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A145080
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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.
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9
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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; internal format)
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OFFSET
| 0,5
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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.
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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,.;
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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)}
(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)}
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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
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KEYWORD
| nonn,tabl
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Sep 30 2008
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