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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.

A355341 G.f.: A(x) = Sum_{n=-oo..+oo} x^(n*(n+1)/2) * C(x)^n, where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

Original entry on oeis.org

2, 1, -2, 1, -3, 0, 1, -4, 2, 0, 1, -5, 5, 0, 0, 1, -6, 9, -2, 0, 0, 1, -7, 14, -7, 0, 0, 0, 1, -8, 20, -16, 2, 0, 0, 0, 1, -9, 27, -30, 9, 0, 0, 0, 0, 1, -10, 35, -50, 25, -2, 0, 0, 0, 0, 1, -11, 44, -77, 55, -11, 0, 0, 0, 0, 0, 1, -12, 54, -112, 105, -36, 2, 0, 0, 0, 0, 0, 1, -13, 65, -156, 182, -91, 13, 0, 0, 0, 0, 0, 0, 1, -14, 77, -210, 294, -196, 49, -2, 0, 0, 0, 0, 0, 0, 1, -15, 90, -275, 450, -378, 140, -15, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Paul D. Hanna, Jul 21 2022

Keywords

Examples

			G.f.: A(x) = 2 + x - 2*x^2 + x^3 - 3*x^4 + x^6 - 4*x^7 + 2*x^8 + x^10 - 5*x^11 + 5*x^12 + x^15 - 6*x^16 + 9*x^17 - 2*x^18 + x^21 - 7*x^22 + 14*x^23 - 7*x^24 + x^28 - 8*x^29 + 20*x^30 - 16*x^31 + 2*x^32 + x^36 - 9*x^37 + 27*x^38 - 30*x^39 + 9*x^40 + x^45 - 10*x^46 + 35*x^47 - 50*x^48 + 25*x^49 - 2*x^50 + ...
such that
A(x) = ... + x^6/C(x)^4 + x^3/C(x)^3 + x/C(x)^2 + 1/C(x) + 1 + x*C(x) + x^3*C(x)^2 + x^6*C(x)^3 + x^10*C(x)^4 + ...
where
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + 1430*x^8 + 4862*x^9 + 16796*x^10 + ... + A000108(n)*x^n + ...
The coefficients of x^k in x^(n*(n+1)/2) * (C(x)^n + 1/C(x)^(n+1)) begin:
n = 0: [2, -1, -1, -2, -5, -14, -42, -132, -429, -1430, -4862, -16796, ...];
n = 1: [0,  2, -1,  1,  3,   9,  28,   90,  297,  1001,  3432,  11934, ...];
n = 2: [0,  0,  0,  2, -1,   5,  13,   39,  123,   401,  1340,   4565, ...];
n = 3: [0,  0,  0,  0,  0,   0,   2,   -1,   11,    28,    89,    293, ...];
n = 4: [0,  0,  0,  0,  0,   0,   0,    0,    0,     0,     2,     -1, ...]; ...
forming a table the column sums of which yield this sequence.
The g.f. may also be written as
A(x) = 2 + (-2*x + 1)*x + (-3*x + 1)*x^3 + (2*x^2 - 4*x + 1)*x^6 + (5*x^2 - 5*x + 1)*x^10 + (-2*x^3 + 9*x^2 - 6*x + 1)*x^15 + (-7*x^3 + 14*x^2 - 7*x + 1)*x^21 + (2*x^4 - 16*x^3 + 20*x^2 - 8*x + 1)*x^28 + (9*x^4 - 30*x^3 + 27*x^2 - 9*x + 1)*x^36 + (-2*x^5 + 25*x^4 - 50*x^3 + 35*x^2 - 10*x + 1)*x^45 + ...
compare to
(1 - 2*y*x)/(1-x + y*x^2) = 1 + (-2*y + 1)*x + (-3*y + 1)*x^2 + (2*y^2 - 4*y + 1)*x^3 + (5*y^2 - 5*y + 1)*x^4 + (-2*y^3 + 9*y^2 - 6*y + 1)*x^5 + (-7*y^3 + 14*y^2 - 7*y + 1)*x^6 + (2*y^4 - 16*y^3 + 20*y^2 - 8*y + 1)*x^7 + (9*y^4 - 30*y^3 + 27*y^2 - 9*y + 1)*x^8 + (-2*y^5 + 25*y^4 - 50*y^3 + 35*y^2 - 10*y + 1)*x^9 + ...
The terms of this sequence may be written as a triangle (see triangle A244422):
2,
1, -2,
1, -3, 0,
1, -4, 2, 0,
1, -5, 5, 0, 0,
1, -6, 9, -2, 0, 0,
1, -7, 14, -7, 0, 0, 0,
1, -8, 20, -16, 2, 0, 0, 0,
1, -9, 27, -30, 9, 0, 0, 0, 0,
1, -10, 35, -50, 25, -2, 0, 0, 0, 0,
1, -11, 44, -77, 55, -11, 0, 0, 0, 0, 0,
1, -12, 54, -112, 105, -36, 2, 0, 0, 0, 0, 0,
1, -13, 65, -156, 182, -91, 13, 0, 0, 0, 0, 0, 0,
1, -14, 77, -210, 294, -196, 49, -2, 0, 0, 0, 0, 0, 0,
1, -15, 90, -275, 450, -378, 140, -15, 0, 0, 0, 0, 0, 0, 0,
1, -16, 104, -352, 660, -672, 336, -64, 2, 0, 0, 0, 0, 0, 0, 0,
...

