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.

A286798 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section. .

Original entry on oeis.org

1, 1, 4, 2, 27, 22, 248, 264, 30, 2830, 3610, 830, 8, 38232, 55768, 18746, 1078, 593859, 961740, 414720, 46986, 576, 10401712, 18326976, 9457788, 1593664, 62682, 112, 202601898, 382706674, 226526362, 49941310, 3569882, 45296, 4342263000, 8697475368, 5740088706, 1540965514, 160998750, 4909674, 16896, 101551822350, 213865372020, 154271354280, 48205014786, 6580808784, 337737294, 4200032, 2560
Offset: 0

Views

Author

Gheorghe Coserea, May 21 2017

Keywords

Comments

Row n>0 contains floor(2*(n+1)/3) terms.

Examples

			A(x;t) = 1 + x + (4 + 2*t)*x^2 + (27 + 22*t)*x^3 + (248 + 264*t + 30*t^2)*x^4 +
Triangle starts:
n\k  [0]        [1]        [2]        [3]       [4]      [5]
[0]  1;
[1]  1;
[2]  4,         2;
[3]  27,        22;
[4]  248,       264,       30;
[5]  2830,      3610,      830,       8;
[6]  38232,     55768,     18746,     1078;
[7]  593859,    961740,    414720,    46986,    576;
[8]  10401712,  18326976,  9457788,   1593664,  62682,   112;
[9]  202601898, 382706674, 226526362, 49941310, 3569882, 45296;
[10] ...

Links

Crossrefs

Cf. A286781, A286782, A286783, A286784, A286785.

Programs

  • Mathematica
    max = 12; y0[x_, t_] = 1; y1[x_, t_] = 0; For[n = 1, n <= max, n++, y1[x_, t_] = 1 + x y0[x, t]^2 + 3 t x^3 y0[x, t]^2 D[y0[x, t], x] + x^2 (2 y0[x, t] D[y0[x, t], x] + t (2 y0[x, t]^3 - D[y0[x, t], x] + y0[x, t] D[y0[x, t], x])) + O[x]^n // Normal // Simplify; y0[x_, t_] = y1[x, t]];
P[n_, t_] := Coefficient[y0[x, t] , x, n];
row[n_] := CoefficientList[P[n, t], t];
Table[row[n], {n, 0, max}] // Flatten (* Jean-François Alcover, May 24 2017, adapted from PARI *)
  • PARI
    A286795_ser(N, t='t) = {
  my(x='x+O('x^N), y0=1, y1=0, n=1);
  while(n++,
    y1 = (1 + x*(1 + 2*t + x*t^2)*y0^2 + t*(1-t)*x^2*y0^3 + 2*x^2*y0*y0');
    y1 = y1 / (1+2*x*t); if (y1 == y0, break()); y0 = y1;); y0;
};
A286798_ser(N,t='t) = {
  my(v = A286795_ser(N,t)); subst(v, 'x, serreverse(x/(1-x*t*v)));
};
concat(apply(p->Vecrev(p), Vec(A286798_ser(12))))
\\ test: y=A286798_ser(50); x^2*y' == (1 - y + x*y^2 + 2*x^2*t*y^3)/(t - (2+t)*y - 3*x*t*y^2)

Formula

y(x;t) = Sum_{n>=0} P_n(t)*x^n satisfies x^2*deriv(y,x) = (1 - y + x*y^2 + 2*x^2*t*y^3)/(t - (2+t)*y - 3*x*t*y^2), with y(0;t) = 1, where P_n(t) = Sum_{k=0..floor((2*n-1)/3)} T(n,k)*t^k for n>0.
A000699(n+1)=T(n,0), A000108(n)=P_n(-1), A286799(n)=P_n(1).

A287029 Row sums of A286800.

