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.

A159315 E.g.f. satisfies: d/dx log(A(x)) = A(2*x)^(1/2).

Original entry on oeis.org

1, 1, 2, 7, 41, 406, 7127, 235147, 15191966, 1953128401, 501361942127, 257110692345262, 263513099974512041, 539923433830720468321, 2212048542930121133510402, 18123271334339868892408048927
Offset: 0

Views

Author

Paul D. Hanna, Apr 19 2009

Keywords

Comments

Row 0 of array A159314.

Examples

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 41*x^4/4! + 406*x^5/5! +...
Related expansions:
log(A(x)) = x +x^2/2! +3*x^3/3! +19*x^4/4! +225*x^5/5! +4801*x^6/6! +...
A(2*x)^(1/2) = 1 + x + 3*x^2/2! +19*x^3/3! +225*x^4/4! +4801*x^5/5! +...
in which the coefficients are given by A126444.

Links

Crossrefs

Cf. A159314, A126444, A159316, A159317.

Programs

  • PARI
    {a(n)=local(A=vector(n+2, j, 1+j*x)); for(i=0, n+1, for(j=0, n, m=n+1-j; A[m]=exp(intformal((A[m+1]+x*O(x^n))^(2^(m-1)))))); n!*polcoeff(A[1], n, x)}

Formula

E.g.f. satisfies: A'(x) = A(x)*A(2*x)^(1/2).
a(n) = Sum_{i=0..n-1} C(n-1,i)*A126444(i)*a(n-1-i) for n>0 with a(0)=1.
E.g.f.: A(x) = G(x/2)^2 where G(x) = e.g.f. of A126444.
E.g.f.: A(x) = F(x/4)^4 where F(x) = e.g.f. of A159316.
a(n) ~ c * 2^(n*(n-3)/2), where c = 14.6416352593041803546... - Vaclav Kotesovec, Feb 23 2014

A159316 E.g.f. A(x) satisfies: d/dx log(A(x)) = A(2*x)^2.

Original entry on oeis.org

1, 1, 5, 61, 1481, 66361, 5390285, 803252341, 224927827601, 121129543555441, 127545238071714965, 265238370995975176621, 1095520296374502654008921, 9015241470782090221556516521, 148067303294213271502974778276445
Offset: 0

Views

Author

Paul D. Hanna, Apr 19 2009

Keywords

Comments

Row 2 of array A159314.

Examples

			E.g.f.: A(x) = 1 +x +5*x^2/2! +61*x^3/3!+1481*x^4/4!+66361*x^5/5! +...
Related expansions:
log(A(x)) = x + 4*x^2/2! + 48*x^3/3! + 1216*x^4/4! + 57600*x^5/5! +...
A(2*x)^2 = 1 + 4*x + 48*x^2/2! + 1216*x^3/3! + 57600*x^4/4! +...
A(x)*A(2*x)^2 = 1 + 5*x +61*x^2/2! +1481*x^3/3! +66361*x^4/4! +...

Crossrefs

Cf. A159314, A159315, A126444.

Programs

  • PARI
    {a(n)=local(A=vector(n+4, j, 1+j*x)); for(i=0, n+3, for(j=0, n+2, m=n+3-j; A[m]=exp(intformal((A[m+1]+x*O(x^n))^(2^(m-1)))))); n!*polcoeff(A[3], n, x)}

Formula

E.g.f. satisfies: A'(x) = A(x)*A(2*x)^2.
a(n) = Sum_{i=0..n-1} C(n-1,i)*4^i*A126444(i)*a(n-1-i) for n>0 with a(0)=1.
E.g.f.: A(x) = G(2*x)^(1/2) where G(x) = e.g.f. of A126444.
E.g.f.: A(x) = F(4*x)^(1/4) where F(x) = e.g.f. of A159315.

A159317 a(n)/2^(n^2) is the coefficient of x^n/n! in F(x)^(1/2^n) where F(x) is the e.g.f. of A159315.

Original entry on oeis.org

1, 1, 5, 217, 81761, 240072001, 5184101454785, 817326468545940097, 958739380619551186754561, 8575669073854524479684954572801, 596451091280508109580869521043477279745
Offset: 0

Views

Author

Paul D. Hanna, Apr 19 2009

Keywords

Comments

Equals main diagonal of array A159314; A159315 equals row 0 of array A159314.

Examples

			E.g.f.: 1 + 1/2*x + 5/2^4*x^2/2! + 217/2^9*x^3/3! + 81761/2^16*x^4/4! +...
The e.g.f. of A159315 is:
F(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 41*x^4/4! + 406*x^5/5! + 7127*x^6/6! +...

Crossrefs

Cf. A159314, A159315, A126444.

Programs

  • PARI
    {a(n)=local(A=vector(2*n+2, j, 1+j*x)); for(i=0, 2*n+1, for(j=0, 2*n, m=2*n+1-j; A[m]=exp(intformal((A[m+1]+x*O(x^n))^(2^(m-1)))))); n!*polcoeff(A[n+1], n, x)}

Formula

E.g.f.: Sum_{n>=0} a(n)/2^(n^2)*x^n/n! = Sum_{n>=0} log(F(x/2^n))^n/n! where F(x) is the e.g.f. of A159315.
F(x)^(1/2^n) = R(n,x/2^n) where F(x)=R(0,x) and R(n,x) is the e.g.f. of row n of array A159314.
Showing 1-3 of 3 results.

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