Links

Crossrefs

Cf. A244422, A355342, A355343.

Programs

  • PARI
    {a(n) = my(A,C = serreverse(x-x^2 +x^2*O(x^n))/x);
A = sum(m=-n-1,n+1, x^(m*(m+1)/2) * C^m); polcoeff(A,n)}
for(n=0,70,print1(a(n),", "))
  • PARI
    {a(n) = my(A,C = serreverse(x-x^2 +x^2*O(x^n))/x, M = sqrtint(2*n+9));
A = sum(m=0,M, x^(m*(m+1)/2) * (C^m + 1/C^(m+1))); polcoeff(A,n)}
for(n=0,70,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n is equal to the following expressions; here, C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
(1) A(x) = 1/C(x) * Product_{n>=1} (1 + x^n/C(x)) * (1 + x^(n-1)*C(x)) * (1-x^n), by the Jacobi triple product identity.
(2) A(x) = Sum_{n=-oo..+oo} x^(n*(n+1)/2) * C(x)^n.
(3) A(x) = Sum_{n>=0} x^(n*(n+1)/2) * (C(x)^n + 1/C(x)^(n+1)).
(4) A(x) = 1 + Sum_{n>=0} x^(n*(n+1)/2) * ( [y^n] (1 - 2*y*x)/(1-y + x*y^2) ).
(5) A(x) = 1 + Sum_{n>=1} x^(n*(n-1)/2) * Sum_{k=0..n} A244422(n,k) * x^k.

A162514 Triangle of coefficients of polynomials defined by the Binet form P(n,x) = U^n + L^n, where U = (x + d)/2, L = (x - d)/2, d = (4 + x^2)^(1/2). Decreasing powers of x.

Original entry on oeis.org

2, 1, 0, 1, 0, 2, 1, 0, 3, 0, 1, 0, 4, 0, 2, 1, 0, 5, 0, 5, 0, 1, 0, 6, 0, 9, 0, 2, 1, 0, 7, 0, 14, 0, 7, 0, 1, 0, 8, 0, 20, 0, 16, 0, 2, 1, 0, 9, 0, 27, 0, 30, 0, 9, 0, 1, 0, 10, 0, 35, 0, 50, 0, 25, 0, 2, 1, 0, 11, 0, 44, 0, 77, 0, 55, 0, 11, 0, 1, 0, 12, 0, 54, 0, 112, 0, 105, 0, 36, 0, 2, 1, 0, 13, 0
Offset: 0

Views

Author

Clark Kimberling, Jul 05 2009

Keywords

Comments

For a signed version of this triangle corresponding to the row reversed version of the triangle A127672 see A244422. - Wolfdieter Lang, Aug 07 2014
The row reversed triangle is A114525. - Paolo Bonzini, Jun 23 2016

Examples

			Triangle begins
   2;  == 2
   1, 0;  == x + 0
   1, 0,  2;  == x^2 + 2
   1, 0,  3, 0;  == x^3 + 3*x + 0
   1, 0,  4, 0,  2;
   1, 0,  5, 0,  5, 0;
   1, 0,  6, 0,  9, 0,  2;
   1, 0,  7, 0, 14, 0,  7, 0;
   1, 0,  8, 0, 20, 0, 16, 0,  2;
   1, 0,  9, 0, 27, 0, 30, 0,  9, 0;
   1, 0, 10, 0, 35, 0, 50, 0, 25, 0, 2;
   ...
From _Wolfdieter Lang_, Aug 07 2014: (Start)
The row polynomials R(n, x) are:
  R(0, x) = 2, R(1, x) = 1 =   x*P(1,1/x),  R(2, x) = 1 + 2*x^2 = x^2*P(2,1/x), R(3, x) = 1 + 3*x^2 = x^3*P(3,1/x), ...
(End)

Links

Crossrefs

Cf. A000032, A114525, A162515, A162516, A162517.

