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 1-4 of 4 results.

A120020 Coefficients of x^n in the n-th iteration of the g.f. of A120010: a(n) = [x^n] { (1-sqrt(1-4*x))/2 o x/(1-n*x) o (x-x^2) } for n>=1.

Original entry on oeis.org

1, 2, 9, 68, 710, 9348, 148085, 2740672, 58033953, 1383923040, 36705564368, 1071911496576, 34179790156473, 1181725089179936, 44035415728886145, 1759481180119564288, 75042973200676887772, 3402984761691650083008
Offset: 1

Views

Author

Paul D. Hanna, Jun 14 2006

Keywords

Comments

Main diagonal of A120019, the table of self-compositions of A120010.
For n>=1, n divides a(n): a(n)/n = A120022(n).

Examples

			Successive iterations of F(x), the g.f. of A120010, begin:
F(x) = (1)x + x^2 + x^3 + 2x^4 + 6x^5 + 18x^6 + 53x^7 + 158x^8 +...
F(F(x)) = x + (2)x^2 + 4x^3 + 10x^4 + 32x^5 + 116x^6 + 440x^7 +...
F(F(F(x))) = x + 3x^2 + (9)x^3 + 30x^4 + 114x^5 + 480x^6 + 2157x^7 +...
F(F(F(F(x)))) = x + 4x^2 + 16x^3 + (68)x^4 + 312x^5 + 1536x^6 +...
F(F(F(F(F(x))))) = x + 5x^2 + 25x^3 + 130x^4 + (710)x^5 + 4070x^6 +...
F(F(F(F(F(F(x)))))) = x + 6x^2 + 36x^3 + 222x^4 + 1416x^5 + (9348)x^6+..

Crossrefs

Cf. A120010, A120019, A120021, A120022.

Programs

  • PARI
    {a(n)=sum(j=1, n, binomial(2*n-2*j, n-j)/(n-j+1)* sum(i=1, j,(-1)^(j-i)*binomial(n-j+i, j-i)*binomial(n-j+i-1, i-1)*n^(i-1)))}
for(n=1,25,print1(a(n),", "))

Formula

a(n) = Sum_{j=1..n} Catalan(n-j) * [ Sum_{i=1..j} (-1)^(j-i) * n^(i-1) * C(n-j+i, j-i) * C(n-j+i-1, i-1) ];
a(n) = Sum_{j=0..n-1} n^j * [ Sum_{i=j..n-1} (-1)^(i-j) * Catalan(n-i-1) * C(n-i+j, i-j) * C(n-i+j-1, j) ], where Catalan(n) = A000108(n) = C(2n, n)/(n+1).

A120018 The third self-composition of A120010; g.f.: A(x) = G(G(G(x))), where G(x) = g.f. of A120010.

Original entry on oeis.org

1, 3, 9, 30, 114, 480, 2157, 10092, 48525, 238143, 1187952, 6006171, 30710553, 158535975, 825143145, 4325320191, 22814398392, 120999555588, 644878190175, 3451975941243, 18550877091063, 100047282676491, 541314936448764
Offset: 1

Views

Author

Paul D. Hanna, Jun 14 2006

Keywords

Comments

Row 3 of A120019, the square table of self-compositions of A120010.

Examples

			A(x) = x + 3*x^2 + 9*x^3 + 30*x^4 + 114*x^5 + 480*x^6 + 2157*x^7 +...
G(x) = x + x^2 + x^3 + 2*x^4 + 6*x^5 + 18*x^6 + 53*x^7 + 158*x^8 +...
where G(x) is the g.f. of A120010 and G(G(G(x))) = A(x).

Links

Crossrefs

Cf. A120010, A120017 (2nd self-composition), A120019.

Programs

  • Mathematica
    CoefficientList[Series[(1 - Sqrt[1 - 4 x (1-x) / (1 -3 x + 3 x^2)]) / x / 2,  {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *)
  • PARI
    {a(n)=polcoeff((1 - sqrt(1 - 4*x*(1-x)/(1-3*x+3*x^2+x*O(x^n)) ))/2, n)}

Formula

G.f.: A(x) = (1 - sqrt(1 - 4*x*(1-x)/(1-3*x+3*x^2) ))/2.
Recurrence: n*a(n) = 2*(5*n-6)*a(n-1) - (31*n-66)*a(n-2) + 42*(n-3)*a(n-3) - 21*(n-4)*a(n-4). - Vaclav Kotesovec, Oct 24 2012
a(n) ~ sqrt(14*sqrt(21)-42)*((7+sqrt(21))/2)^n/(16*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 24 2012

Extensions

Typo in Mma program fixed by Vincenzo Librandi, May 22 2013

A120021 Coefficients of x^n in the (n+1)-th self-composition of the g.f. of A120010: a(n) = [x^n] { (1-sqrt(1-4*x))/2 o x/(1-(n+1)*x) o (x-x^2) } for n>=1.

