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%I #4 Mar 30 2012 18:37:01
%S 1,1,0,1,1,0,1,2,2,0,1,3,7,5,0,1,4,15,30,16,0,1,5,26,91,159,66,0,1,6,
%T 40,204,666,1056,348,0,1,7,57,385,1899,5955,8812,2321,0,1,8,77,650,
%U 4345,21180,65794,92062,19437,0,1,9,100,1015,8616,57876,287568,901881
%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 for n>=0, with R_0(y)=1.
%C Antidiagonal sums equal row 1 (A124551).
%F Let G_n(y) be the g.f. of row n in table A124560, then R_n(y) = G_n(y)^n and thus G_n(y) = Sum_{k>=0} y^k * R_{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) + y^2*R_{2n}(y) + y^3*R_{3n}(y) +...]^n
%e more explicitly,
%e R_0 = [1 + y + y^2 + y^3 +... ]^0 = 1,
%e R_1 = [1 + y*R_1 + y^2*R_2 + y^3*R_3 + y^4*R_4 +...]^1,
%e R_2 = [1 + y*R_2 + y^2*R_4 + y^3*R_6 + y^4*R_8 +...]^2,
%e R_3 = [1 + y*R_3 + y^2*R_6 + y^3*R_9 + y^4*R_12 +...]^3,
%e R_4 = [1 + y*R_4 + y^2*R_8 + y^3*R_12 + y^4*R_16 +...]^4,
%e etc., for all rows.
%e Table begins:
%e 1,0,0,0,0,0,0,0,0,0,...
%e 1,1,2,5,16,66,348,2321,19437,203554,2661035,43399794,883165898,...
%e 1,2,7,30,159,1056,8812,92062,1200415,19512990,395379699,9991017068,...
%e 1,3,15,91,666,5955,65794,901881,15346419,324465907,8535776700,...
%e 1,4,26,204,1899,21180,287568,4802716,99084889,2531896840,...
%e 1,5,40,385,4345,57876,926340,18088835,434349525,12879458545,...
%e 1,6,57,650,8616,133212,2447115,54419202,1481595429,49675372516,...
%e 1,7,77,1015,15449,271677,5621371,139777303,4236941723,157754261392,...
%e 1,8,100,1496,25706,506376,11637540,319211576,10629219251,...
%e 1,9,126,2109,40374,880326,22228296,665618589,24097683942,...
%e 1,10,155,2870,60565,1447752,39814650,1290831110,50395939380,...
%e 1,11,187,3795,87516,2275383,67666852,2359273213,98672395096,...
%e 1,12,222,4900,122589,3443748,110082100,4104444564,182882370066,...
%e 1,13,260,6201,167271,5048472,172579056,6848496031,323591733868,...
%e 1,14,301,7714,223174,7201572,262109169,11025158762,550236760920,...
%e 1,15,345,9455,292035,10032753,387284805,17206288875,903909656190,...
%e 1,16,392,11440,375716,13690704,558624184,26132289904,1440743993738,...
%e 1,17,442,13685,476204,18344394,788813124,38746675145,2235979092419,...
%e 1,18,495,16206,595611,24184368,1092983592,56235032046,3388787136045,...
%e 1,19,551,19019,736174,31424043,1489009062,80068650785,5027951628273,...
%e 1,20,610,22140,900255,40301004,1997816680,112053079180,7318490555455,...
%e 1,21,672,25585,1090341,51078300,2643716236,154381866075,10469322413655,..
%e 1,22,737,29370,1309044,64045740,3454745943,209695755346,14742078039007,..
%e 1,23,805,33511,1559101,79521189,4463035023,281147592671,20461165963557,..
%e 1,24,876,38024,1843374,97851864,5705183100,372473207208,28025203801701,..
%o (PARI) {T(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,T(n+i*n,j-1-i)) ) ))^n)[k+1], Vec(subst(Ser(concat(concat(0, Vec(subst(Ser(vector(k+1,j,T(j-1,k))),x,x/(1+x))/(1+x))),vector(n-k+1)) ),x,x/(1-x))/(1-x +x*O(x^(n))))[n]))))}
%Y Rows: A124551, A124552, A124553, A124554, A124555, A124556; diagonals: A124557, A124558, A124559; variants: A124560, A124460, A124530, A124540.
%K nonn,tabl
%O 0,8
%A _Paul D. Hanna_, Nov 07 2006
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