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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

LINKS

Table of n, a(n) for n=0..51.

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.

EXAMPLE

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).

PROG

(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)}

CROSSREFS

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

Sequence in context: A145085 A228904 A144512 * A135701 A051467 A243716

Adjacent sequences:  A159311 A159312 A159313 * A159315 A159316 A159317

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Apr 19 2009

STATUS

approved

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Last modified September 23 20:42 EDT 2021. Contains 347617 sequences. (Running on oeis4.)