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 1-7 of 7 results.

A367720 E.g.f. satisfies A(x) = exp(x*A(x^4)).

Original entry on oeis.org

1, 1, 1, 1, 1, 121, 721, 2521, 6721, 196561, 3659041, 29993041, 159762241, 1686639241, 60298558321, 987112886761, 9315623640961, 76611297104161, 2454331471018561, 69805324167893281, 1086439146068753281, 62621251106366355481, 1358219171406244427281
Offset: 0

Views

Author

Seiichi Manyama, Nov 28 2023

Keywords

Comments

This sequence is different from A354554.

Crossrefs

Cf. A138292, A367719.
Cf. A354554.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=0, (i-1)\4, (4*j+1)*v[j+1]*v[i-4*j]/(j!*(i-1-4*j)!))); v;

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..floor((n-1)/4)} (4*k+1) * a(k) * a(n-1-4*k) / (k! * (n-1-4*k)!).

A143566 E.g.f. satisfies A(x) = exp(x*A(x^2/2!)).

Original entry on oeis.org

1, 1, 1, 4, 13, 46, 241, 1471, 9409, 67348, 564841, 4771801, 45459481, 463867834, 5060656693, 58878140686, 730612429681, 9556314730456, 131627520296929, 1912237000523623, 29032781640572881, 462811831018070206, 7687624300327129621, 133275225843052767244
Offset: 0

Views

Author

Alois P. Heinz, Aug 24 2008

Keywords

Links

Crossrefs

2nd column of A143565.
Cf. A138292.

Programs

  • Maple
    A:= proc(n) option remember; if n<=0 then 1 else unapply(convert(
       series(exp(x*A(n-2)(x^2/2)), x,n+1), polynom),x) fi
    end:
a:= n-> coeff(A(n)(x), x,n)*n!:
seq(a(n), n=0..28);
  • Mathematica
    A[n_] := A[n] = If[n <= 0, 1&, Function[Normal[Series[Exp[y*A[n-2][y^2/2]], {y, 0, n+1}] /. y -> #]]]; a[n_] := Coefficient[A[n][x], x, n]*n!; Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Feb 13 2014, after Maple *)

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..floor((n-1)/2)} (2*k+1) * a(k) * a(n-1-2*k) / (2^k * k! * (n-1-2*k)!). - Seiichi Manyama, Nov 28 2023

A367719 E.g.f. satisfies A(x) = exp(x*A(x^3)).

Original entry on oeis.org

1, 1, 1, 1, 25, 121, 361, 3361, 42001, 275185, 1819441, 30777121, 371238121, 9284332201, 131442054745, 1454933712961, 34120902859681, 851562584890081, 12300037440760801, 187928965721651905, 6019555345508794681, 130768735411230580441
Offset: 0

Views

Author

Seiichi Manyama, Nov 28 2023

Keywords

Comments

This sequence is different from A354553.

Crossrefs

Cf. A138292, A367720.
Cf. A354553.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=0, (i-1)\3, (3*j+1)*v[j+1]*v[i-3*j]/(j!*(i-1-3*j)!))); v;

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..floor((n-1)/3)} (3*k+1) * a(k) * a(n-1-3*k) / (k! * (n-1-3*k)!).

A367747 E.g.f. satisfies A(x) = exp(x * (1 + x) * A(x^2)).

Original entry on oeis.org

1, 1, 3, 13, 73, 561, 4771, 49813, 562353, 7340833, 102829411, 1627648221, 27294311353, 502042022353, 9759264753603, 205434011254501, 4544894700204001, 107346788357502273, 2657668122191037763, 69701762677026498733, 1909106308252976007081
Offset: 0

Views

Author

Seiichi Manyama, Nov 29 2023

Keywords

Crossrefs

Cf. A000272, A367748, A367749, A367751.
Cf. A127680, A138292.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=0, i-1, (j+1)*v[j\2+1]*v[i-j]/((j\2)!*(i-1-j)!))); v;

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..n-1} (k+1) * a(floor(k/2)) * a(n-1-k) / (floor(k/2)! * (n-1-k)!).

A294638 E.g.f. satisfies: A'(x) = A(x) * A(x^2).

