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 1-6 of 6 results.

A005446 Denominators of expansion of -W_{-1}(-e^{-1-x^2/2}) where W_{-1} is Lambert W function.

Original entry on oeis.org

1, 1, 3, 36, 270, 4320, 17010, 5443200, 204120, 2351462400, 1515591000, 2172751257600, 354648294000, 10168475885568000, 7447614174000, 1830325659402240000, 1595278956070800000, 2987091476144455680000
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

See A299430/A299431 for more formulas; given g.f. A(x) = Sum_{n>=0} A005447(n)/A005446(n)*x^n, then A(x)^2 = Sum_{n>=0} A299430(n)/A299431(n)*x^n.

Examples

			1, 1/3, 1/36, -1/270, 1/4320, 1/17010, -139/5443200, 1/204120, -571/2351462400, ...
G.f.: A(x) = 1 + x + 1/3*x^2 + 1/36*x^3 - 1/270*x^4 + 1/4320*x^5 + 1/17010*x^6 - 139/5443200*x^7 + 1/204120*x^8 + ... + A005447(n)/A005446(n)x^n + ...

References

  • E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, 1935, Oxford University Press, p. 221.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A005447, A090804/A065973.
Cf. A299430 / A299431 (A(x)^2), A299432 / A299433.

Programs

  • Maple
    Maple program from N. J. A. Sloane, Jun 23 2011, based on J. Marsaglia's 1986 paper:
a[1]:=1;
M:=25;
for n from 2 to M do
t1:=a[n-1]/(n+1)-add(a[k]*a[n+1-k],k=2..floor(n/2));
if n mod 2 = 1 then t1:=t1-a[(n+1)/2]^2/2; fi;
a[n]:=t1;
od:
s1:=[seq(a[n],n=1..M)];
  • Mathematica
    terms = 18; Assuming[x > 0, -ProductLog[-1, -Exp[-1 - x^2/2]] + O[x]^terms] // CoefficientList[#, x]& // Take[#, terms]& // Denominator (* Jean-François Alcover, Jun 20 2013, updated Feb 21 2018 *)
  • PARI
    a(n)=local(A); if(n<1,n==0,A=vector(n,k,1); for(k=2,n,A[k]=(A[k-1]-sum(i=2,k-1,i*A[i]*A[k+1-i]))/(k+1)); denominator(A[n])) /* Michael Somos, Jun 09 2004 */
  • PARI
    a(n)=if(n<1,n==0,denominator(polcoeff(serreverse(sqrt(2*(x-log(1+x+x^2*O(x^n))))),n))) /* Michael Somos, Jun 09 2004 */
  • SageMath
    @CachedFunction
def b(n): return 1 if (n<2) else (1/(n+1))*( b(n-1) - sum( j*b(n-j+1)*b(j) for j in range(2,n) ))
def A005446(n): return denominator((-1)^n*b(n))
[A005446(n) for n in range(31)] # G. C. Greubel, Nov 21 2022

Formula

G.f.: A(x) = Sum_{n>=0} A005447(n)/A005446(n)*x^n satisfies log(A(x)) = A(x) - 1 - x^2/2.
a(n) = denominator of ((-1)^n * b(n)), where b(n) = (1/(n+1))*( b(n-1) - Sum_{j=2..n-1} j*b(j)*b(n-j+1) ) with b(0) = b(1) = 1 (from Borwein and Corless). - G. C. Greubel, Nov 21 2022

Extensions

Edited by Michael Somos, Jul 21 2002

A005447 Numerators of the expansion of -W_{-1}(-e^(-1 - x^2/2)) where x > 0 and W_{-1} is the Lambert W function.

