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 A274570 Triangle, read by rows, that transforms diagonals in the array A274390 of coefficients in successive iterations of Euler's tree function (A000169). 6
 1, 1, 1, 7, 2, 1, 127, 20, 3, 1, 4377, 470, 39, 4, 1, 245481, 19912, 1125, 64, 5, 1, 20391523, 1326382, 56505, 2188, 95, 6, 1, 2354116899, 127677580, 4354923, 127056, 3755, 132, 7, 1, 360734454993, 16767030632, 476265591, 11117244, 247465, 5922, 175, 8, 1, 70865037282673, 2880746218304, 70056231213, 1360983976, 24228925, 436632, 8785, 224, 9, 1, 17367953099244051, 627213971899610, 13329387478113, 221585119536, 3281909155, 47290506, 716457, 12440, 279, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS This triangle also transforms diagonals in the array A274391 into each other, if we omit column 0 from those diagonals. The e.g.f. of row n of array A274391 equals exp(T^n(x)), where T^n(x) denotes the n-th iteration of Euler's tree function (A000169). LINKS EXAMPLE This triangle T(n,k), n>=0, k=0..n, begins: 1; 1, 1; 7, 2, 1; 127, 20, 3, 1; 4377, 470, 39, 4, 1; 245481, 19912, 1125, 64, 5, 1; 20391523, 1326382, 56505, 2188, 95, 6, 1; 2354116899, 127677580, 4354923, 127056, 3755, 132, 7, 1; 360734454993, 16767030632, 476265591, 11117244, 247465, 5922, 175, 8, 1; 70865037282673, 2880746218304, 70056231213, 1360983976, 24228925, 436632, 8785, 224, 9, 1; 17367953099244051, 627213971899610, 13329387478113, 221585119536, 3281909155, 47290506, 716457, 12440, 279, 10, 1; ... Let D denote the triangular matrix defined by D(n,k) = T(n,k)/(n-k)!, such that D begins: 1; 1, 1; 7/2!, 2, 1; 127/3!, 20/2!, 3, 1; 4377/4!, 470/3!, 39/2!, 4, 1; 245481/5!, 19912/4!, 1125/3!, 64/2!, 5, 1; 20391523/6!, 1326382/5!, 56505/4!, 2188/3!, 95/2!, 6, 1; ... then D transforms diagonals in the array A274390 into each other: D * [1, 2/2, 30/3!, 948/4!, 50680/5!, 4090980/6!, ...]~ = [1, 4/2!, 63/3!, 2056/4!, 112625/5!, 9266706/6!, ...]~; D * [1, 4/2!, 63/3!, 2056/4!, 112625/5!, 9266706/6!, ...]~ = [1, 6/2!, 108/3!, 3800/4!, 219000/5!, 18704322/6!, ...]~; D * [1, 6/2!, 108/3!, 3800/4!, 219000/5!, 18704322/6!, ...]~ = [1, 8/2!, 165/3!, 6324/4!, 387205/5!, 34617288/6!, ...]; ... where array A274390 consists of coefficients in the iterations of Euler's tree function (A000169), and begins: 1,  0,   0,     0,       0,        0,          0, ...; 1,  2,   9,    64,     625,     7776,     117649, ...; 1,  4,  30,   332,    4880,    89742,    1986124, ...; 1,  6,  63,   948,   18645,   454158,   13221075, ...; 1,  8, 108,  2056,   50680,  1537524,   55494712, ...; 1, 10, 165,  3800,  112625,  4090980,  176238685, ...; 1, 12, 234,  6324,  219000,  9266706,  463975764, ...; 1, 14, 315,  9772,  387205, 18704322, 1067280319, ...; 1, 16, 408, 14288,  637520, 34617288, 2217367600, ...; ... Note that this triangle also transforms the diagonals of table A274391 into each other, if we omit column 0 from those diagonals. After truncating column 0, table A274391 begins: 1,  1,   1,     1,       1,         1,          1, ...; 1,  3,  16,   125,    1296,     16807,     262144, ...; 1,  5,  43,   525,    8321,    162463,    3774513, ...; 1,  7,  82,  1345,   28396,    734149,   22485898, ...; 1,  9, 133,  2729,   71721,   2300485,   87194689, ...; 1, 11, 196,  4821,  151376,   5787931,  261066156, ...; 1, 13, 271,  7765,  283321,  12567187,  656778529, ...; 1, 15, 358, 11705,  486396,  24539593, 1457297878, ...; ... for which the e.g.f. of row n equals exp(T^n(x)) - 1, where T^n(x) denotes the n-th iteration of Euler's tree function (A000169). For example: D * [1, 3/2!, 43/3!, 1345/4!, 71721/5!, 5787931/6!, ...]~ = [1, 5/2!, 82/3!, 2729/4!, 151376/5!, 12567187/6!, ...]; D * [1, 5/2!, 82/3!, 2729/4!, 151376/5!, 12567187/6!, ...] = [1, 7/2!, 133/3!, 4821/4!, 283321/5!, 24539593/6!, ...]; D * [1, 7/2!, 133/3!, 4821/4!, 283321/5!, 24539593/6!, ... = [1, 9/2!, 196/3!, 7765/4!, 486396/5!, 44223529/6!, ...]; ... The matrix inverse of triangle D, as shown with elements [D^-1][n,k] * (n-k)!, begins: 1; -1, 1; -3, -2, 1; -40, -8, -3, 1; -1155, -140, -15, -4, 1; -57696, -5040, -324, -24, -5, 1; -4417175, -302092, -13923, -616, -35, -6, 1; -479964528, -26990720, -970848, -30720, -1040, -48, -7, 1; -70186001319, -3352727646, -98952435, -2439864, -58995, -1620, -63, -8, 1; -13284014648320, -551688200000, -13810202640, -279099200, -5254000, -102960, -2380, -80, -9, 1; -3158467118697099, -116039984093000, -2522473482375, -43202840076, -666167975, -10157796, -167475, -3344, -99, -10, 1; ... The matrix square of triangle D, as shown with elements [D^2][n,k] * (n-k)!, begins: 1; 2, 1; 18, 4, 1; 377, 52, 6, 1; 14304, 1414, 102, 8, 1; 859977, 65904, 3411, 168, 10, 1; 75306424, 4699274, 188496, 6668, 250, 12, 1; 9061819643, 476161840, 15542811, 426144, 11485, 348, 14, 1; 1435831150784, 65093379838, 1788015528, 39885108, 833280, 18162, 462, 16, 1; 289948340816657, 11551390491440, 273593165397, 5134299808, 87266525, 1474704, 26999, 592, 18, 1; ... PROG (PARI) {T(n, k)=local(F=x, LW=serreverse(x*exp(-x+x*O(x^(n+2)))), M, N, P, m=max(n, k)); M=matrix(m+3, m+3, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, LW)); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, M[r, c]); P=matrix(m+1, m+1, r, c, M[r+1, c]); (n-k)!*(P~*N~^-1)[n+1, k+1]} /* Print this triangle: */ for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print("")) CROSSREFS Cf. A274390, A274571, A274572, A274573, A274574. Sequence in context: A120455 A268895 A108433 * A176704 A289917 A198415 Adjacent sequences:  A274567 A274568 A274569 * A274571 A274572 A274573 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Jun 28 2016 STATUS approved

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Last modified October 21 22:47 EDT 2019. Contains 328315 sequences. (Running on oeis4.)