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 A124550 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. 18
 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, 40, 204, 666, 1056, 348, 0, 1, 7, 57, 385, 1899, 5955, 8812, 2321, 0, 1, 8, 77, 650, 4345, 21180, 65794, 92062, 19437, 0, 1, 9, 100, 1015, 8616, 57876, 287568, 901881 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS Antidiagonal sums equal row 1 (A124551). LINKS FORMULA 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. EXAMPLE The g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = [1 + y*R_{n}(y) + y^2*R_{2n}(y) + y^3*R_{3n}(y) +...]^n more explicitly, R_0 = [1 + y + y^2 + y^3 +... ]^0 = 1, R_1 = [1 + y*R_1 + y^2*R_2 + y^3*R_3 + y^4*R_4 +...]^1, R_2 = [1 + y*R_2 + y^2*R_4 + y^3*R_6 + y^4*R_8 +...]^2, R_3 = [1 + y*R_3 + y^2*R_6 + y^3*R_9 + y^4*R_12 +...]^3, R_4 = [1 + y*R_4 + y^2*R_8 + y^3*R_12 + y^4*R_16 +...]^4, etc., for all rows. Table begins: 1,0,0,0,0,0,0,0,0,0,... 1,1,2,5,16,66,348,2321,19437,203554,2661035,43399794,883165898,... 1,2,7,30,159,1056,8812,92062,1200415,19512990,395379699,9991017068,... 1,3,15,91,666,5955,65794,901881,15346419,324465907,8535776700,... 1,4,26,204,1899,21180,287568,4802716,99084889,2531896840,... 1,5,40,385,4345,57876,926340,18088835,434349525,12879458545,... 1,6,57,650,8616,133212,2447115,54419202,1481595429,49675372516,... 1,7,77,1015,15449,271677,5621371,139777303,4236941723,157754261392,... 1,8,100,1496,25706,506376,11637540,319211576,10629219251,... 1,9,126,2109,40374,880326,22228296,665618589,24097683942,... 1,10,155,2870,60565,1447752,39814650,1290831110,50395939380,... 1,11,187,3795,87516,2275383,67666852,2359273213,98672395096,... 1,12,222,4900,122589,3443748,110082100,4104444564,182882370066,... 1,13,260,6201,167271,5048472,172579056,6848496031,323591733868,... 1,14,301,7714,223174,7201572,262109169,11025158762,550236760920,... 1,15,345,9455,292035,10032753,387284805,17206288875,903909656190,... 1,16,392,11440,375716,13690704,558624184,26132289904,1440743993738,... 1,17,442,13685,476204,18344394,788813124,38746675145,2235979092419,... 1,18,495,16206,595611,24184368,1092983592,56235032046,3388787136045,... 1,19,551,19019,736174,31424043,1489009062,80068650785,5027951628273,... 1,20,610,22140,900255,40301004,1997816680,112053079180,7318490555455,... 1,21,672,25585,1090341,51078300,2643716236,154381866075,10469322413655,.. 1,22,737,29370,1309044,64045740,3454745943,209695755346,14742078039007,.. 1,23,805,33511,1559101,79521189,4463035023,281147592671,20461165963557,.. 1,24,876,38024,1843374,97851864,5705183100,372473207208,28025203801701,.. PROG (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]))))} CROSSREFS Rows: A124551, A124552, A124553, A124554, A124555, A124556; diagonals: A124557, A124558, A124559; variants: A124560, A124460, A124530, A124540. Sequence in context: A144074 A261780 A124540 * A306024 A237018 A290605 Adjacent sequences:  A124547 A124548 A124549 * A124551 A124552 A124553 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Nov 07 2006 STATUS approved

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Last modified March 22 01:06 EDT 2019. Contains 321406 sequences. (Running on oeis4.)