Original entry on oeis.org

1, 1, 1, 1, -1, 1, 1, -139, 1, -571, -281, 163879, -5221, 5246819, 5459, -534703531, 91207079, -4483131259, -2650986803, 432261921612371, -6171801683, 6232523202521089, 4283933145517, -25834629665134204969, 11963983648109
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Numerators of the expansion of -W_0(-e^(-1 - x^2/2)) where x < 0 an W_0 is the principal branch of the Lambert W function. - Michael Somos, Oct 06 2017
See A299430/A299431 for more formulas; given g.f. A(x) = Sum_{n>=0} A005447(n)/A005446(n)*x^n, then A(x)^2 = Sum_{n>=0} A299430(n)/A299431(n)*x^n.

Examples

			G.f.: A(x) = 1 + x + (1/3)*x^2 + (1/36)*x^3 - (1/270)*x^4 + (1/4320)*x^5 + (1/17010)*x^6 - (139/5443200)*x^7 + (1/204120)*x^8 + ... + (A005447(n)/A005446(n))*x^n + ...

References

  • E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, 1935, Oxford University Press, p. 221.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A005446 (denominators), A090804/A065973.
Cf. A299430 / A299431 (A(x)^2), A299432 / A299433.

Programs

  • Maple
    h := proc(k) option remember; local j; `if`(k<=0,1,(h(k-1)/k-add((h(k-j)*h(j))/(j+1),j=1..k-1))/(1+1/(k+1))) end:
A005447 := n -> `if`(n<4,1,`if`(n=4,-1,numer(h(n-1))));
seq(A005447(i),i=0..24); # Peter Luschny, Feb 08 2011
# other program
a[1]:=1;
M:=25;
for n from 2 to M do
t1:=a[n-1]/(n+1)-add(a[k]*a[n+1-k],k=2..floor(n/2));
if n mod 2 = 1 then t1:=t1-a[(n+1)/2]^2/2; fi;
a[n]:=t1;
od:
s1:=[seq(a[n],n=1..M)]; # N. J. A. Sloane, Jun 23 2011, based on J. Marsaglia's 1986 paper
  • Mathematica
    terms = 25; Assuming[x > 0, -ProductLog[-1, -Exp[-1 - x^2/2]] + O[x]^terms] // CoefficientList[#, x]& // Take[#, terms]& // Numerator (* Jean-François Alcover, Jun 20 2013, updated Feb 21 2018 *)
a[ n_] := If[ n < 0, 0, Block[ {$Assumptions = x < 0}, SeriesCoefficient[ -ProductLog[ -Exp[-1 - x^2/2]], {x, 0, n}] // Numerator]]; (* Michael Somos, Oct 06 2017 *)
  • PARI
    {a(n) = my(A); if( n<1, n==0, A = vector(n, k, 1); for(k=2, n, A[k] = (A[k-1] - sum(i=2, k-1, i * A[i] * A[k+1-i])) / (k+1)); numerator(A[n]))}; /* Michael Somos, Jun 09 2004 */
  • PARI
    {a(n) = if( n<1, n==0, numerator( polcoeff( serreverse(sqrt( 2 * (x - log(1 + x + x^2 * O(x^n))))), n)))}; /* Michael Somos, Jun 09 2004 */
  • SageMath
    @CachedFunction
def h(n): return 1 if (n<1) else ((n+1)/(n+2))*( h(n-1)/n - sum( h(n-j)*h(j)/(j+1) for j in range(1,n) ))
def A005447(n):
    if (n<4): return 1
    elif (n==4): return -1
    else: return numerator(h(n-1))
[A005447(n) for n in range(31)] # G. C. Greubel, Nov 21 2022

Formula

G.f.: A(x) = Sum_{n>=0} A005447(n)/A005446(n)*x^n satisfies log(A(x)) = A(x) - 1 - x^2/2.

Extensions

Edited by Michael Somos, Jul 21 2002

A299430 Numerators of coefficients in C(x) where: C(x)^(1/2) - S(x)^(1/2) = 1 such that C'(x)*S(x)^(1/2) = S'(x)*C(x)^(1/2) = 2*x*C(x).

