A361550 - OEIS

%I #11 Jan 05 2024 23:33:08

%S 1,0,1,0,5,1,0,18,10,1,0,55,61,20,1,0,149,290,215,35,1,0,371,1172,

%T 1660,555,56,1,0,867,4212,10311,5850,1254,84,1,0,1923,13833,54688,

%U 47460,17773,2555,120,1,0,4086,42262,256815,319409,188300,46844,4810,165,1,0,8374,121625,1093790,1864445,1621116,621915,111348,8505,220,1,0,16634,332764,4297370,9717550,11913160,6557572,1818022,243795,14290,286,1

%N Expansion of g.f. A(x,y) satisfying x*y = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x,y)^(3*n) - 1/A(x,y)^(3*n+1)), as a triangle read by rows.

%C A359920(n) = Sum_{k=0..n} T(n,k) for n >= 0.

%C A361552(n) = Sum_{k=0..n} T(n,k) * 2^k for n >= 0.

%C A361553(n) = Sum_{k=0..n} T(n,k) * 3^k for n >= 0.

%C A361554(n) = Sum_{k=0..n} T(n,k) * 4^k for n >= 0.

%C A361555(n) = Sum_{k=0..n} T(n,k) * 5^k for n >= 0.

%C A361556(n) = T(2*n,n) for n >= 0.

%C A360191(n) = T(n+1,1) for n >= 0.

%C A361535(n) = T(n+2,2) for n >= 0.

%H Paul D. Hanna, <a href="/A361550/b361550.txt">Table of n, a(n) for n = 0..1325</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuintupleProductIdentity.html">Quintuple Product Identity</a>.

%F G.f. A(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^n*y^k satisfies the following.

%F (1) x*y = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x,y)^(3*n) - 1/A(x,y)^(3*n+1)).

%F (2) x*y = Sum_{n=-oo..+oo} x^(n*(3*n-1)/2) * A(x,y)^(3*n) * (x^n - 1/A(x,y)).

%F (3) x*y = Product_{n>=1} (1 - x^n) * (1 - x^n*A(x,y)) * (1 - x^(n-1)/A(x,y)) * (1 - x^(2*n-1)*A(x,y)^2) * (1 - x^(2*n-1)/A(x,y)^2), by the Watson quintuple product identity.

%F (4) Sum_{n>=0} T(n+1,1) * x^n = 1 / Product_{n>=1} (1 - x^n)^3 * (1 - x^(2*n-1))^2, which is the g.f. of A360191.

%F (5) Sum_{n>=0} T(n+2,2) * x^n = 1 / Product_{n>=1} (1 - x^n)^6 * (1 - x^(2*n-1))^4, which is the g.f. of A361535.

%e G.f.: A(x,y) = 1 + y*x + (5*y + y^2)*x^2 + (18*y + 10*y^2 + y^3)*x^3 + (55*y + 61*y^2 + 20*y^3 + y^4)*x^4 + (149*y + 290*y^2 + 215*y^3 + 35*y^4 + y^5)*x^5 + (371*y + 1172*y^2 + 1660*y^3 + 555*y^4 + 56*y^5 + y^6)*x^6 + (867*y + 4212*y^2 + 10311*y^3 + 5850*y^4 + 1254*y^5 + 84*y^6 + y^7)*x^7 + (1923*y + 13833*y^2 + 54688*y^3 + 47460*y^4 + 17773*y^5 + 2555*y^6 + 120*y^7 + y^8)*x^8 + (4086*y + 42262*y^2 + 256815*y^3 + 319409*y^4 + 188300*y^5 + 46844*y^6 + 4810*y^7 + 165*y^8 + y^9)*x^9 + (8374*y + 121625*y^2 + 1093790*y^3 + 1864445*y^4 + 1621116*y^5 + 621915*y^6 + 111348*y^7 + 8505*y^8 + 220*y^9 + y^10)*x^10 + ...

%e This triangle of coefficients T(n,k) of x^n*y^k in g.f. A(x,y) begins:

%e 1;

%e 0, 1;

%e 0, 5, 1;

%e 0, 18, 10, 1;

%e 0, 55, 61, 20, 1;

%e 0, 149, 290, 215, 35, 1;

%e 0, 371, 1172, 1660, 555, 56, 1;

%e 0, 867, 4212, 10311, 5850, 1254, 84, 1;

%e 0, 1923, 13833, 54688, 47460, 17773, 2555, 120, 1;

%e 0, 4086, 42262, 256815, 319409, 188300, 46844, 4810, 165, 1;

%e 0, 8374, 121625, 1093790, 1864445, 1621116, 621915, 111348, 8505, 220, 1;

%e 0, 16634, 332764, 4297370, 9717550, 11913160, 6557572, 1818022, 243795, 14290, 286, 1;

%e 0, 32152, 871641, 15771148, 46148620, 77162284, 58002140, 23152872, 4811721, 499180, 23012, 364, 1;

%e ...

%o (PARI) {T(n,k) = my(A=[1]); for(i=1, n, A = concat(A, 0);

%o A[#A] = polcoeff(x*y - prod(m=1, #A, (1 - x^m) * (1 - x^m*Ser(A)) * (1 - x^(m-1)/Ser(A)) * (1 - x^(2*m-1)*Ser(A)^2) * (1 - x^(2*m-1)/Ser(A)^2) ), #A-1) );

%o polcoeff(polcoeff(Ser(A),n,x),k,y)}

%o for(n=0, 12, for(k=0,n, print1(T(n,k), ", "));print(""))

%Y Cf. A359920 (y=1), A361552 (y=2), A361553 (y=3), A361554 (y=4), A361555 (y=5).

%Y Cf. A360191 (column 1), A361535 (column 2), A361556 (central terms).

%Y Cf. A361050 (related triangle).

%K nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Mar 19 2023