A124560 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).
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, 35, 158, 429, 348, 1, 1, 1, 7, 51, 317, 1455, 3716, 2321, 1, 1, 1, 8, 70, 556, 3634, 16918, 40272, 19437, 1, 1, 1, 9, 92, 891, 7581, 52199, 244644, 541655, 203554, 1
Offset: 0
Examples
The g.f. of row n, R_n(y), simultaneously satisfies:
R_n(y) = 1 + y*R_{n}(y)^n + y^2*R_{2n}(y)^(2n) + y^3*R_{3n}(y)^(3n) +...
more explicitly,
R_0 = 1 + y + y^2 + y^3 +... = 1/(1-y),
R_1 = 1 + y*(R_1)^1 + y^2*(R_2)^2 + y^3*(R_3)^3 + y^4*(R_4)^4 +...,
R_2 = 1 + y*(R_2)^2 + y^2*(R_4)^4 + y^3*(R_6)^6 + y^4*(R_8)^8 +...,
R_3 = 1 + y*(R_3)^3 + y^2*(R_6)^6 + y^3*(R_9)^9 + y^4*(R_12)^12 +...,
R_4 = 1 + y*(R_4)^4 + y^2*(R_8)^8 + y^3*(R_12)^12 + y^4*(R_16)^16 +...,
etc., for all rows.
Table begins:
1,1,1,1,1,1,1,1,1,1,...
1,1,2,5,16,66,348,2321,19437,203554,2661035,43399794,883165898,...
1,1,3,12,63,429,3716,40272,541655,9022405,186233087,4771577072,...
1,1,4,22,158,1455,16918,244644,4361883,95746603,2592416878,...
1,1,5,35,317,3634,52199,928608,20282765,543008771,17866390922,...
1,1,6,51,556,7581,128532,2689248,68880819,2155007000,82603481941,...
1,1,7,70,891,14036,272914,6525900,190604859,6781448755,...
1,1,8,92,1338,23864,521662,13975298,456468525,18121964864,...
1,1,9,117,1913,38055,921709,27263527,981599065,42880525630,...
1,1,10,145,2632,57724,1531900,49474783,1941904513,92344174075,...
1,1,11,176,3511,84111,2424288,84736940,3594121407,184465174294,...
1,1,12,210,4566,118581,3685430,138423924,6299505191,346530455866,...
1,1,13,247,5813,162624,5417683,217374894,10551425445,618507018238,...
1,1,14,287,7268,217855,7740500,330130230,17007128087,1057156741967,...
1,1,15,330,8947,286014,10791726,487184328,26523926691,1741018836674,...
1,1,16,376,10866,368966,14728894,701255202,40200085065,2776362938533,..
1,1,17,425,13041,468701,19730521,987570893,59420653233,4304220653087,..
Crossrefs
Programs
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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])}
Formula
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.