Original entry on oeis.org

1, 3, 16, 130, 1416, 19236, 312512, 5906502, 127313320, 3082645951, 82848394752, 2447576485341, 78846484722208, 2750891289611235, 103344880800464896, 4159577854374314795, 178587276548655542112, 8147334149686335230068
Offset: 1

Views

Author

Paul D. Hanna, Jun 14 2006

Keywords

Comments

Secondary diagonal of A120019, the table of self-compositions of A120010.

Examples

			Successive self-compositions of F(x), the g.f. of A120010, begin:
F(x) = x + x^2 + x^3 + 2x^4 + 6x^5 + 18x^6 + 53x^7 + 158x^8 +...
F(F(x)) = (1)x + 2x^2 + 4x^3 + 10x^4 + 32x^5 + 116x^6 + 440x^7 +...
F(F(F(x))) = x + (3)x^2 + 9x^3 + 30x^4 + 114x^5 + 480x^6 + 2157x^7 +...
F(F(F(F(x)))) = x + 4x^2 + (16)x^3 + 68x^4 + 312x^5 + 1536x^6 +...
F(F(F(F(F(x))))) = x + 5x^2 + 25x^3 + (130)x^4 + 710x^5 + 4070x^6 +...
F(F(F(F(F(F(x)))))) = x + 6x^2 + 36x^3 + 222x^4 + (1416)x^5 + 9348x^6+..

Crossrefs

Cf. A120010, A120019, A120020.

Programs

  • PARI
    a(n)=polcoeff((1-sqrt(1-4*x*(1-x)/(1-(n+1)*x*(1-x)+x*O(x^n))))/2, n, x)
  • PARI
    /* Alternative Formula: */ a(n)=sum(j=1, n, binomial(2*n-2*j, n-j)/(n-j+1)*sum(i=1, j,(-1)^(j-i)*(n+1)^(i-1)*binomial(n-j+i, j-i)*binomial(n-j+i-1, i-1)))

Formula

a(n) = Sum_{j=1..n} Catalan(n-j)*[Sum_{i=1..j} (-1)^(j-i)*(n+1)^(i-1)*C(n-j+i, j-i)*C(n-j+i-1, i-1)], where Catalan(n) = A000108(n) = C(2n, n)/(n+1).

A120022 a(n) = A120020(n)/n = coefficient of x^n in the n-th self-composition of the g.f. of A120010, divided by n, for n>=1.

Original entry on oeis.org

1, 1, 3, 17, 142, 1558, 21155, 342584, 6448217, 138392304, 3336869488, 89325958048, 2629214627421, 84408934941424, 2935694381925743, 109967573757472768, 4414292541216287516, 189054708982869449056
Offset: 1

Views

Author

Paul D. Hanna, Jun 14 2006

Keywords

Crossrefs

Cf. A120010, A120019, A120020, A120021, A120016.

Programs

  • Mathematica
    a[n_] := Sum[((-1)^(j-i) n^(i-2) Binomial[2n-2j, n-j] Binomial[n+i-j, j-i] Binomial[n+i-j-1, i-1])/(n-j+1), {j, 1, n}, {i, 1, j}]; Array[a, 18] (* Jean-François Alcover, Nov 14 2016 *)
  • PARI
    a(n)=polcoeff((1-sqrt(1-4*x*(1-x)/(1-(n+1)*x*(1-x)+x*O(x^n))))/2, n)/n
  • PARI
    /* Alternate Formula: */ a(n)=sum(j=1, n, binomial(2*n-2*j, n-j)/(n-j+1)* sum(i=1, j,(-1)^(j-i)*binomial(n-j+i, j-i)*binomial(n-j+i-1, i-1)*n^(i-2)))
Showing 1-4 of 4 results.

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