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.

Previous Showing 11-16 of 16 results.

A215648 G.f. satisfies: A(x) = 1 + x*A(x)^2 + 3*x^2*A'(x)*A(x).

Original entry on oeis.org

1, 1, 5, 44, 539, 8337, 154632, 3332640, 81711479, 2244563555, 68272834865, 2278102125040, 82749748994500, 3250966816344604, 137371215935579892, 6213234210869600376, 299527133488944917631, 15332761842086151881175, 830648056455231849827895
Offset: 0

Views

Author

Paul D. Hanna, Mar 09 2013

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 44*x^3 + 539*x^4 + 8337*x^5 + 154632*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 11*x^2 + 98*x^3 + 1191*x^4 + 18192*x^5 + 333264*x^6 +...
A'(x)*A(x) = 1 + 11*x + 147*x^2 + 2382*x^3 + 45480*x^4 + 999792*x^5 +...
where A(x) = 1 + x*A(x)^2 + 3*x^2*A'(x)*A(x).

Links

Crossrefs

Cf. A112936, A112938, A218168.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+x*deriv(x*A^3)/(A+x*O(x^n)));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. satisfies: A(x) = 1 + x*[d/dx x*A(x)^3]/A(x).
a(n) ~ n! * 3^(n+1) / (Pi*exp(1)). - Vaclav Kotesovec, Aug 24 2017
a(0) = 1; a(n) = Sum_{k=0..n-1} (3*k+1) * a(k) * a(n-k-1). - Ilya Gutkovskiy, Sep 05 2024

A355794 Row 1 of A355793.

Original entry on oeis.org

1, 3, 24, 282, 4236, 76548, 1608864, 38488152, 1032125136, 30670171248, 1000637672064, 35571839009952, 1368990872569536, 56720594992438848, 2517761078627172864, 119222916630934484352, 5999613754698100628736, 319763269764299852744448, 17994913747767982690289664
Offset: 0

Views

Author

Peter Bala, Jul 19 2022

Keywords

Links

Crossrefs

Cf. A355793 (table).
Cf. A112936 (row 0), A355795 (row 2), A355796 (row 3), A355797 (row 4).
Cf. A008544, A111528, A355721.

Programs

  • Maple
    n := 1: seq(coeff(series( hypergeom([n+2/3, 1], [], 3*x)/hypergeom([n-1/3, 1], [], 3*x ), x, 21), x, k), k = 0..20);

Formula

O.g.f.: A(x) = ( Sum_{k >= 0} t(k+1)/t(1)*x^k )/( Sum_{k >= 0} t(k)/t(0)*x^k ), where t(n) = Product_{k = 1..n} 3*k-1 = A008544(n) (triple factorial numbers).
A(x)/(1 - 2*x*A(x)) = Sum_{k >= 0} t(k+1)/t(1)*x^k.
A(x) = 1/(1 + 2*x - 5*x/(1 + 5*x - 8*x/(1 + 8*x - 11*x/(1 + 11*x - ... )))) (continued fraction).
A(x) satisfies the Riccati differential equation 3*x^2*A(x)' + 2*x*A(x)^2 - (1 - x)*A(x) + 1 = 0 with A(0) = 1.
Hence by Stokes, A(x) = 1/(1 - 3*x/(1 - 5*x/(1 - 6*x/(1 - 8*x/(1 - 9*x/(1 - 11*x/(1 - 12*x/(1 - ... )))))))), a continued fraction of Stieltjes type.

A355795 Row 2 of A355793.

Original entry on oeis.org

1, 3, 33, 507, 9609, 212835, 5350785, 149961675, 4628365305, 155913036915, 5692874399025, 224034935130075, 9456933847187625, 426402330032719875, 20460268520575152225, 1041301103429870128875, 56040353252589013121625, 3180443637298592493577875, 189863589771186976073108625
Offset: 0

Views

Author

Peter Bala, Jul 21 2022

Keywords

Links

Crossrefs

Cf. A355793 (table).
Cf. A112936 (row 0), A355794 (row 1), A355796 (row 3), A355797 (row 4).
Cf. A008544, A111528, A355721.

Programs

  • Maple
    n := 2: seq(coeff(series( hypergeom([n+2/3, 1], [], 3*x)/hypergeom([n-1/3, 1], [], 3*x ), x, 21), x, k), k = 0..20);

Formula

O.g.f.: A(x) = ( Sum_{k >= 0} t(k+2)/t(2)*x^k )/( Sum_{k >= 0} t(k+1)/t(1)*x^k ), where t(n) = Product_{k = 1..n} 3*k-1 = A008544(n) (triple factorial numbers).
A(x)/(1 - 5*x*A(x)) = Sum_{k >= 0} t(k+2)/t(2)*x^k.
A(x) = 1/(1 + 5*x - 8*x/(1 + 8*x - 11*x/(1 + 11*x - 14*x/(1 + 14*x - ... )))) (continued fraction).
A(x) satisfies the Riccati differential equation 3*x^2*A(x)' + 5*x*A(x)^2 - (1 + 2*x)*A(x) + 1 = 0 with A(0) = 1.
Hence by Stokes, A(x) = 1/(1 - 3*x/(1 - 8*x/(1 - 6*x/(1 - 11*x/(1 - 9*x/(1 - 14*x/(1 - 12*x/(1 - ... )))))))), a continued fraction of Stieltjes type.

A355796 Row 3 of A355793.

Original entry on oeis.org

1, 3, 42, 786, 17736, 459768, 13333488, 425600976, 14791250688, 555381292800, 22398626084352, 965768866650624, 44347055502428160, 2161455366606034944, 111489317304231616512, 6069676735484389779456, 347921629212782938472448, 20950823605616500202323968, 1322561808699778749456678912
Offset: 0

Views

Author

Peter Bala, Jul 21 2022

Keywords

Links

Crossrefs

Cf. A355793 (table).
Cf. A112936 (row 0), A355794 (row 1), A355795 (row 2), A355797 (row 4).
Cf. A008544, A111528, A355721.

