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!)
A145080 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. 10
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, 4029, 18710, 31851, 1, 7, 78, 965, 10756, 88491, 412402, 680265, 1, 8, 105, 1602, 23635, 288764, 2450085, 11253638, 17947631, 1, 9, 136, 2471, 45606, 750905 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..1325 (this square table read by antidiagonals).

Emma Colaric, Ryan DeMuse, Jeremy L. Martin, and Mei Yin, Interval parking functions, arXiv:2006.09321 [math.CO], 2020.

FORMULA

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

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

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

EXAMPLE

Table begins:

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

1,2,10,86,1090,18710,412402,11253638,370191682,14385490550,...;

1,3,21,243,4029,88491,2450085,82648611,3313381293,154912893243,...;

1,4,36,524,10756,288764,9667476,390576684,18591797156,1023871865244,...;

1,5,55,965,23635,750905,29542255,1393871405,77048083675,...;

1,6,78,1602,45606,1674546,75766566,4093500690,258080963526,...;

1,7,105,2471,80185,3341975,171024777,10417654023,738359218457,...;

1,8,136,3608,131464,6132536,350289640,23758306136,1870826922568,...;

1,9,171,5049,204111,10537029,664636851,49684658769,4304660616711,...;

1,10,210,6830,303370,17172110,1185578010,96858862110,9158464815050,...;

1,11,253,8987,435061,26794691,2009911981,178179417515,18260617937221,..;

1,12,300,11556,605580,40316340,3265094652,312177663108,34472306461932,.;

PROG

(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)}

for(n=1, 10, for(k=0, 10, print1(T(n, k), ", ")); print(""))

(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)}

for(n=1, 10, for(k=0, 10, print1(T(n, k), ", ")); print(""))

CROSSREFS

Cf. A145085; rows: A145081, A145082, A145083, A145084.

Sequence in context: A010027 A151880 A108990 * A065078 A203989 A126744

Adjacent sequences:  A145077 A145078 A145079 * A145081 A145082 A145083

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Sep 30 2008

STATUS

approved

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 8 17:01 EST 2021. Contains 349596 sequences. (Running on oeis4.)