cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

A361050 Expansion of g.f. A(x,y) satisfying y/x = 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.

Original entry on oeis.org

1, 0, 1, 0, 5, 4, 0, 18, 40, 22, 0, 55, 244, 335, 140, 0, 149, 1160, 2924, 2875, 969, 0, 371, 4688, 19090, 32745, 25081, 7084, 0, 867, 16848, 103110, 272250, 352814, 221397, 53820, 0, 1923, 55332, 485356, 1839075, 3565548, 3709244, 1971775, 420732, 0, 4086, 169048, 2054520, 10674985, 28909300, 44146487, 38344384, 17682895, 3362260
Offset: 1

Views

Author

Paul D. Hanna, Mar 18 2023

Keywords

Comments

A359921(n) = Sum_{k=0..n-1} T(n,k) for n >= 1.
A359924(n) = Sum_{k=0..n-1} T(n,k) * 2^k for n >= 1.
A361051(n) = Sum_{k=0..n-1} T(n,k) * 3^k for n >= 1.
A361052(n) = Sum_{k=0..n-1} T(n,k) * 4^k for n >= 1.
A361538(n) = T(2*n-1,n-1) for n >= 1.
A360191(n) = T(n+2,1) for n >= 0.
A361535(n) = T(n+3,2)/4 for n >= 0.
A002293(n) = T(n+1,n) for n >= 0.

Examples

			G.f.: A(x,y) = x + y*x^2 + (5*y + 4*y^2)*x^3 + (18*y + 40*y^2 + 22*y^3)*x^4 + (55*y + 244*y^2 + 335*y^3 + 140*y^4)*x^5 + (149*y + 1160*y^2 + 2924*y^3 + 2875*y^4 + 969*y^5)*x^6 + (371*y + 4688*y^2 + 19090*y^3 + 32745*y^4 + 25081*y^5 + 7084*y^6)*x^7 + (867*y + 16848*y^2 + 103110*y^3 + 272250*y^4 + 352814*y^5 + 221397*y^6 + 53820*y^7)*x^8 + (1923*y + 55332*y^2 + 485356*y^3 + 1839075*y^4 + 3565548*y^5 + 3709244*y^6 + 1971775*y^7 + 420732*y^8)*x^9 + (4086*y + 169048*y^2 + 2054520*y^3 + 10674985*y^4 + 28909300*y^5 + 44146487*y^6 + 38344384*y^7 + 17682895*y^8 + 3362260*y^9)*x^10 + ...
This triangle of coefficients T(n,k) of x^n*y^k, n >= 1, k = 0..n-1, in g.f. A(x,y) begins:
1;
0, 1;
0, 5, 4;
0, 18, 40, 22;
0, 55, 244, 335, 140;
0, 149, 1160, 2924, 2875, 969;
0, 371, 4688, 19090, 32745, 25081, 7084;
0, 867, 16848, 103110, 272250, 352814, 221397, 53820;
0, 1923, 55332, 485356, 1839075, 3565548, 3709244, 1971775, 420732;
0, 4086, 169048, 2054520, 10674985, 28909300, 44146487, 38344384, 17682895, 3362260;
0, 8374, 486500, 7984667, 55085875, 199363606, 417661860, 525322468, 391561335, 159463876, 27343888;
0, 16634, 1331056, 28909580, 258486830, 1211896230, 3335033317, 5680806120, 6069336891, 3961602925, 1444601027, 225568798;
...

Links

Crossrefs

Cf. A360191 (column 1), A361535 (column 2), A002293 (diagonal), A361538 (central terms).
Cf. A359921 (y=1), A359924 (y=2), A361051 (y=3), A361052 (y=4).
Cf. A002293, A356500 (related table), A361550 (related triangle).

Programs

  • PARI
    {T(n,k) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(y/x - 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-4) );
polcoeff(polcoeff(H=Ser(A),n,x),k,y)}
for(n=1, 12, for(k=0,n-1, print1(T(n,k), ", "));print(""))

Formula

G.f. A(x,y) = Sum_{n>=1} Sum_{k=0..n-1} T(n,k)*x^n*y^k satisfies the following.
(1) y/x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x,y)^(3*n) - 1/A(x,y)^(3*n+1)).
(2) y/x = 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.
(3) Sum_{n>=0} T(n+2,1) * x^n = 1 / Product_{n>=1} (1 - x^n)^3 * (1 - x^(2*n-1))^2, which is the g.f. of A360191.
(4) Sum_{n>=0} T(n+3,2) * x^n = 4*F(x) where F(x) = 1/Product_{n>=1} (1 - x^n)^6 * (1 - x^(2*n-1))^4, which is the g.f. of A361535.
(5) Sum_{n>=0} T(n+1,n) * x^n = D(x) where D(x) = 1 + x*D(x)^4 is the g.f. of A002293.
(6) T(n+1,n) = binomial(4*n, n)/(3*n + 1) for n >= 0.

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.

Original entry on oeis.org

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

Views

Author

Paul D. Hanna, Mar 19 2023

Keywords

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.

Examples

			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;
...

Links

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).

Programs

  • 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(""))

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.

A361535 Expansion of g.f. 1 / Product_{n>=1} ((1 - x^n)^6 * (1 - x^(2*n-1))^4).

Original entry on oeis.org

1, 10, 61, 290, 1172, 4212, 13833, 42262, 121625, 332764, 871641, 2197936, 5359005, 12679730, 29200593, 65617892, 144189054, 310400110, 655669910, 1360910666, 2779007594, 5589070978, 11081585154, 21679798590, 41883282555, 79958881544, 150943109191, 281926365224
Offset: 0

Views

Author

Paul D. Hanna, Mar 18 2023

Keywords

Examples

			G.f.: A(x) = 1 + 10*x + 61*x^2 + 290*x^3 + 1172*x^4 + 4212*x^5 + 13833*x^6 + 42262*x^7 + 121625*x^8 + 332764*x^9 + 871641*x^10 + ...
A related series begins
A(x)^(1/2) = 1 + 5*x + 18*x^2 + 55*x^3 + 149*x^4 + 371*x^5 + 867*x^6 + 1923*x^7 + 4086*x^8 + 8374*x^9 + ... + A360191(n)*x^n + ...

Links

Crossrefs

Cf. A360191, A361050, A361550.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[1/((1 - x^k)^6 * (1 - x^(2*k-1))^4), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 19 2023 *)
  • PARI
    {a(n) = polcoeff( 1/prod(m=1,n, (1 - x^m)^6 * (1 - x^(2*m-1))^4 + x*O(x^n)), n)}
for(n=0,30,print1(a(n),", "))

Formula

a(n) ~ exp(4*Pi*sqrt(n/3)) / (2^(5/2) * 3^(7/4) * n^(9/4)). - Vaclav Kotesovec, Mar 19 2023
Showing 1-3 of 3 results.

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