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 1-5 of 5 results.

A200537 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} A200536(n,k)^2 * x^k] / A(x)^n * x^n/n ), where A200536(n,k) is the coefficient of x^k in (1+3*x+2*x^2)^n.

Original entry on oeis.org

1, 1, 9, 13, 40, 72, 144, 252, 432, 720, 1152, 1872, 2880, 4608, 6912, 10944, 16128, 25344, 36864, 57600, 82944, 129024, 184320, 285696, 405504, 626688, 884736, 1363968, 1916928, 2949120, 4128768, 6340608, 8847360, 13565952, 18874368, 28901376, 40108032, 61341696, 84934656, 129761280
Offset: 0

Views

Author

Paul D. Hanna, Nov 18 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 9*x^2 + 13*x^3 + 40*x^4 + 72*x^5 + 144*x^6 +...
The logarithm of the g.f. A(x) equals the series:
log(A(x)) = (1 + 3^2*x + 2^2*x^2)/A(x) * x +
(1 + 6^2*x + 13^2*x^2 + 12^2*x^3 + 4^2*x^4)/A(x)^2 * x^2/2 +
(1 + 9^2*x + 62^2*x^2 + 63^2*x^3 + 66^2*x^4 + 36^2*x^5 + 8^2*x^6)/A(x)^3 * x^3/3 +
(1 + 12^2*x + 62^2*x^2 + 180^2*x^3 + 321^2*x^4 + 360^2*x^5 + 248^2*x^6 + 96^2*x^7 + 16^2*x^8)/A(x)^4 * x^4/4 +
(1 + 15^2*x + 100^2*x^2 + 390^2*x^3 + 985^2*x^4 + 1683^2*x^5 + 1970^2*x^6 + 1560^2*x^7 + 800^2*x^8 + 240^2*x^9 + 32^2*x^10)/A(x)^5 * x^5/5 +...
which involves the squares of coefficients A200536(n,k) in (1+3*x+2*x^2)^n.

Crossrefs

Cf. A200536, A200535, A199257, A251689.

Programs

  • Mathematica
    CoefficientList[Series[(1+x)*(1+x^2)*(1+4*x^2)*(1+4*x^3) / (1-2*x^2)^2, {x, 0, 40}], x] (* Vaclav Kotesovec, Feb 11 2015 *)
Flatten[{1,1,9,13,40,Table[FullSimplify[9 * 2^(n/2-4) * (-8 - 7*Sqrt[2] + 4*n + 3*Sqrt[2]*n + (-1)^(n+1)*(8 - 7*Sqrt[2] + (-4 + 3*Sqrt[2])*n))],{n,5,40}]}] (* Vaclav Kotesovec, Feb 11 2015 *)
  • PARI
    {a(n)=polcoeff((1+x)*(1+x^2)*(1+4*x^2)*(1+4*x^3) / ((1-2*x^2)^2+x*O(x^n)),n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1, n, sum(k=0, n, polcoeff((1+3*x+2*x^2+x*O(x^k))^m, k)^2 *x^k) *x^m/(A+x*O(x^n))^m/m)+x*O(x^n)));polcoeff(A, n)}

Formula

G.f.: (1+x)*(1+x^2)*(1+4*x^2)*(1+4*x^3) / (1-2*x^2)^2.

A199248 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} A027907(n,k)^2 * x^k * A(x)^k]* x^n/n ), where A027907 is the triangle of trinomial coefficients.

Original entry on oeis.org

1, 1, 2, 6, 20, 69, 248, 923, 3523, 13706, 54152, 216710, 876607, 3578405, 14722432, 60986158, 254145337, 1064712328, 4481577078, 18943753140, 80381689202, 342254333393, 1461864544896, 6262021627055, 26894816382199, 115792035533779, 499648608539714, 2160504474956390
Offset: 0

