OFFSET
1,2
COMMENTS
Related identity: Sum_{n=-oo..+oo} x^n*(y - x^n)^n = 0, which holds formally for all y.
LINKS
FORMULA
G.f. A(x,y) = Sum_{n>=1} Sum_{k=0..n-1} T(n,k)*x^n*y^k satisfies the following formulas.
(1) 1/x = Sum_{n=-oo..+oo} A(x,y)^n * (A(x,y)^n + y)^(n+1).
(2) 1/x = Sum_{n=-oo..+oo} A(x,y)^(2*n) * (A(x,y)^n - y)^n.
(3) A(x,y) = x * Sum_{n=-oo..+oo} A(x,y)^(n^2) / (1 + y*A(x,y)^(n+1))^n.
(4) A(x,y) = x * Sum_{n=-oo..+oo} A(x,y)^(n^2) / (1 - y*A(x,y)^(n+1))^(n+1).
(5) A(B(x,y), y) = x where B(x,y) = 1/( Sum_{n=-oo..+oo} x^n * (x^n + y)^(n+1) ).
EXAMPLE
G.f.: A(x,y) = x*(1) + x^2*(2 + y) + x^3*(4 + 4*y + 2*y^2) + x^4*(8 + 13*y + 12*y^2 + 5*y^3) + x^5*(18 + 40*y + 52*y^2 + 40*y^3 + 14*y^4) + x^6*(52 + 130*y + 204*y^2 + 215*y^3 + 140*y^4 + 42*y^5) + x^7*(184 + 472*y + 813*y^2 + 1004*y^3 + 896*y^4 + 504*y^5 + 132*y^6) + x^8*(688 + 1863*y + 3430*y^2 + 4588*y^3 + 4816*y^4 + 3738*y^5 + 1848*y^6 + 429*y^7) + x^9*(2512 + 7536*y + 15016*y^2 + 21472*y^3 + 24540*y^4 + 22656*y^5 + 15576*y^6 + 6864*y^7 + 1430*y^8) + x^10*(8866 + 30144*y + 65880*y^2 + 102177*y^3 + 124830*y^4 + 126801*y^5 + 104940*y^6 + 64779*y^7 + 25740*y^8 + 4862*y^9) + ...
where 1/x = Sum_{n=-oo..+oo} A(x,y)^n * (A(x,y)^n + y)^(n+1).
TRIANGLE.
This triangle of coefficients T(n,k) of x^n*y^k in A(x,y), for n >= 1, k=0..n-1, begins
n = 1: [1];
n = 2: [2, 1];
n = 3: [4, 4, 2];
n = 4: [8, 13, 12, 5];
n = 5: [18, 40, 52, 40, 14];
n = 6: [52, 130, 204, 215, 140, 42];
n = 7: [184, 472, 813, 1004, 896, 504, 132];
n = 8: [688, 1863, 3430, 4588, 4816, 3738, 1848, 429];
n = 9: [2512, 7536, 15016, 21472, 24540, 22656, 15576, 6864, 1430];
n =10: [8866, 30144, 65880, 102177, 124830, 126801, 104940, 64779, 25740, 4862];
n =11: [30824, 118420, 284305, 483300, 636750, 693528, 638825, 479908, 268840, 97240, 16796];
n =12: [108088, 460746, 1205402, 2242581, 3213584, 3758727, 3731794, 3154866, 2171312, 1113398, 369512, 58786];
...
RELATED SEQUENCES.
A000108(n) = T(n+1,n) for n >= 0 (Catalan numbers).
A028329(n) = T(n+2,n) for n >= 0.
A166952(n) = T(n+1,0) for n >= 0 (g.f. F(x) = theta_3(x*F(x))).
A379201(n) = T(n,1) for n >= 2 (column 1).
A379206(n) = T(2*n-1,n-1) for n >= 1 (central terms).
A378264(n) = Sum_{k=0..n-1} T(n,k) for n >= 1.
A379199(n) = Sum_{k=0..n-1} T(n,k) * (-1)^k for n >= 1.
A379202(n) = Sum_{k=0..n-1} T(n,k) * 2^k for n >= 1.
A379203(n) = Sum_{k=0..n-1} T(n,k) * 3^k for n >= 1.
A379204(n) = Sum_{k=0..n-1} T(n,k) * 4^k for n >= 1.
A379205(n) = Sum_{k=0..n-1} T(n,k) * 5^k for n >= 1.
ALTERNATIVE FORMAT.
This triangle may also be presented as a rectangular table like so:
[ 1, 1, 2, 5, 14, 42, 132, ...];
[ 2, 4, 12, 40, 140, 504, 1848, ...];
[ 4, 13, 52, 215, 896, 3738, 15576, ...];
[ 8, 40, 204, 1004, 4816, 22656, 104940, ...];
[ 18, 130, 813, 4588, 24540, 126801, 638825, ...];
[ 52, 472, 3430, 21472, 124830, 693528, 3731794, ...];
[184, 1863, 15016, 102177, 636750, 3758727, 21365548, ...];
...
PROG
(PARI) {T(n, k) = my(V=[0, 1], A); for(i=1, n, V=concat(V, 0); A = Ser(V);
V[#V] = polcoef( sum(m=-#A, #A, A^m*(A^m + y)^(m+1) ), #V-3); ); polcoef(polcoef(A, n, x), k, y)}
for(n=1, 12, for(k=0, n-1, print1(T(n, k), ", ")); print(""))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Dec 20 2024
STATUS
approved