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 A268652 G.f. satisfies: A(x,y) = 1 + x*y*A(x,y+1)^2. 3
 1, 0, 1, 0, 2, 2, 0, 9, 14, 5, 0, 64, 124, 74, 14, 0, 624, 1388, 1074, 352, 42, 0, 7736, 18964, 17292, 7520, 1588, 132, 0, 116416, 307088, 314356, 163728, 46561, 6946, 429, 0, 2060808, 5760704, 6434394, 3807910, 1311172, 266116, 29786, 1430, 0, 41952600, 122980872, 147159406, 95921164, 37846790, 9373620, 1438006, 126008, 4862, 0, 965497440, 2945806672, 3729264888, 2623904244, 1147995184, 327833296, 61731036, 7455440, 527900, 16796, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Column 1 equals A128577. Row sums equal A128318. Main diagonal equals the Catalan numbers, A000108. LINKS FORMULA The g.f. of the row sums, A(x,1), equals the limit of nested squares given by A(x,1) = 1 + x*(1 + 2*x*(1 + 3*x*(1 + 4*x*(...(1 + n*x*(...)^2)^2...)^2)^2)^2)^2. EXAMPLE This triangle of coefficients in g.f. A(x,y) begins: 1; 0, 1; 0, 2, 2; 0, 9, 14, 5; 0, 64, 124, 74, 14; 0, 624, 1388, 1074, 352, 42; 0, 7736, 18964, 17292, 7520, 1588, 132; 0, 116416, 307088, 314356, 163728, 46561, 6946, 429; 0, 2060808, 5760704, 6434394, 3807910, 1311172, 266116, 29786, 1430; 0, 41952600, 122980872, 147159406, 95921164, 37846790, 9373620, 1438006, 126008, 4862; 0, 965497440, 2945806672, 3729264888, 2623904244, 1147995184, 327833296, 61731036, 7455440, 527900, 16796; 0, 24786054816, 78270032288, 103887986400, 77816220888, 36954748286, 11761455804, 2565654006, 382043344, 37445610, 2195580, 58786; ... where the g.f. A(x,y) = 1 + x*y*A(x,y+1)^2 begins: A(x,y) = 1 + x*(y) + x^2*(2*y + 2*y^2) + x^3*(9*y + 14*y^2 + 5*y^3) + x^4*(64*y + 124*y^2 + 74*y^3 + 14*y^4) + x^5*(624*y + 1388*y^2 + 1074*y^3 + 352*y^4 + 42*y^5) + x^6*(7736*y + 18964*y^2 + 17292*y^3 + 7520*y^4 + 1588*y^5 + 132*y^6) + x^7*(116416*y + 307088*y^2 + 314356*y^3 + 163728*y^4 + 46561*y^5 + 6946*y^6 + 429*y^7) + x^8*(2060808*y + 5760704*y^2 + 6434394*y^3 + 3807910*y^4 + 1311172*y^5 + 266116*y^6 + 29786*y^7 + 1430*y^8) +... RELATED TRIANGLES. The triangle T1 of coefficients in A(x,y+1) begins: 1; 1, 1; 4, 6, 2; 28, 52, 29, 5; 276, 590, 430, 130, 14; 3480, 8240, 7142, 2902, 562, 42; 53232, 136352, 133820, 65892, 17440, 2380, 132; 955524, 2606056, 2811333, 1588813, 515738, 97246, 9949, 429; 19672320, 56489536, 65680352, 41222664, 15498120, 3613454, 514658, 41226, 1430; 456803328, 1369670752, 1692959656, 1154579428, 485522796, 131955696, 23376294, 2621102, 169766, 4862; 11810032896, 36744177952, 47799342376, 34885949644, 16033889224, 4899599348, 1016573628, 142394476, 12962360, 695860, 16796; ... in which row sums form A128571: [1, 2, 12, 114, 1440, 22368, 409248, 8585088, ...] where A(x,y+1) = 1 + x*(1 + y) + x^2*(4 + 6*y + 2*y^2) + x^3*(28 + 52*y + 29*y^2 + 5*y^3) + x^4*(276 + 590*y + 430*y^2 + 130*y^3 + 14*y^4) + x^5*(3480 + 8240*y + 7142*y^2 + 2902*y^3 + 562*y^4 + 42*y^5) + x^6*(53232 + 136352*y + 133820*y^2 + 65892*y^3 + 17440*y^4 + 2380*y^5 + 132*y^6) + x^7*(955524 + 2606056*y + 2811333*y^2 + 1588813*y^3 + 515738*y^4 + 97246*y^5 + 9949*y^6 + 429*y^7) +... The triangle T2 of coefficients in A(x,y)^2 begins: 1; 0, 2; 0, 4, 5; 0, 18, 32, 14; 0, 128, 270, 184, 42; 0, 1248, 2940, 2488, 928, 132; 0, 15472, 39513, 38364, 18266, 4372, 429; 0, 232832, 633296, 678712, 377332, 117430, 19776, 1430; 0, 4121616, 11800512, 13648092, 8478840, 3119480, 692086, 87112, 4862; 0, 83905200, 250768144, 308424612, 208690548, 86565216, 22913292, 3836896, 376736, 16796; 0, 1930994880, 5987236848, 7750642944, 5617656996, 2555316840, 767744018, 154465024, 20330760, 1607720, 58786; ... in which row sums form A128577: [1, 2, 9, 64, 624, 7736, 116416, 2060808, 41952600, ...] where A(x,y)^2 = 1 + x*(2*y) + x^2*(4*y + 5*y^2) + x^3*(18*y + 32*y^2 + 14*y^3) + x^4*(128*y + 270*y^2 + 184*y^3 + 42*y^4) + x^5*(1248*y + 2940*y^2 + 2488*y^3 + 928*y^4 + 132*y^5) + x^6*(15472*y + 39513*y^2 + 38364*y^3 + 18266*y^4 + 4372*y^5 + 429*y^6) + x^7*(232832*y + 633296*y^2 + 678712*y^3 + 377332*y^4 + 117430*y^5 + 19776*y^6 + 1430*y^7) +... PROG (PARI) /* Print this triangle of coefficients in A(x, y): */ {T(n, k) = my(A=1); for(i=1, n, A = 1 + x*y*subst(A, y, y+1)^2 +x*O(x^n)); polcoeff(polcoeff(A, n, x), k, y)} for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print("")) (PARI) /* Print triangle of coefficients in A(x, y+1): */ {T1(n, k) = my(A=1); for(i=1, n, A = 1 + x*y*subst(A, y, y+1)^2 +x*O(x^n)); polcoeff(polcoeff(subst(A, y, y+1), n, x), k, y)} for(n=0, 12, for(k=0, n, print1(T1(n, k), ", ")); print("")) (PARI) /* Print triangle of coefficients in A(x, y)^2: */ {T2(n, k) = my(A=1); for(i=1, n, A = 1 + x*y*subst(A, y, y+1)^2 +x*O(x^n)); polcoeff(polcoeff(A^2, n, x), k, y)} for(n=0, 12, for(k=0, n, print1(T2(n, k), ", ")); print("")) CROSSREFS Cf. A128577 (column 1), A128318 (row sums), A128570, A000108 (diagonal), A128571. Sequence in context: A179198 A117739 A243203 * A111810 A019265 A285539 Adjacent sequences:  A268649 A268650 A268651 * A268653 A268654 A268655 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Mar 16 2016 STATUS approved

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Last modified January 15 18:14 EST 2019. Contains 319153 sequences. (Running on oeis4.)