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Rectangular table, read by antidiagonals, such that the g.f. of row n, R_n(y), satisfies: R_n(y) = Sum_{k>=0} y^k * [R_{n*k}(y)]^(n*k) for n>=0, with R_0(y)=1/(1-y).
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%I #4 Mar 30 2012 18:37:01

%S 1,1,1,1,1,1,1,1,2,1,1,1,3,5,1,1,1,4,12,16,1,1,1,5,22,63,66,1,1,1,6,

%T 35,158,429,348,1,1,1,7,51,317,1455,3716,2321,1,1,1,8,70,556,3634,

%U 16918,40272,19437,1,1,1,9,92,891,7581,52199,244644,541655,203554,1

%N Rectangular table, read by antidiagonals, such that the g.f. of row n, R_n(y), satisfies: R_n(y) = Sum_{k>=0} y^k * [R_{n*k}(y)]^(n*k) for n>=0, with R_0(y)=1/(1-y).

%F Let F_n(y) be the g.f. of row n in table A124550, then F_n(y) = R_n(y)^n and thus R_n(y) = Sum_{k>=0} y^k * F_{n*k}(y) for n>=0, where R_n(y) is the g.f. of row n in this table.

%e The g.f. of row n, R_n(y), simultaneously satisfies:

%e R_n(y) = 1 + y*R_{n}(y)^n + y^2*R_{2n}(y)^(2n) + y^3*R_{3n}(y)^(3n) +...

%e more explicitly,

%e R_0 = 1 + y + y^2 + y^3 +... = 1/(1-y),

%e R_1 = 1 + y*(R_1)^1 + y^2*(R_2)^2 + y^3*(R_3)^3 + y^4*(R_4)^4 +...,

%e R_2 = 1 + y*(R_2)^2 + y^2*(R_4)^4 + y^3*(R_6)^6 + y^4*(R_8)^8 +...,

%e R_3 = 1 + y*(R_3)^3 + y^2*(R_6)^6 + y^3*(R_9)^9 + y^4*(R_12)^12 +...,

%e R_4 = 1 + y*(R_4)^4 + y^2*(R_8)^8 + y^3*(R_12)^12 + y^4*(R_16)^16 +...,

%e etc., for all rows.

%e Table begins:

%e 1,1,1,1,1,1,1,1,1,1,...

%e 1,1,2,5,16,66,348,2321,19437,203554,2661035,43399794,883165898,...

%e 1,1,3,12,63,429,3716,40272,541655,9022405,186233087,4771577072,...

%e 1,1,4,22,158,1455,16918,244644,4361883,95746603,2592416878,...

%e 1,1,5,35,317,3634,52199,928608,20282765,543008771,17866390922,...

%e 1,1,6,51,556,7581,128532,2689248,68880819,2155007000,82603481941,...

%e 1,1,7,70,891,14036,272914,6525900,190604859,6781448755,...

%e 1,1,8,92,1338,23864,521662,13975298,456468525,18121964864,...

%e 1,1,9,117,1913,38055,921709,27263527,981599065,42880525630,...

%e 1,1,10,145,2632,57724,1531900,49474783,1941904513,92344174075,...

%e 1,1,11,176,3511,84111,2424288,84736940,3594121407,184465174294,...

%e 1,1,12,210,4566,118581,3685430,138423924,6299505191,346530455866,...

%e 1,1,13,247,5813,162624,5417683,217374894,10551425445,618507018238,...

%e 1,1,14,287,7268,217855,7740500,330130230,17007128087,1057156741967,...

%e 1,1,15,330,8947,286014,10791726,487184328,26523926691,1741018836674,...

%e 1,1,16,376,10866,368966,14728894,701255202,40200085065,2776362938533,..

%e 1,1,17,425,13041,468701,19730521,987570893,59420653233,4304220653087,..

%o (PARI) {A124550(n,k)=if(k==0,1,if(n==0,0,if(k==1,n,if(n<=k, Vec(( 1+x*Ser( vector(k,j,sum(i=0,j-1,A124550(n+i*n,j-1-i)) ) ))^n)[k+1], Vec(subst(Ser(concat(concat(0, Vec(subst(Ser(vector(k+1,j,A124550(j-1,k))),x,x/(1+x))/(1+x))),vector(n-k+1)) ),x,x/(1-x))/(1-x +x*O(x^(n))))[n]))))} /* Determined Elements from A124550: */ {T(n,k)=if(n==0|k==0,1,Vec((Ser(vector(k+1,j,A124550(n,j-1)))+x*O(x^k))^(1/n))[k+1])}

%Y Rows: A124551, A124562, A124563, A124564, A124565, A124566; diagonals: A124567, A124568, A124569; A124561 (antidiagonal sums); variants: A124550, A124460, A124530, A124540.

%K nonn,tabl

%O 0,9

%A _Paul D. Hanna_, Nov 07 2006