A361550 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.

1, 0, 1, 0, 5, 1, 0, 18, 10, 1, 0, 55, 61, 20, 1, 0, 149, 290, 215, 35, 1, 0, 371, 1172, 1660, 555, 56, 1, 0, 867, 4212, 10311, 5850, 1254, 84, 1, 0, 1923, 13833, 54688, 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

Offset

0,5

Comments

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

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

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

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

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

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

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

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

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1325

Eric Weisstein's World of Mathematics, Quintuple Product Identity.

Formula

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

(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)).

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

(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.

(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.

(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.

Example

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 + ...
This triangle of coefficients T(n,k) of x^n*y^k in g.f. A(x,y) begins:
1;
0, 1;
0, 5, 1;
0, 18, 10, 1;
0, 55, 61, 20, 1;
0, 149, 290, 215, 35, 1;
0, 371, 1172, 1660, 555, 56, 1;
0, 867, 4212, 10311, 5850, 1254, 84, 1;
0, 1923, 13833, 54688, 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;
0, 32152, 871641, 15771148, 46148620, 77162284, 58002140, 23152872, 4811721, 499180, 23012, 364, 1;
...

Prog

(PARI) {T(n, k) = my(A=[1]); for(i=1, n, A = concat(A, 0);
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) );
polcoeff(polcoeff(Ser(A), n, x), k, y)}
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))

Crossrefs

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

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

Cf. A361050 (related triangle).

Sequence in context: A200420 A176324 A081817 * A123260 A019113 A289620

Adjacent sequences: A361547 A361548 A361549 * A361551 A361552 A361553

Keyword

nonn,tabl

Author

_Paul D. Hanna_, Mar 19 2023

Status

approved