Views

Author

Paul D. Hanna, Nov 04 2011

Keywords

Comments

Trinomial coefficients satisfy: Sum_{k=0..2*n} A027907(n,k)*x^k = (1+x+x^2)^n.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 20*x^4 + 69*x^5 + 248*x^6 + 923*x^7 +...
such that A(x) = G(x*A(x)) where G(x) is given by:
G(x) = (1 - x + x^2)*(1 - x^2 + x^4)/(1-x)^2 = (1-x^5)/(1-x) + x^3/(1-x)^2:
G(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 6*x^8 + 7*x^9 +...
...
Let A = x*A(x), then the logarithm of the g.f. A(x) equals the series:
log(A(x)) = (1 + A + A^2)*x +
(1 + 2^2*A + 3^2*A^2 + 2^2*A^3 + A^4)*x^2/2 +
(1 + 3^2*A + 6^2*A^2 + 7^2*A^3 + 6^2*A^4 + 3^2*A^5 + A^6)*x^3/3 +
(1 + 4^2*A + 10^2*A^2 + 16^2*A^3 + 19^2*A^4 + 16^2*A^5 + 10^2*A^6 + 4^2*A^7 + A^8)*x^4/4 +
(1 + 5^2*A + 15^2*A^2 + 30^2*A^3 + 45^2*A^4 + 51^2*A^5 + 45^2*A^6 + 30^2*A^7 + 15^2*A^8 + 5^2*A^9 + A^10)*x^5/5 +...
which involves the squares of the trinomial coefficients A027907(n,k).

Crossrefs

Cf. A186236, A199257, A027907.

Programs

  • PARI
    {a(n)=local(A=1+x); A=1/x*serreverse(x*(1-x)*(1-x^3)*(1-x^4)/(1-x^12+x*O(x^n))); polcoeff(A, n)}
  • PARI
    /* G.f. A(x) using the squares of the trinomial coefficients */
{A027907(n, k)=polcoeff((1+x+x^2)^n, k)}
{a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1, n, sum(k=0, 2*m, A027907(m, k)^2 *x^k*A^k) *x^m/m)+x*O(x^n)));polcoeff(A, n)}

Formula

G.f. satisfies: A(x) = G(x*A(x)) where A(x/G(x)) = G(x) = (1 - x + x^2)*(1 - x^2 + x^4)/(1-x)^2.
G.f.: A(x) = (1/x)*Series_Reversion( x*(1-x)*(1-x^3)*(1-x^4)/(1-x^12) ).

A200475 G.f. satisfies: A(x) = exp( Sum_{n>=1} (Sum_{k=0..2*n} A027907(n,k)^2 * x^k * A(x)^(2*k)) * x^n*A(x)^n/n ), where A027907 is the triangle of trinomial coefficients.

Original entry on oeis.org

1, 1, 3, 13, 65, 350, 1981, 11627, 70132, 432090, 2707595, 17202779, 110563543, 717547090, 4695774335, 30952628861, 205318395288, 1369539030021, 9180527051187, 61813112864984, 417850301293691, 2834802846097200, 19294989810689802, 131723105933867817, 901709774424393614
Offset: 0

Views

Author

Paul D. Hanna, Nov 18 2011

Keywords

Comments

Trinomial coefficients satisfy: Sum_{k=0..2*n} A027907(n,k)*x^k = (1+x+x^2)^n.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 65*x^4 + 350*x^5 + 1981*x^6 +...
Let A = g.f. A(x), then the logarithm of the g.f. equals the series:
log(A(x)) = (1 + x*A^2 + x^2*A^4)*x*A +
(1 + 2^2*x*A^2 + 3^2*x^2*A^4 + 2^2*x^3*A^6 + x^4*A^8)*x^2*A^2/2 +
(1 + 3^2*x*A^2 + 6^2*x^2*A^4 + 7^2*x^3*A^6 + 6^2*x^4*A^8 + 3^2*x^5*A^10 + x^6*A^12)*x^3*A^3/3 +
(1 + 4^2*x*A^2 + 10^2*x^2*A^4 + 16^2*x^3*A^6 + 19^2*x^4*A^8 + 16^2*x^5*A^10 + 10^2*x^6*A^12 + 4^2*x^7*A^14 + x^8*A^16)*x^4*A^4/4 +
(1 + 5^2*x*A^2 + 15^2*x^2*A^4 + 30^2*x^3*A^6 + 45^2*x^4*A^8 + 51^2*x^5*A^10 + 45^2*x^6*A^12 + 30^2*x^7*A^14 + 15^2*x^8*A^16 + 5^2*x^9*A^18 + x^10*A^20)*x^5*A^5/5 +...
which involves the squares of the trinomial coefficients A027907(n,k).

Crossrefs

Cf. A200377, A199248, A186236, A199257, A168592, A027907.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=(1-x*A^2+x^3*A^6-x^5*A^10+x^6*A^12)/(1-x*A^2+x*O(x^n))^2);polcoeff(A,n)}
  • PARI
    /* G.f. A(x) using the squares of the trinomial coefficients */
{A027907(n, k)=polcoeff((1+x+x^2)^n, k)}
{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, 2*m, A027907(m, k)^2 *x^k*(A+x*O(x^n))^(2*k))*x^m*A^m/m))); polcoeff(A, n)}

Formula

G.f. satisfies: A(x) = (1 + x^3*A(x)^6)*(1 + x^6*A(x)^12)/((1 - x*A(x)^2)*(1 - x^4*A(x)^8)).

