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.

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

A159314 Rectangular array, read by antidiagonals, where row e.g.f.s, R(n,x), satisfy: d/dx log( R(n,x) ) = R(n+1,x)^(2^n) with R(n,0) = 1; that is, the logarithmic derivative of the e.g.f. of row n equals the e.g.f. of row n+1 to the 2^n power, for n>=0.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 7, 1, 1, 5, 19, 41, 1, 1, 9, 61, 225, 406, 1, 1, 17, 217, 1481, 4801, 7127, 1, 1, 33, 817, 10737, 66361, 185523, 235147, 1, 1, 65, 3169, 81761, 988561, 5390285, 13298659, 15191966, 1, 1, 129, 12481, 638145, 15269281, 164637369
Offset: 0

Views

Author

Paul D. Hanna, Apr 19 2009

Keywords

Examples

			Array begins:
1,1,2,7,41,406,7127,235147,15191966,1953128401,501361942127,...;
1,1,3,19,225,4801,185523,13298659,1815718305,481790947681,...;
1,1,5,61,1481,66361,5390285,803252341,224927827601,...;
1,1,9,217,10737,988561,164637369,49987302697,28333326990177,...;
1,1,17,817,81761,15269281,5149256177,3155353490257,...;
1,1,33,3169,638145,240072001,162919458273,200565037419169,...;
1,1,65,12481,5042561,3807826561,5184101454785,12792473234253121,...;
1,1,129,49537,40092417,60660860161,165425163421569,...;
1,1,257,197377,319751681,968467745281,5286172203486977,...;
1,1,513,787969,2554072065,15478671283201,169038775947894273,...;
1,1,1025,3148801,20416829441,247524381173761,5407342625815542785,...;
...
where row e.g.f.s begin:
R(0,x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 41*x^4/4! + 406*x^5/5! +...;
R(1,x) = 1 + x + 3*x^2/2! +19*x^3/3! +225*x^4/4! +4801*x^5/5! +...;
R(2,x) = 1 + x + 5*x^2/2! +61*x^3/3!+1481*x^4/4!+66361*x^5/5! +...;
...
Row e.g.f.s satisfy: R(n+1,x)^(2^n) = d/dx log( R(n,x) ):
R(1,x)^1 = d/dx log(1+x +2*x^2/2! +7*x^3/3! +41*x^4/4! +...);
R(2,x)^2 = d/dx log(1+x +3*x^2/2! +19*x^3/3! +225*x^4/4! +...);
R(3,x)^4 = d/dx log(1+x +5*x^2/2! +61*x^3/3! +1481*x^4/4! +...);
R(4,x)^8 = d/dx log(1+x +9*x^2/2! +217*x^3/3! +10737*x^4/4! +...);
...
Examples of R(n,x) = R(n+m,x/2^m)^(2^m):
R(n-1,x) = R(n,x/2)^2 and R(n+1,x) = R(n,2x)^(1/2);
R(0,x) = R(n,x/2^n)^(2^n) and R(n,x) = R(0,2^n*x)^(1/2^n).

Crossrefs

Cf. rows: A159315, A126444, A159316, diagonal: A159317, variant: A145085.

Programs

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

Formula

T(n,k) = Sum_{i=0..k-1} C(k-1,i)*2^(n*i)*T(1,i)*T(n,k-1-i) for k>0 with T(n,0)=1, for n>=0.
Row e.g.f.s, R(n,x), satisfy:
(1) R'(n,x)/R(n,x) = R(n+1,x)^(2^n) with R(n,0) = 1;
(2) R(n,x) = R(n+m,x/2^m)^(2^m) for m >= -n.

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.

A385983 a(0) = 1; a(n) = Sum_{k=0..n-1} 3^k * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 4, 46, 1432, 123808, 30876832, 22731703408, 49898049707776, 327831911519538304, 6455998409280026369536, 381291302353791118798096384, 67549186687935750257213597283328, 35899285521583612190120694413539704832, 57235559192922896714567337515980987820597248
Offset: 0

Views

Author

Seiichi Manyama, Jul 14 2025

Keywords

Crossrefs

Cf. A000142, A126444, A155585, A385984.

Programs

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

Formula

E.g.f. A(x) satisfies A'(x) = A(x) * A(3*x).

A385984 a(0) = 1; a(n) = Sum_{k=0..n-1} (-2)^k * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, -1, -9, 57, 1353, -38313, -2796417, 339169041, 90178580529, -45316930884849, -46917802526957721, 95533688640942728073, 392558870984301366092217, -3210372581644929567134113497, -52647023496165910533698485658193, 1724296469950918188679460249845485729
Offset: 0

Views

Author

Seiichi Manyama, Jul 14 2025

Keywords

Crossrefs

Cf. A000142, A126444, A155585, A385983.

Programs

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

Formula

E.g.f. A(x) satisfies A'(x) = A(x) * A(-2*x).
Showing 1-6 of 6 results.

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