|
%I #13 Sep 20 2020 19:17:33
%S 1,1,1,1,2,3,1,3,10,17,1,4,21,86,151,1,5,36,243,1090,1901,1,6,55,524,
%T 4029,18710,31851,1,7,78,965,10756,88491,412402,680265,1,8,105,1602,
%U 23635,288764,2450085,11253638,17947631,1,9,136,2471,45606,750905
%N Square table, read by antidiagonals, where row e.g.f.s, R(n,x), satisfy: d/dx log( R(n,x) )/n = R(n+1,x) with R(n,0) = 1; that is, the logarithmic derivative of the e.g.f. of row n, divided by n, equals the e.g.f. of row n+1, for n>=1.
%H Paul D. Hanna, <a href="/A145080/b145080.txt">Table of n, a(n) for n = 0..1325 (this square table read by antidiagonals).</a>
%H Emma Colaric, Ryan DeMuse, Jeremy L. Martin, and Mei Yin, <a href="https://arxiv.org/abs/2006.09321">Interval parking functions</a>, arXiv:2006.09321 [math.CO], 2020.
%F Row e.g.f.s satisfy: R(n,x) = exp( n*Integral R(n+1,x) dx ).
%F Row e.g.f.s satisfy: R(n,x) = 1 + n*Integral R(n,x)*R(n+1,x) dx.
%F Row e.g.f.s satisfy: R'(n,x)/R(n,x) = n*R(n+1,x) with R(n,0) = 1.
%e Table begins:
%e 1,1,3,17,151,1901,31851,680265,17947631,571101141,21507723971,...;
%e 1,2,10,86,1090,18710,412402,11253638,370191682,14385490550,...;
%e 1,3,21,243,4029,88491,2450085,82648611,3313381293,154912893243,...;
%e 1,4,36,524,10756,288764,9667476,390576684,18591797156,1023871865244,...;
%e 1,5,55,965,23635,750905,29542255,1393871405,77048083675,...;
%e 1,6,78,1602,45606,1674546,75766566,4093500690,258080963526,...;
%e 1,7,105,2471,80185,3341975,171024777,10417654023,738359218457,...;
%e 1,8,136,3608,131464,6132536,350289640,23758306136,1870826922568,...;
%e 1,9,171,5049,204111,10537029,664636851,49684658769,4304660616711,...;
%e 1,10,210,6830,303370,17172110,1185578010,96858862110,9158464815050,...;
%e 1,11,253,8987,435061,26794691,2009911981,178179417515,18260617937221,..;
%e 1,12,300,11556,605580,40316340,3265094652,312177663108,34472306461932,.;
%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(m*intformal(A[m+1]+x*O(x^k))))); k!*polcoeff(A[n],k,x)}
%o for(n=1,10, for(k=0,10, print1(T(n,k),", "));print(""))
%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]^n,k,x)}
%o for(n=1,10, for(k=0,10, print1(T(n,k),", "));print(""))
%Y Cf. A145085; rows: A145081, A145082, A145083, A145084.
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Sep 30 2008
|