Original entry on oeis.org

1, 3, 13, 147, 1965, 30979, 559357, 11289219, 250794109, 6066778627, 158533572861, 4447703062787, 133309656009469, 4251322261512195, 143749952968507389, 5137921526511802371, 193589838004887201789, 7670544451820808601603, 318892867844484240154621, 13881730766388536085356547
Offset: 1

Views

Author

Gheorghe Coserea, May 22 2017

Keywords

Examples

			A(x) = x + 3*x^2 + 13*x^3 + 147*x^4 + 1965*x^5 + 30979*x^6 + ...

Links

Crossrefs

Cf. A049464, A286799, A286800, A287039.

Programs

  • Mathematica
    terms = 20; y[, ] = 0; Do[y[x_, t_] = (1/(-1 + y[x, t])) x (-1 - y[x, t]^2 - 2 y[x, t] (-1 + D[y[x, t], x]) + t x (-1 + y[x, t]) (2 (-1 + y[x, t])^2 + (x (-1 + y[x, t]) + y[x, t]) D[y[x, t], x])) + O[x]^n // Normal // Simplify, {n, terms+1}];
Total[CoefficientList[#, t]]& /@ CoefficientList[y[x, t], x] // Rest
  • PARI
    A286795_ser(N, t='t) = {
  my(x='x+O('x^N), y0=1, y1=0, n=1);
  while(n++,
    y1 = (1 + x*(1 + 2*t + x*t^2)*y0^2 + t*(1-t)*x^2*y0^3 + 2*x^2*y0*y0');
    y1 = y1 / (1+2*x*t); if (y1 == y0, break()); y0 = y1;); y0;
};
A286798_ser(N,t='t) = {
  my(v = A286795_ser(N,t)); subst(v, 'x, serreverse(x/(1-x*t*v)));
};
A286800_ser(N, t='t) = {
  my(v = A286798_ser(N,t)); 1-1/subst(v, 'x, serreverse(x*v^2));
};
A287029_ser(N) = A286800_ser(N+1, 1);
Vec(A287029_ser(20))

Formula

a(n) = Sum_{k=0..floor((2*n-1)/3)} A286800(n,k) for n>=1.
a(n) ~ 4*exp(-7/2)/sqrt(Pi) * n^(3/2) * 2^n * n! * (1 - 15/(8*n) - 503/(128*n^2) + O(1/n^3)). (see Borinsky link) - Gheorghe Coserea, Oct 21 2017

A294158 Row sums of A291844.

Original entry on oeis.org

1, 1, 6, 52, 602, 8223, 128917, 2273716, 44509914, 957408649, 22449011336, 570032756328, 15587503694363, 456793916757139, 14284890417759141, 474896318288651220, 16726743380843538668, 622282429409944248297, 24385251974172090147514, 1004017088910699487855180
Offset: 0

Views

Author

Gheorghe Coserea, Oct 24 2017

Keywords

Links

Crossrefs

Cf. A049464(y), A287039(x), A286799(z), A287029(u), A291844.

Programs

  • PARI
    A291843_ser(N, t='t) = {
  my(x='x+O('x^N), y=1, y1=0, n=1,
  dn = 1/(-2*t^2*x^4 - (2*t^2+3*t)*x^3 - (2*t+1)*x^2 + (2*t-1)*x + 1));
  while (n++,
   y1 = (2*x^2*y'*((-t^2 + t)*x + (-t + 1) + (t^2*x^2 + (t^2 + t)*x + t)*y) +
        (t*x^2 + t*x)*y^2 - (2*t^2*x^3 + 3*t*x^2 + (-t + 1)*x - 1))*dn;
   if (y1 == y, break); y = y1;); y;
};
A291844_ser(N, t='t) = {
  my(z = A291843_ser(N+1,t));
  ((1+x)*z - 1)*(1 + t*x)/((1-t + t*(1+x)*z)*x*z^2);
};
Vec(A291844_ser(20,t=1))

Formula

a(n) = Sum_{k=0..floor((2*n-1)/3)} A291844(n,k), n > 0.
Showing 1-3 of 3 results.

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