Original entry on oeis.org

1, 2, 5, 13, 43, 5, -19, 41, 1, -7243, 923, 23183, -21401, 64243259, -73, -471741703, 328578883, -11162934047, -103170103, 405632121933911, -811862551, 6065578925646109, 2180051549129, -261709011005693843, 36779766732769, -1552304807519349743393, -33230318647573, 727642520296392573166003, -85602927913097260249, 3885938896026347271301517
Offset: 0

Views

Author

Paul D. Hanna, Feb 09 2018

Keywords

Comments

Given g.f. C(x) = Sum_{n>=0} A299430(n)/A299431(n)*x^n, then C(x)^(1/2) = Sum_{n>=0} A005447(n)/A005446(n)*x^n.

Examples

			G.f.: C(x) = 1 + 2*x + 5/3*x^2 + 13/18*x^3 + 43/270*x^4 + 5/432*x^5 - 19/17010*x^6 + 41/2721600*x^7 + 1/40824*x^8 - 7243/1175731200*x^9 + 923/1515591000*x^10 + ...
Related power series begin:
S(x) = x^2 + 2/3*x^3 + 1/6*x^4 + 1/90*x^5 - 1/810*x^6 + 1/15120*x^7 + 1/68040*x^8 - 139/24494400*x^9 + 1/1020600*x^10 - 571/12933043200*x^11 + ...
sqrt(C) = 1 + x + 1/3*x^2 + 1/36*x^3 - 1/270*x^4 + 1/4320*x^5 + 1/17010*x^6 - 139/5443200*x^7 + 1/204120*x^8 - 571/2351462400*x^9 - 281/1515591000*x^10 + ... + A005447(n)/A005446(n)*x^n + ...

Crossrefs

Cf. A299431 (denominators in C), A299432/A299433 (S), A005447/A005446 (sqrt(C)).

Programs

  • Mathematica
    terms = 30; Assuming[x>0, ProductLog[-1, -Exp[-1 - x^2/2]]^2 + O[x]^terms] // CoefficientList[#, x]& // Numerator (* Jean-François Alcover, Feb 22 2018 *)
  • PARI
    {a(n) = my(C=1, S=x^2); for(i=0,n, C = (1 + sqrt(S +O(x^(n+2))))^2; S = intformal( 2*x*sqrt(C) ) ); numerator(polcoeff(C,n))}
for(n=0,30,print1(a(n),", "))

Formula

The functions C = C(x) and S = S(x) satisfy:
(1) sqrt(C) - sqrt(S) = 1.
(2a) C'*sqrt(S) = S'*sqrt(C) = 2*x*C.
(2b) C' = 2*x*C/sqrt(S).
(2c) S' = 2*x*sqrt(C).
(3a) C = 1 + Integral 2*x*C/sqrt(S) dx.
(3b) S = Integral 2*x*sqrt(C) dx.
(4a) sqrt(C) = exp( Integral x/(sqrt(C) - 1) dx ).
(4b) sqrt(S) = exp( Integral x/sqrt(S) dx ) - 1.
(5a) C - S = exp( Integral 2*x*C/(C*sqrt(S) + S*sqrt(C)) dx ).
(5b) C - S = exp( Integral C'*S'/(C*S' + S*C') dx).
(6a) sqrt(C) = exp( sqrt(C) - 1 - x^2/2 ).
(6b) sqrt(C) = 1 + x^2/2 + Integral x/(sqrt(C) - 1) dx.

A299431 Denominators of coefficients in C(x) where: C(x)^(1/2) - S(x)^(1/2) = 1 such that C'(x)*S(x)^(1/2) = S'(x)*C(x)^(1/2) = 2*x*C(x).

