login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A145085 Square table, read by antidiagonals, where row e.g.f.s, R(n,x), satisfy: d/dx log( R(n,x) ) = R(n+1,x)^(n+1) 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 n+1 power, for n>=0. 11

%I

%S 1,1,1,1,1,2,1,1,3,7,1,1,4,17,39,1,1,5,31,151,322,1,1,6,49,373,1901,

%T 3723,1,1,7,71,741,6250,31851,57577,1,1,8,97,1291,15457,136711,680265,

%U 1147188,1,1,9,127,2059,32186,416661,3740137,17947631,28557909,1,1,10,161,3081,59677,1030491,13908049,124143598,571101141,866222535

%N Square table, read by antidiagonals, where row e.g.f.s, R(n,x), satisfy: d/dx log( R(n,x) ) = R(n+1,x)^(n+1) 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 n+1 power, for n>=0.

%H Paul D. Hanna, <a href="/A145085/b145085.txt">Table of n, a(n) for n = 0..1080</a>

%F Row e.g.f.s satisfy: R(n,x) = exp( Integral R(n+1,x)^(n+1) dx ).

%F Row e.g.f.s satisfy: R(n,x) = 1 + Integral R(n,x)*R(n+1,x)^(n+1) dx.

%F Row e.g.f.s satisfy: R'(n,x)/R(n,x) = R(n+1,x)^(n+1) with R(n,0) = 1.

%e Table begins:

%e 1,1,2,7,39,322,3723,57577,1147188,28557909,866222535,31362744620,...;

%e 1,1,3,17,151,1901,31851,680265,17947631,571101141,21507723971,...;

%e 1,1,4,31,373,6250,136711,3740137,124143598,4887140221,224203589593,...;

%e 1,1,5,49,741,15457,416661,13908049,557865765,26296627233,...;

%e 1,1,6,71,1291,32186,1030491,40606281,1911466016,105145651821,...;

%e 1,1,7,97,2059,59677,2211823,100479577,5431432483,341787359269,...;

%e 1,1,8,127,3081,101746,4283511,220384585,13453788426,953539677861,...;

%e 1,1,9,161,4393,162785,7672041,440897697,30000376553,2365207145121,...;

%e 1,1,10,199,6031,247762,12921931,820341289,61561430380,5344379824933,...;

%e 1,1,11,241,8031,362221,20710131,1439328361,118089834231,11194348009941,...;

%e 1,1,12,287,10429,512282,31860423,2405825577,214232473478,22019097106029,..;

%e 1,1,13,337,13261,704641,47357821,3860734705,370824076621,41076472798081,..;

%e 1,1,14,391,16563,946570,68362971,5983992457,616668950808,73237232298621,..;

%o (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]^m+x*O(x^k))))); k!*polcoeff(A[n+1],k,x)}

%o for(n=0, 10, for(k=0, 10, print1(T(n, k), ", ")); print(""))

%Y Cf. A145080; rows: A145086, A145087, A145088, A145089.

%K nonn,tabl

%O 0,6

%A _Paul D. Hanna_, Oct 01 2008

%E Entry corrected by _Paul D. Hanna_, Sep 22 2020

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 7 22:00 EST 2021. Contains 349589 sequences. (Running on oeis4.)