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.

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A222052 a(n) = A222051(n)/binomial(2*n,n), the central terms in rows of triangle A220178 divided by the central binomial coefficients.

Original entry on oeis.org

1, 3, 25, 210, 1881, 17303, 162214, 1540710, 14776281, 142774455, 1387743525, 13553773500, 132906406950, 1307654814222, 12902933709922, 127632756058610, 1265251299930585, 12566655467547195, 125025126985317013, 1245750306517239978, 12429515281592007781
Offset: 0

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Author

Paul D. Hanna, Feb 06 2013

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 25*x^2 + 210*x^3 + 1881*x^4 + 17303*x^5 +...
Illustrate a(n) = [x^n] 1/(sqrt(1-2*x-3*x^2))^(2*n+1):
Let G(x) = 1/sqrt(1-2*x-3*x^2) be the g.f. of A002426, then
the array of coefficients of x^k in G(x)^(2*n+1) begins:
G(x)^1 : [1,  1,   3,    7,    19,    51,    141,     393,...];
G(x)^3 : [1,  3,  12,   40,   135,   441,   1428,    4572,...];
G(x)^5 : [1,  5,  25,  105,   420,  1596,   5880,   21120,...];
G(x)^7 : [1,  7,  42,  210,   966,  4158,  17094,   67782,...];
G(x)^9 : [1,  9,  63,  363,  1881,  9009,  40755,  176319,...];
G(x)^11: [1, 11,  88,  572,  3289, 17303,  85228,  398684,...];
G(x)^13: [1, 13, 117,  845,  5330, 30498, 162214,  814606,...];
G(x)^15: [1, 15, 150, 1190,  8160, 50388, 287470, 1540710,...]; ...
in which the main diagonal forms this sequence.

Crossrefs

Cf. A222050, A222051.

Programs

  • PARI
    {a(n)=polcoeff(1/sqrt(1-2*x-3*x^2+x*O(x^n))^(2*n+1),n)}
for(n=0,25,print1(a(n),", "))

Formula

a(n) = [x^n] 1/(sqrt(1-2*x-3*x^2))^(2*n+1).
a(n) = (2*n+1)*A222050(n), where g.f. G(x) of A222050 satisfies: G(x) = sqrt(1 + 2*x*G(x)^4 + 3*x^2*G(x)^6).

A222050 G.f. satisfies: A(x) = sqrt(1 + 2*x*A(x)^4 + 3*x^2*A(x)^6).

Original entry on oeis.org

1, 1, 5, 30, 209, 1573, 12478, 102714, 869193, 7514445, 66083025, 589294500, 5316256278, 48431659786, 444928748618, 4117185679310, 38340948482745, 359047299072777, 3379057486089649, 31942315551724102, 303158909307122141, 2887629443604011421, 27595392738011189028
Offset: 0

Views

Author

Paul D. Hanna, Feb 06 2013

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 30*x^3 + 209*x^4 + 1573*x^5 + 12478*x^6 +...
Related expansions.
A(x)^2 = 1 + 2*x + 11*x^2 + 70*x^3 + 503*x^4 + 3864*x^5 + 31092*x^6 +...
A(x)^4 = 1 + 4*x + 26*x^2 + 184*x^3 + 1407*x^4 + 11280*x^5 + 93606*x^6 +...
A(x)^6 = 1 + 6*x + 45*x^2 + 350*x^3 + 2844*x^4 + 23814*x^5 + 204149*x^6 +...
where A(x)^2 = 1 + 2*x*A(x)^4 + 3*x^2*A(x)^6.
Let G(x) = 1/sqrt(1-2*x-3*x^2) denote the g.f. of A002426,
then the array of coefficients of x^k in G(x)^(2*n+1) begins:
G(x)^1 : [1,  1,   3,    7,    19,    51,    141,     393,...];
G(x)^3 : [1,  3,  12,   40,   135,   441,   1428,    4572,...];
G(x)^5 : [1,  5,  25,  105,   420,  1596,   5880,   21120,...];
G(x)^7 : [1,  7,  42,  210,   966,  4158,  17094,   67782,...];
G(x)^9 : [1,  9,  63,  363,  1881,  9009,  40755,  176319,...];
G(x)^11: [1, 11,  88,  572,  3289, 17303,  85228,  398684,...];
G(x)^13: [1, 13, 117,  845,  5330, 30498, 162214,  814606,...];
G(x)^15: [1, 15, 150, 1190,  8160, 50388, 287470, 1540710,...]; ...
in which the main diagonal (A222052) forms this sequence like so:
[1/1, 3/3, 25/5, 210/7, 1881/9, 17303/11, 162214/13, 1540710/15,...].

Crossrefs

Cf. A222051, A222052, A002426.

Programs

  • PARI
    {a(n)=polcoeff(sqrt(1/x*serreverse(x*(1-2*x-3*x^2)+x^2*O(x^n))),n)}
for(n=0,25,print1(a(n),", "))
  • PARI
    {a(n)=polcoeff(1/sqrt(1-2*x-3*x^2+x*O(x^n))^(2*n+1),n)/(2*n+1)}
for(n=0,25,print1(a(n),", "))

Formula

G.f.: sqrt( (1/x)*Series_Reversion( x*(1-2*x-3*x^2) ) ).
a(n) = [x^n] sqrt( 1/(1-2*x-3*x^2)^(2*n+1) ) / (2*n+1).
a(n) = A222052(n)/(2*n+1).
Showing 1-2 of 2 results.

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