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.
1, 1, 1, 2, 5, 3, 5, 21, 28, 12, 14, 84, 180, 165, 55, 42, 330, 990, 1430, 1001, 273, 132, 1287, 5005, 10010, 10920, 6188, 1428, 429, 5005, 24024, 61880, 92820, 81396, 38760, 7752, 1430, 19448, 111384, 352716, 678300, 813960, 596904, 245157, 43263, 4862, 75582, 503880, 1899240, 4476780, 6864396, 6864396, 4326300, 1562275, 246675
Offset: 0
Examples
The triangle T(n, k) begins:
[0] 1;
[1] 1, 1;
[2] 2, 5, 3;
[3] 5, 21, 28, 12;
[4] 14, 84, 180, 165, 55;
[5] 42, 330, 990, 1430, 1001, 273;
[6] 132, 1287, 5005, 10010, 10920, 6188, 1428;
[7] 429, 5005, 24024, 61880, 92820, 81396, 38760, 7752;
[8] 1430, 19448, 111384, 352716, 678300, 813960, 596904, 245157, 43263;
...
The array A(n, k) begins:
[0] 1, 1, 3, 12, 55, 273, 1428, ... [A001764]
[1] 1, 5, 28, 165, 1001, 6188, 38760, ... [A025174]
[2] 2, 21, 180, 1430, 10920, 81396, 596904, ... [A383450]
[3] 5, 84, 990, 10010, 92820, 813960, 6864396, ... [A383451]
[4] 14, 330, 5005, 61880, 678300, 6864396, 65615550, ...
[5] 42, 1287, 24024, 352716, 4476780, 51482970, 551170620, ...
[6] 132, 5005, 111384, 1899240, 27457584, 354323970, 4206302100, ...
[A000108] | [A074922][A383452]
[A002054]
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
- N. J. Wildberger and Dean Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, Amer. Math. Monthly (2025). See sections 8 and 12.
- Jian Zhou, Fat and Thin Emergent Geometries of Hermitian One-Matrix Models, arXiv:1810.03883 [math-ph], 2018.
Crossrefs
Programs
-
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
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Maple
From Peter Luschny, May 04 2025: (Start) T := (n, k) -> (k + 2*n)!/(k!*(n - k)!*(n + k + 1)!): seq(print(seq(T(n, k), k = 0..n)), n = 0..10); # Alternatively the array: A := (n, k) -> (3*k + 2*n)!/(k!*n!*(n + 2*k + 1)!); for n from 0 to 8 do seq(A(n, k), k = 0..7) od; (End)
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Mathematica
T[n_, k_]:= Binomial[2n+k, n+2k]*Binomial[n+2k, k]/(n+k+1); Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, Jan 27 2019 *) -
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) for(n=0, 10, for(k=0, n, print1(T(n,k),", ")); print("")) -
PARI
Dy(n, F)=local(D=F); for(i=1, n, D=deriv(D,y)); D 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) for(n=0,10,for(k=0,n,print1(T(n,k),", "));print()) \\ Paul D. Hanna, Jun 22 2012
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PARI
x='x; y='y; z='z; Fxyz = 1 - z + x*z^2 + x*y*z^3; seq(N) = { my(z0 = 1 + O((x*y)^N), z1 = 0); for (k = 1, N^2, z1 = z0 - subst(Fxyz, z, z0)/subst(deriv(Fxyz, z), z, z0); if (z0 == z1, break()); z0 = z1); vector(N, n, Vecrev(polcoeff(z0, n-1, 'x))); }; concat(seq(9)) \\ Gheorghe Coserea, Nov 30 2016 -
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
Formula
T(n, k) = binomial(2*n+k, n+2*k)*binomial(n+2*k, k)/(n+k+1).
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
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
A(n, k) = T(n + k, k) = (3*k + 2*n)! / (k!*n!*(n + 2*k + 1)!). - Peter Luschny, May 04 2025