Original entry on oeis.org

1, 1, 3, 18, 270, 432, 17010, 2721600, 40824, 1175731200, 1515591000, 217275125760, 354648294000, 5084237942784000, 114578679600, 915162829701120000, 1595278956070800000, 298709147614445568000, 818378104464320400000, 168561571998831634022400000, 982053725357184480000, 45745017385529077294694400000, 279517041579788632620000000
Offset: 0

Views

Author

Paul D. Hanna, Feb 09 2018

Keywords

Comments

Given g.f. C(x) = Sum_{n>=0} A299430(n)/A299431(n)*x^n, then C(x)^(1/2) = Sum_{n>=0} A005447(n)/A005446(n)*x^n.

Examples

			G.f.: C(x) = 1 + 2*x + 5/3*x^2 + 13/18*x^3 + 43/270*x^4 + 5/432*x^5 - 19/17010*x^6 + 41/2721600*x^7 + 1/40824*x^8 - 7243/1175731200*x^9 + 923/1515591000*x^10 + ...
Related power series begin:
S(x) = x^2 + 2/3*x^3 + 1/6*x^4 + 1/90*x^5 - 1/810*x^6 + 1/15120*x^7 + 1/68040*x^8 - 139/24494400*x^9 + 1/1020600*x^10 - 571/12933043200*x^11 + ...
sqrt(C) = 1 + x + 1/3*x^2 + 1/36*x^3 - 1/270*x^4 + 1/4320*x^5 + 1/17010*x^6 - 139/5443200*x^7 + 1/204120*x^8 - 571/2351462400*x^9 - 281/1515591000*x^10 + ...
+ A005447(n)/A005446(n)*x^n + ...

Crossrefs

Cf. A299430 (numerators in C), A299432/A299433 (S), A005447/A005446 (sqrt(C)).

Programs

  • Mathematica
    terms = 30; Assuming[x>0, ProductLog[-1, -Exp[-1 - x^2/2]]^2 + O[x]^terms] // CoefficientList[#, x]& // Denominator (* Jean-François Alcover, Feb 22 2018 *)
  • PARI
    {a(n) = my(C=1, S=x^2); for(i=0,n, C = (1 + sqrt(S +O(x^(n+2))))^2; S = intformal( 2*x*sqrt(C) ) ); denominator(polcoeff(C,n))}
for(n=0,30,print1(a(n),", "))

Formula

The functions C = C(x) and S = S(x) satisfy:
(1) sqrt(C) - sqrt(S) = 1.
(2a) C'*sqrt(S) = S'*sqrt(C) = 2*x*C.
(2b) C' = 2*x*C/sqrt(S).
(2c) S' = 2*x*sqrt(C).
(3a) C = 1 + Integral 2*x*C/sqrt(S) dx.
(3b) S = Integral 2*x*sqrt(C) dx.
(4a) sqrt(C) = exp( Integral x/(sqrt(C) - 1) dx ).
(4b) sqrt(S) = exp( Integral x/sqrt(S) dx ) - 1.
(5a) C - S = exp( Integral 2*x*C/(C*sqrt(S) + S*sqrt(C)) dx ).
(5b) C - S = exp( Integral C'*S'/(C*S' + S*C') dx).
(6a) sqrt(C) = exp( sqrt(C) - 1 - x^2/2 ).
(6b) sqrt(C) = 1 + x^2/2 + Integral x/(sqrt(C) - 1) dx.

A299433 Denominators of coefficients in S(x) where: C(x)^(1/2) - S(x)^(1/2) = 1 such that C'(x)*S(x)^(1/2) = S'(x)*C(x)^(1/2) = 2*x*C(x).