Programs

  • Maple
    n := 3: seq(coeff(series( hypergeom([n+2/3, 1], [], 3*x)/hypergeom([n-1/3, 1], [], 3*x ), x, 21), x, k), k = 0..20);

Formula

Let t(n) = Product_{k = 1..n} 3*k-1 = A008544(n) (triple factorial numbers).
O.g.f.: A(x) = ( Sum_{k >= 0} t(k+3)/t(3)*x^k )/( Sum_{k >= 0} t(k+2)/t(2)*x^k ).
A(x)/(1 - 8*x*A(x)) = Sum_{k >= 0} t(k+3)/t(3)*x^k.
A(x) = 1/(1 + 8*x - 11*x/(1 + 11*x - 14*x/(1 + 14*x - 17*x/(1 + 17*x - ... )))) (continued fraction).
A(x) satisfies the Riccati differential equation 3*x^2*d/dx(A(x)) + 8*x*R(n,x)^2 - (1 + 5*x)*R(n,x) + 1 = 0 with A(0) = 1.
Applying Stokes 1982 gives A(x) = 1/(1 - 3*x/(1 - 11*x/(1 - 6*x/(1 - 14*x/(1 - 9*x/(1 - 17*x/(1 - 12*x/(1 - ...)))))))), a continued fraction of Stieltjes type.

A355797 Row 4 of A355793.

Original entry on oeis.org

1, 3, 51, 1119, 29103, 859143, 28091463, 1002057591, 38606468343, 1595167432599, 70315835952471, 3293268346004439, 163337193581191575, 8554718468806548951, 471976737725208306327, 27369722655919760451159, 1664858070989667129693975, 106029602841882346657155543
Offset: 0

Views

Author

Peter Bala, Jul 21 2022

Keywords

Links

Crossrefs

Cf. A355793 (table).
Cf. A112936 (row 0), A355794 (row 1), A355795 (row 2), A355796 (row 3).
Cf. A008544, A111528, A355721.

Programs

  • Maple
    n := 4: seq(coeff(series( hypergeom([n+2/3, 1], [], 3*x)/hypergeom([n-1/3, 1], [], 3*x ), x, 21), x, k), k = 0..20);

Formula

Let t(n) = Product_{k = 1..n} 3*k-1 = A008544(n) (triple factorial numbers).
O.g.f.: A(x) = ( Sum_{k >= 0} t(k+4)/t(4)*x^k )/( Sum_{k >= 0} t(k+3)/t(3)*x^k ).
A(x)/(1 - 11*x*A(x)) = Sum_{k >= 0} t(k+4)/t(4)*x^k.
A(x) = 1/(1 + 11*x - 14*x/(1 + 14*x -17*x/(1 + 17*x - 20*x/(1 + 20*x - ... )))) (continued fraction).
A(x) satisfies the Riccati differential equation 3*x^2*d/dx(A(x)) + 11*x*A(x)^2 - (1 + 8*x)*A(x) + 1 = 0 with A(0) = 1.
Applying Stokes 1982 gives A(x) = 1/(1 - 3*x/(1 - 14*x/(1 - 6*x/(1 - 17*x/(1 - 9*x/(1 - 20*x/(1 - 12*x/(1 - 23*x/(1 - ...))))))))), a continued fraction of Stieltjes type.

A218168 G.f. satisfies: A(x) = 1 + x*[d/dx x*A(x)^3]/A(x)^4.

Original entry on oeis.org

1, 1, 2, 5, 16, 77, 630, 7956, 132480, 2664389, 62140078, 1643056625, 48503833280, 1580192424580, 56292381309608, 2176251789724500, 90726545080501440, 4056755120695005717, 193649641793538259950, 9828338731644277922679, 528459404673048692682384
Offset: 0

Views

Author

Paul D. Hanna, Mar 04 2013

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 16*x^4 + 77*x^5 + 630*x^6 + 7956*x^7 +...
Related expansions:
A'(x) = 1 + 4*x + 15*x^2 + 64*x^3 + 385*x^4 + 3780*x^5 + 55692*x^6 +...
A(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 46*x^4 + 206*x^5 + 1503*x^6 +...
A(x)^3 = 1 + 3*x + 9*x^2 + 28*x^3 + 96*x^4 + 414*x^5 + 2735*x^6 +...
A(x)^4 = 1 + 4*x + 14*x^2 + 48*x^3 + 173*x^4 + 736*x^5 + 4486*x^6 +...
d/dx x*A(x)^3 =  1 + 6*x + 27*x^2 + 112*x^3 + 480*x^4 + 2484*x^5 +...
where A(x) = 1 + x*[d/dx x*A(x)^3]/A(x)^4.
1/A(x) = 1 - x - x^2 - 2*x^3 - 7*x^4 - 45*x^5 - 468*x^6 - 6624*x^7 -...
A'(x)/A(x)^2 = 1 + 2*x + 6*x^2 + 28*x^3 + 225*x^4 + 2808*x^5 +...
where A(x) = 1 + x/A(x) + 3*x^2*A'(x)/A(x)^2.

Links

Crossrefs

Cf. A112936, A112938.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+x*deriv(x*A^3)/(A+x*O(x^n))^4);polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. satisfies: A(x)^3 = A(x)^2 + x*A(x) + 3*x^2*A'(x).
a(n) ~ c * 3^n * n! / n^2, where c = 0.0220701841954367321830819129913... - Vaclav Kotesovec, Aug 24 2017
Previous Showing 11-16 of 16 results.

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