Programs

  • Mathematica
    Table[Reverse[CoefficientList[LucasL[n, x], x]], {n, 0, 12}]//Flatten  (* G. C. Greubel, Nov 05 2018 *)
  • PARI
    P(n)=
{
    local(U, L, d, r, x);
    if ( n<0, return(0) );
    x = 'x+O('x^(n+1));
    d=(4 + x^2)^(1/2);
    U=(x+d)/2;  L=(x-d)/2;
    r = U^n+L^n;
    r = truncate(r);
    return( r );
}
for (n=0, 10, print(Vec(P(n))) ); /* show triangle */
/* Joerg Arndt, Jul 24 2011 */

Formula

P(n,x) = x*P(n-1,x) + P(n-2,x) for n >= 2, P(0,x) = 2, P(1,x) = x.
From Wolfdieter Lang, Aug 07 2014: (Start)
T(n,m) = [x^(n-m)] P(n,x), m = 0, 1, ..., n and n >= 0.
G.f. of polynomials P(n,x): (2 - x*z)/(1 - x*z - z^2).
G.f. of row polynomials R(n,x) = Sum_{m=0..n} T(n,m)*x^m: (2 - z)/(1 - z - (x*z)^2) (rows for P(n,x) reversed).
(End)
For n > 0, T(n,2*m+1) = 0, T(n,2*m) = A034807(n,m). - Paolo Bonzini, Jun 23 2016

Extensions

Name clarified by Wolfdieter Lang, Aug 07 2014

A355342 G.f.: A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * C(x)^n, where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

Original entry on oeis.org

0, 1, -2, -1, 3, 0, 1, -4, 2, 0, -1, 5, -5, 0, 0, 1, -6, 9, -2, 0, 0, -1, 7, -14, 7, 0, 0, 0, 1, -8, 20, -16, 2, 0, 0, 0, -1, 9, -27, 30, -9, 0, 0, 0, 0, 1, -10, 35, -50, 25, -2, 0, 0, 0, 0, -1, 11, -44, 77, -55, 11, 0, 0, 0, 0, 0, 1, -12, 54, -112, 105, -36, 2, 0, 0, 0, 0, 0, -1, 13, -65, 156, -182, 91, -13, 0, 0, 0, 0, 0, 0, 1, -14, 77, -210, 294, -196, 49, -2, 0, 0, 0, 0, 0, 0, -1, 15, -90, 275, -450, 378, -140, 15, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Paul D. Hanna, Jul 22 2022