A200377 G.f.: A(x) = exp( Sum_{n>=1} (Sum_{k=0..2*n} A027907(n,k)^2 * x^k / A(x)^k) * x^n/n ).

Original entry on oeis.org

1, 1, 2, 4, 7, 11, 19, 34, 61, 106, 181, 311, 543, 955, 1668, 2885, 4980, 8650, 15114, 26391, 45845, 79385, 137718, 239866, 418338, 727926, 1263097, 2191463, 3810775, 6638258, 11556361, 20078960, 34855400, 60567092, 105405431, 183483906, 319039355, 554158992, 962743619, 1674359119, 2913758685, 5068194691
Offset: 0

Views

Author

Paul D. Hanna, Nov 17 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 7*x^4 + 11*x^5 + 19*x^6 + 34*x^7 +...
Let A = g.f. A(x), then the logarithm of the g.f. equals the series:
log(A(x)) = (1 + x/A + x^2/A^2)*x +
(1 + 2^2*x/A + 3^2*x^2/A^2 + 2^2*x^3/A^3 + x^4/A^4)*x^2/2 +
(1 + 3^2*x/A + 6^2*x^2/A^2 + 7^2*x^3/A^3 + 6^2*x^4/A^4 + 3^2*x^5/A^5 + x^6/A^6)*x^3/3 +
(1 + 4^2*x/A + 10^2*x^2/A^2 + 16^2*x^3/A^3 + 19^2*x^4/A^4 + 16^2*x^5/A^5 + 10^2*x^6/A^6 + 4^2*x^7/A^7 + x^8/A^8)*x^4/4 +
(1 + 5^2*x/A + 15^2*x^2/A^2 + 30^2*x^3/A^3 + 45^2*x^4/A^4 + 51^2*x^5/A^5 + 45^2*x^6/A^6 + 30^2*x^7/A^7 + 15^2*x^8/A^8 + 5^2*x^9/A^9 + x^10/A^10)*x^5/5 +...
which involves the squares of the trinomial coefficients A027907(n,k).

Crossrefs

Cf. A199248, A186236, A199257, A168592, A027907.

Programs

  • PARI
    /* G.f. A(x) using the squares of the trinomial coefficients */
{A027907(n, k)=polcoeff((1+x+x^2)^n, k)}
{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, 2*m, A027907(m, k)^2 *x^k/(A+x*O(x^n))^k) *x^m/m))); polcoeff(A, n)}

A200535 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} C(2*n,k)^2 * x^k] / A(x)^n * x^n/n ).

Original entry on oeis.org

1, 1, 4, 5, 9, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232, 236, 240, 244, 248, 252, 256, 260
Offset: 0

Views

Author

Paul D. Hanna, Nov 18 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 5*x^3 + 9*x^4 + 12*x^5 + 16*x^6 + 20*x^7 +...
where A(x) = G(x/A(x)) and G(x) = A(x*G(x)) is the g.f. of A199257:
G(x) = 1 + x + 5*x^2 + 18*x^3 + 86*x^4 + 408*x^5 + 2075*x^6 +...
...
The logarithm of the g.f. A(x) equals the series:
log(A(x)) = (1 + 2^2*x + x^2)/A(x) * x +
(1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)/A(x)^2 * x^2/2 +
(1 + 6^2*x + 15^2*x^2 + 20^2*x^3 + 15^2*x^4 + 6^2*x^5 + x^6)/A(x)^3 * x^3/3 +
(1 + 8^2*x + 28^2*x^2 + 56^2*x^3 + 70^2*x^4 + 56^2*x^5 + 28^2*x^6 + 8^2*x^7 + x^8)/A(x)^4 * x^4/4 +
(1 + 10^2*x + 45^2*x^2 + 120^2*x^3 + 210^2*x^4 + 252^2*x^5 + 210^2*x^6 + 120^2*x^7 + 45^2*x^8 + 10^2*x^9 + x^10)/A(x)^5 * x^5/5 +...
which involves the squares of binomial coefficients C(2*n,k).

Links

Crossrefs

Cf. A199257.

Programs

  • PARI
    {a(n)=polcoeff((1+x^2)^2*(1+x^3)/((1-x)*(1-x^2) +x*O(x^n)),n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1, n, sum(k=0, n, binomial(2*m, k)^2 *x^k)/(A+x*O(x^n))^m *x^m/m)));polcoeff(A, n)}

Formula

G.f.: (1+x^2)^2*(1+x^3)/((1-x)*(1-x^2)).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.