92820,81396,
%U A104978 38760,7752,1430,19448,111384,352716,678300,813960,596904,245157,43263,4862,75582,503880,1899240,4476780,6864396,6864396,4326300,1562275,246675
%N A104978 Triangle read by rows, where the g.f. satisfies A(x, y) = 1 + x*A(x, y)^2 + x*y*A(x, y)^3.
%H A104978 G. C. Greubel, <a href="/A104978/b104978.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A104978 N. J. Wildberger and Dean Rubine, <a href="https://doi.org/10.1080/00029890.2025.2460966">A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode</a>, Amer. Math. Monthly (2025). See sections 8 and 12.
%H A104978 Jian Zhou, <a href="https://arxiv.org/abs/1810.03883">Fat and Thin Emergent Geometries of Hermitian One-Matrix Models</a>, arXiv:1810.03883 [math-ph], 2018.
%F A104978 T(n, k) = binomial(2*n+k, n+2*k)*binomial(n+2*k, k)/(n+k+1).
%F A104978 G.f.: A(x, y) = Sum_{n>=0} x^n/y^(n+1) * d^(n-1)/dy^(n-1) (y^2 + y^3)^n / n!. - _Paul D. Hanna_, Jun 22 2012
%F A104978 G.f. of row n: 1/y^(n+1) * d^(n-1)/dy^(n-1) (y^2+y^3)^n / n!. - _Paul D. Hanna_, Jun 22 2012
%F A104978 A(n, k) = T(n + k, k) = (3*k + 2*n)! / (k!*n!*(n + 2*k + 1)!). - _Peter Luschny_, May 04 2025
%e A104978 The triangle T(n, k) begins:
%e A104978 [0] 1;
%e A104978 [1] 1, 1;
%e A104978 [2] 2, 5, 3;
%e A104978 [3] 5, 21, 28, 12;
%e A104978 [4] 14, 84, 180, 165, 55;
%e A104978 [5] 42, 330, 990, 1430, 1001, 273;
%e A104978 [6] 132, 1287, 5005, 10010, 10920, 6188, 1428;
%e A104978 [7] 429, 5005, 24024, 61880, 92820, 81396, 38760, 7752;
%e A104978 [8] 1430, 19448, 111384, 352716, 678300, 813960, 596904, 245157, 43263;
%e A104978 ...
%e A104978 The array A(n, k) begins:
%e A104978 [0] 1, 1, 3, 12, 55, 273, 1428, ... [A001764]
%e A104978 [1] 1, 5, 28, 165, 1001, 6188, 38760, ... [A025174]
%e A104978 [2] 2, 21, 180, 1430, 10920, 81396, 596904, ... [A383450]
%e A104978 [3] 5, 84, 990, 10010, 92820, 813960, 6864396, ... [A383451]
%e A104978 [4] 14, 330, 5005, 61880, 678300, 6864396, 65615550, ...
%e A104978 [5] 42, 1287, 24024, 352716, 4476780, 51482970, 551170620, ...
%e A104978 [6] 132, 5005, 111384, 1899240, 27457584, 354323970, 4206302100, ...
%e A104978 [A000108] | [A074922][A383452]
%e A104978 [A002054]
%p A104978 From _Peter Luschny_, May 04 2025: (Start)
%p A104978 T := (n, k) -> (k + 2*n)!/(k!*(n - k)!*(n + k + 1)!):
%p A104978 seq(print(seq(T(n, k), k = 0..n)), n = 0..10);
%p A104978 # Alternatively the array:
%p A104978 A := (n, k) -> (3*k + 2*n)!/(k!*n!*(n + 2*k + 1)!);
%p A104978 for n from 0 to 8 do seq(A(n, k), k = 0..7) od; (End)
%t A104978 T[n_, k_]:= Binomial[2n+k, n+2k]*Binomial[n+2k, k]/(n+k+1);
%t A104978 Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* _Jean-François Alcover_, Jan 27 2019 *)
%o A104978 (PARI) T(n,k) = local(A=1+x+x*y+x*O(x^n)+y*O(y^k)); for(i=1,n,A=1+x*A^2+x*y*A^3); polcoeff(polcoeff(A,n,x),k,y)
%o A104978 for(n=0, 10, for(k=0, n, print1(T(n,k),", ")); print(""))
%o A104978 (PARI) Dy(n, F)=local(D=F); for(i=1, n, D=deriv(D,y)); D
%o A104978 T(n,k)=local(A=1); A=1+sum(m=1, n+1, x^m/y^(m+1) * Dy(m-1, (y^2+y^3)^m/m!)) +x*O(x^n)+y*O(y^k); polcoeff(polcoeff(A, n,x),k,y)
%o A104978 for(n=0,10,for(k=0,n,print1(T(n,k),", "));print()) \\ _Paul D. Hanna_, Jun 22 2012
%o A104978 (PARI)
%o A104978 x='x; y='y; z='z; Fxyz = 1 - z + x*z^2 + x*y*z^3;
%o A104978 seq(N) = {
%o A104978 my(z0 = 1 + O((x*y)^N), z1 = 0);
%o A104978 for (k = 1, N^2,
%o A104978 z1 = z0 - subst(Fxyz, z, z0)/subst(deriv(Fxyz, z), z, z0);
%o A104978 if (z0 == z1, break()); z0 = z1);
%o A104978 vector(N, n, Vecrev(polcoeff(z0, n-1, 'x)));
%o A104978 };
%o A104978 concat(seq(9)) \\ _Gheorghe Coserea_, Nov 30 2016
%o A104978 (Magma) [Binomial(2*n+k, n+2*k)*Binomial(n+2*k, k)/(n+k+1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 08 2021
%o A104978 (Sage) flatten([[binomial(2*n+k, n+2*k)*binomial(n+2*k, k)/(n+k+1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 08 2021
%Y A104978 Columns of array: A000108, A002054, A074922, A383452.
%Y A104978 Rows of array: A001764, A025174, A383450, A383451.
%Y A104978 Cf. A001002 (antidiagonal sums), A001764 (semidiagonal sums), A027307 (row sums), A104979, A383439 (central terms).
%K A104978 nonn,tabl
%O A104978 0,4
%A A104978 _Paul D. Hanna_, Mar 30 2005