Keywords

Examples

			G.f.: A(x) = x - 2*x^2 - x^3 + 3*x^4 + x^6 - 4*x^7 + 2*x^8 - x^10 + 5*x^11 - 5*x^12 + x^15 - 6*x^16 + 9*x^17 - 2*x^18 - x^21 + 7*x^22 - 14*x^23 + 7*x^24 + x^28 - 8*x^29 + 20*x^30 - 16*x^31 + 2*x^32 - x^36 + 9*x^37 - 27*x^38 + 30*x^39 - 9*x^40 + x^45 - 10*x^46 + 35*x^47 - 50*x^48 + 25*x^49 - 2*x^50 + ...
such that
A(x) = ... + x^6/C(x)^4 - x^3/C(x)^3 + x/C(x)^2 - 1/C(x) + 1 - x*C(x) + x^3*C(x)^2 - x^6*C(x)^3 + x^10*C(x)^4 +- ...
where
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + 1430*x^8 + 4862*x^9 + 16796*x^10 + ... + A000108(n)*x^n + ...
The coefficients of x^k in (-1)^n * x^(n*(n+1)/2) * (C(x)^n - 1/C(x)^(n+1)) begin:
n = 0: [0, 1,  1,  2,  5,  14,  42,  132,  429,  1430,  4862,  16796, ...];
n = 1: [0, 0, -3, -3, -7, -19, -56, -174, -561, -1859, -6292, -21658, ...];
n = 2: [0, 0,  0,  0,  5,   5,  15,   45,  141,   457,  1520,   5159, ...];
n = 3: [0, 0,  0,  0,  0,   0,   0,   -7,   -7,   -28,   -91,   -301,  ...];
n = 4: [0, 0,  0,  0,  0,   0,   0,    0,    0,     0,     0,      9, ...]; ...
forming a table the column sums of which yield this sequence.
The g.f. may also be written as
A(x) = 0 + (-2*x + 1)*x - (-3*x + 1)*x^3 + (2*x^2 - 4*x + 1)*x^6 - (5*x^2 - 5*x + 1)*x^10 + (-2*x^3 + 9*x^2 - 6*x + 1)*x^15 - (-7*x^3 + 14*x^2 - 7*x + 1)*x^21 + (2*x^4 - 16*x^3 + 20*x^2 - 8*x + 1)*x^28 - (9*x^4 - 30*x^3 + 27*x^2 - 9*x + 1)*x^36 + (-2*x^5 + 25*x^4 - 50*x^3 + 35*x^2 - 10*x + 1)*x^45 + ...
compare to
(1 + 2*y*x)/(1+x + y*x^2) = 1 - (-2*y + 1)*x + (-3*y + 1)*x^2 - (2*y^2 - 4*y + 1)*x^3 + (5*y^2 - 5*y + 1)*x^4 - (-2*y^3 + 9*y^2 - 6*y + 1)*x^5 + (-7*y^3 + 14*y^2 - 7*y + 1)*x^6 - (2*y^4 - 16*y^3 + 20*y^2 - 8*y + 1)*x^7 + (9*y^4 - 30*y^3 + 27*y^2 - 9*y + 1)*x^8 - (-2*y^5 + 25*y^4 - 50*y^3 + 35*y^2 - 10*y + 1)*x^9 + ...
The terms of this sequence may be written as a triangle:
0,
1, -2,
-1, 3, 0,
1, -4, 2, 0,
-1, 5, -5, 0, 0,
1, -6, 9, -2, 0, 0,
-1, 7, -14, 7, 0, 0, 0,
1, -8, 20, -16, 2, 0, 0, 0,
-1, 9, -27, 30, -9, 0, 0, 0, 0,
1, -10, 35, -50, 25, -2, 0, 0, 0, 0,
-1, 11, -44, 77, -55, 11, 0, 0, 0, 0, 0,
1, -12, 54, -112, 105, -36, 2, 0, 0, 0, 0, 0,
-1, 13, -65, 156, -182, 91, -13, 0, 0, 0, 0, 0, 0,
1, -14, 77, -210, 294, -196, 49, -2, 0, 0, 0, 0, 0, 0,
-1, 15, -90, 275, -450, 378, -140, 15, 0, 0, 0, 0, 0, 0, 0,
1, -16, 104, -352, 660, -672, 336, -64, 2, 0, 0, 0, 0, 0, 0, 0,
...

Links

Crossrefs

Cf. A244422, A355341, A355343.
Cf. A034807.

Programs

  • PARI
    {a(n) = my(A,C = serreverse(x-x^2 +x^2*O(x^n))/x);
A = sum(m=-n-1,n+1, (-1)^m * x^(m*(m+1)/2) * C^m); polcoeff(A,n)}
for(n=0,70,print1(a(n),", "))
  • PARI
    {a(n) = my(A,C = serreverse(x-x^2 +x^2*O(x^n))/x, M = sqrtint(2*n+9));
A = sum(m=0,M, (-1)^m * x^(m*(m+1)/2) * (C^m - 1/C^(m+1))); polcoeff(A,n)}
for(n=0,70,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n is equal to the following expressions; here, C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
(1) A(x) = -1/C(x) * Product_{n>=1} (1 - x^n/C(x)) * (1 - x^(n-1)*C(x)) * (1-x^n), by the Jacobi triple product identity.
(2) A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * C(x)^n.
(3) A(x) = Sum_{n>=0} (-1)^n * x^(n*(n+1)/2) * (C(x)^n - 1/C(x)^(n+1)).
(4) A(x) = 1 - Sum_{n>=0} x^(n*(n+1)/2) * ( [y^n] (1 + 2*y*x)/(1+x + y*x^2) ).
(5) A(x) = 1 - Sum_{n>=1} (-1)^n * x^(n*(n-1)/2) * Sum_{k=0..n} A244422(n,k) * x^k.
Showing 1-3 of 3 results.

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