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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.

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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.

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.
Showing 1-3 of 3 results.

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