Original entry on oeis.org

1, 1, 1, 3, 6, 90, 810, 15120, 68040, 24494400, 1020600, 12933043200, 9093546000, 14122883174400, 2482538058000, 76263569141760000, 59580913392000, 15557768104919040000, 14357510604637200000, 28377369023372328960000, 8183781044643204000000, 3539793011975464314470400000, 270064774473225732000000, 13677760198273194111113625600000
Offset: 0

Views

Author

Paul D. Hanna, Feb 09 2018

Keywords

Examples

			G.f.: S(x) = x^2 + 2/3*x^3 + 1/6*x^4 + 1/90*x^5 - 1/810*x^6 + 1/15120*x^7 + 1/68040*x^8 - 139/24494400*x^9 + 1/1020600*x^10 - 571/12933043200*x^11 + ...
Related power series begin:
C(x) = 1 + 2*x + 5/3*x^2 + 13/18*x^3 + 43/270*x^4 + 5/432*x^5 - 19/17010*x^6 + 41/2721600*x^7 + 1/40824*x^8 - 7243/1175731200*x^9 + 923/1515591000*x^10 + ...
sqrt(C) = 1 + x + 1/3*x^2 + 1/36*x^3 - 1/270*x^4 + 1/4320*x^5 + 1/17010*x^6 - 139/5443200*x^7 + 1/204120*x^8 - 571/2351462400*x^9 - 281/1515591000*x^10 + ... + A005447(n)/A005446(n)*x^n + ...

Crossrefs

Cf. A299432 (numerators in S), A299430/A299431 (C), A005447/A005446 (sqrt(C)).

Programs

  • Mathematica
    terms = 30; c[x_] = Assuming[x > 0, ProductLog[-1, -Exp[-1 - x^2/2]]^2 + O[x]^terms]; Integrate[2*x*Sqrt[c[x]] + O[x]^terms, x] // CoefficientList[#, x] & // Denominator (* Jean-François Alcover, Feb 22 2018 *)
  • PARI
    {a(n) = my(C=1, S=x^2); for(i=0,n, C = (1 + sqrt(S +O(x^(n+2))))^2; S = intformal( 2*x*sqrt(C) ) ); denominator(polcoeff(S,n))}
for(n=0,30,print1(a(n),", "))

Formula

The functions C = C(x) and S = S(x) satisfy:
(1) sqrt(C) - sqrt(S) = 1.
(2a) C'*sqrt(S) = S'*sqrt(C) = 2*x*C.
(2b) C' = 2*x*C/sqrt(S).
(2c) S' = 2*x*sqrt(C).
(3a) C = 1 + Integral 2*x*C/sqrt(S) dx.
(3b) S = Integral 2*x*sqrt(C) dx.
(4a) sqrt(C) = exp( Integral x/(sqrt(C) - 1) dx ).
(4b) sqrt(S) = exp( Integral x/sqrt(S) dx ) - 1.
(5a) C - S = exp( Integral 2*x*C/(C*sqrt(S) + S*sqrt(C)) dx ).
(5b) C - S = exp( Integral C'*S'/(C*S' + S*C') dx).
(6a) sqrt(C) = exp( sqrt(C) - 1 - x^2/2 ).
(6b) sqrt(C) = 1 + x^2/2 + Integral x/(sqrt(C) - 1) dx.

A299853 G.f. C(x) satisfies C(x)^(1/2) - S(x)^(1/2) = 1 such that C'(x)*S(x)^(1/2) = S'(x)*C(x)^(1/2) = 72*x.

Original entry on oeis.org

1, 12, 12, -24, 96, -504, 3072, -20592, 147456, -1108536, 8650752, -69535440, 572522496, -4808643120, 41070624768, -355839590880, 3121367482368, -27676994061240, 247750893502464, -2236495344667920, 20341652308623360, -186268112277342480, 1716095758400225280, -15898314689790251040, 148031912376784650240, -1384743209480730865584, 13008588976864521879552
Offset: 0

Views

Author

Paul D. Hanna, Feb 20 2018

Keywords

Comments

The functions C = C(x) and S = S(x) such that C(x)^(1/2) - S(x)^(1/2) = 1 may be generated by the following method.
(Start) Set C = 1, S = x^2, then iterate
C = 1 + Integral S'*sqrt(C/S) dx and
S = Integral C'*sqrt(S/C) dx.
The limit will converge to C = C(x) and S = S(x) defined by A299853 and A299854. (End)
Note that different seed values of C and S yield different solutions; see A299430/A299431 and A299432/A299433 for other functions that satisfy C(x)^(1/2) - S(x)^(1/2) = 1.