Original entry on oeis.org

1, 1, 1, 3, 9, 33, 153, 963, 6129, 47457, 393489, 3689379, 36673209, 410924097, 4810169961, 64694478627, 878318278497, 13230037503297, 203967546446241, 3494178651687363, 60117798742663401, 1137159539308348641, 21683284489630748601, 452680959717183978243, 9454328250188008785489, 214087305044257976127393, 4862802200825123466537393, 119970186740330465448543843, 2944202974922987534742898329
Offset: 0

Views

Author

Paul D. Hanna, Nov 05 2017

Keywords

Examples

			E.g.f.: A(x) = 1 + x + x^2/2! + 3*x^3/3! + 9*x^4/4! + 33*x^5/5! + 153*x^6/6! + 963*x^7/7! + 6129*x^8/8! + 47457*x^9/9! + 393489*x^10/10! + 3689379*x^11/11! + 36673209*x^12/12! + 410924097*x^13/13! + 4810169961*x^14/14! + 64694478627*x^15/15! + 878318278497*x^16/16! + 13230037503297*x^17/17! + 203967546446241*x^18/18! + 3494178651687363*x^19/19! + ...
such that A'(x) = A(x) * A(x^2).
Also, A(x) = exp( Integral A(x^2) dx ).
RELATED SERIES.
The logarithm of the e.g.f. is an odd function that begins:
log(A(x)) = x + x^3/3 + x^5/(5*2!) + 3*x^7/(7*3!) + 9*x^9/(9*4!) + 33*x^11/(11*5!) + 153*x^13/(13*6!) + 963*x^15/(15*7!) + 6129*x^17/(17*8!) + 47457*x^19/(19*9!) + 393489*x^21/(21*10!) +...+ a(n) * x^(2*n+1)/((2*n+1)*n!) +...
which equals Integral A(x^2) dx.
Explicitly,
log(A(x)) = x + 2*x^3/3! + 12*x^5/5! + 360*x^7/7! + 15120*x^9/9! + 997920*x^11/11! + 101787840*x^13/13! + 16657280640*x^15/15! + 3180450873600*x^17/17! + 837294557299200*x^19/19! +...+ (2*n)!/n! * a(n) * x^(2*n+1)/(2*n+1)! +...

Links

Crossrefs

Cf. A138292.

Programs

  • PARI
    {a(n) = my(A=1); for(i=1,#binary(n+1), A = exp( intformal( subst(A,x,x^2) +x*O(x^n)) ) ); n!*polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

E.g.f. A(x) = Sum_{n>=0} a(n) * x^n/n! satisfies:
(1) A(x) = exp( Integral A(x^2) dx ).
(2) A(x) = 1/A(-x).
(3) A(x) = exp( Sum_{n>=0} a(n) * x^(2*n+1) / ((2*n+1)*n!) ) .
(4) A(x) = exp( Sum_{n>=0} (2*n)!/n! * a(n) * x^(2*n+1)/(2*n+1)! ).

A367721 E.g.f. satisfies A(x) = exp(x*A(-x^2)).

Original entry on oeis.org

1, 1, 1, -5, -23, 1, 601, 7771, 26545, -401183, -6965999, -42828389, 528611161, 15543020065, 141983039017, -2393449681349, -83586615493919, -708151768946879, 15447932991283105, 635290179334026427, 7146984268771158601, -162583738763505944639
Offset: 0

Views

Author

Seiichi Manyama, Nov 28 2023

Keywords

Crossrefs

Cf. A141369, A367722, A367723.
Cf. A138292.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=0, (i-1)\2, (-1)^j*(2*j+1)*v[j+1]*v[i-2*j]/(j!*(i-1-2*j)!))); v;

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..floor((n-1)/2)} (-1)^k * (2*k+1) * a(k) * a(n-1-2*k) / (k! * (n-1-2*k)!).

A138738 E.g.f. satisfies: A(x) = exp(x*exp(x^2*exp(x^3*exp(...exp(x^n*...))))).

Original entry on oeis.org

1, 1, 1, 7, 25, 121, 1561, 9871, 101137, 1293265, 15765841, 226501111, 3355388521, 55348183177, 981791212585, 19611613003711, 387083895223201, 8241127257050401, 193814428658623777, 4572332597344204135, 116627907470591924281, 3142752742361513926681
Offset: 0

Views

Author

Paul D. Hanna, Mar 27 2008

Keywords

Comments

a(n) == 1 (mod 6) (conjecture).

Examples

			E.g.f: A(x) = 1 + x + 1/2*x^2 + 7/6*x^3 + 25/24*x^4 + 121/120*x^5 +...

Links

Crossrefs

Cf. A138292.

Programs

  • PARI
    {a(n)=local(A=1); for(i=0, n-1, A=exp(x^(n-i)*A+x*O(x^n))); n!*polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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