Examples

			G.f.: C(x) = 1 + 12*x + 12*x^2 - 24*x^3 + 96*x^4 - 504*x^5 + 3072*x^6 - 20592*x^7 + 147456*x^8 - 1108536*x^9 + 8650752*x^10 + ...
RELATED SERIES.
S(x) = 36*x^2 - 144*x^3 + 864*x^4 - 6048*x^5 + 46080*x^6 - 370656*x^7 + 3096576*x^8 - 26604864*x^9 + 233570304*x^10 + ...
C(x)^(1/2) = 1 + 6*x - 12*x^2 + 60*x^3 - 384*x^4 + 2772*x^5 - 21504*x^6 + 175032*x^7 - 1474560*x^8 + 12748164*x^9 - 112459776*x^10 + ...
sqrt(S(x)) = 6*x - 12*x^2 + 60*x^3 - 384*x^4 + 2772*x^5 - 21504*x^6 + 175032*x^7 - 1474560*x^8 + 12748164*x^9 - 112459776*x^10 + ...
where C(x)^(1/2) - S(x)^(1/2) = 1
and C'*sqrt(S) = S'*sqrt(C) = 72*x.

Crossrefs

Cf. A299854, A299855.

Programs

  • PARI
    {a(n) = my(C=1, S=x^2); for(i=0, n, C = 1 + intformal( 72*x/sqrt(S +x^3*O(x^n)) ); S = intformal( 72*x/sqrt(C) ) ); polcoeff(C, n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    {a(n) = if(n==0,1, 2*(-4)^n * binomial(3*n/2,n) / ((3*n-2)*(3*n-4)) )}
for(n=0,30,print1(a(n),", "))

Formula

The functions C = C(x) and S = S(x) satisfy:
(1a) sqrt(C) - sqrt(S) = 1.
(1b) C'*sqrt(S) = S'*sqrt(C) = 72*x.
(1c) C' = 72*x/sqrt(S).
(1d) S' = 72*x/sqrt(C).
Integrals.
(2a) C = 1 + Integral 72*x/sqrt(S) dx.
(2b) S = Integral 72*x/sqrt(C) dx.
(2c) C = 1 + Integral S'*sqrt(C/S) dx.
(2d) S = Integral C'*sqrt(S/C) dx.
Exponentials.
(3a) sqrt(C) = exp( Integral 36*x/(C*sqrt(S)) dx ).
(3b) sqrt(S) = 6*x*exp( Integral 36*x/(S*sqrt(C)) - 1/x dx ).
(3c) C - S = exp( Integral 72*x/(C*sqrt(S) + S*sqrt(C)) dx ).
(3d) C - S = exp( Integral C'*S'/(C*S' + S*C') dx).
Functional equations.
(4a) C = 1/3 - 36*x^2 + (2/3)*C^(3/2).
(4b) S = 36*x^2 - (2/3)*S^(3/2).
Explicit solutions.
(5a) C(x) = 1 + Sum_{n>=1} 2*(-4)^n*binomial(3*n/2,n)/((3*n-2)*(3*n-4)) * x^n.
(5b) S(x) = 36*x^2 + Sum_{n>=3} 18*(-4)^n*(3*n-3)*binomial(3*n/2-2,n)/((3*n-4)*(3*n-6)) * x^n.
(5c) sqrt(C(x)) = 1 + Sum_{n>=1} -(-4)^n * binomial(3*n/2,n)/(3*n-2) * x^n.
Formulas for terms.
a(n) = 2*(-4)^n * binomial(3*n/2,n) / ((3*n-2)*(3*n-4)) for n>=1, with a(0) = 1.
Showing 1-6 of 6 results.

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