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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

%I

%S 1,1,1,7,2,1,127,20,3,1,4377,470,39,4,1,245481,19912,1125,64,5,1,

%T 20391523,1326382,56505,2188,95,6,1,2354116899,127677580,4354923,

%U 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

%N Triangle, read by rows, that transforms diagonals in the array A274390 of coefficients in successive iterations of Euler's tree function (A000169).

%C 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).

%H Paul D. Hanna, <a href="/A274570/b274570.txt">Table of n, a(n) for n = 0..860, of rows 0..40 of flattened triangle.</a>

%e This triangle T(n,k), n>=0, k=0..n, begins:

%e 1;

%e 1, 1;

%e 7, 2, 1;

%e 127, 20, 3, 1;

%e 4377, 470, 39, 4, 1;

%e 245481, 19912, 1125, 64, 5, 1;

%e 20391523, 1326382, 56505, 2188, 95, 6, 1;

%e 2354116899, 127677580, 4354923, 127056, 3755, 132, 7, 1;

%e 360734454993, 16767030632, 476265591, 11117244, 247465, 5922, 175, 8, 1;

%e 70865037282673, 2880746218304, 70056231213, 1360983976, 24228925, 436632, 8785, 224, 9, 1;

%e 17367953099244051, 627213971899610, 13329387478113, 221585119536, 3281909155, 47290506, 716457, 12440, 279, 10, 1;

%e ...

%e Let D denote the triangular matrix defined by D(n,k) = T(n,k)/(n-k)!, such that D begins:

%e 1;

%e 1, 1;

%e 7/2!, 2, 1;

%e 127/3!, 20/2!, 3, 1;

%e 4377/4!, 470/3!, 39/2!, 4, 1;

%e 245481/5!, 19912/4!, 1125/3!, 64/2!, 5, 1;

%e 20391523/6!, 1326382/5!, 56505/4!, 2188/3!, 95/2!, 6, 1;

%e ...

%e then D transforms diagonals in the array A274390 into each other:

%e D * [1, 2/2, 30/3!, 948/4!, 50680/5!, 4090980/6!, ...]~ =

%e [1, 4/2!, 63/3!, 2056/4!, 112625/5!, 9266706/6!, ...]~;

%e D * [1, 4/2!, 63/3!, 2056/4!, 112625/5!, 9266706/6!, ...]~ =

%e [1, 6/2!, 108/3!, 3800/4!, 219000/5!, 18704322/6!, ...]~;

%e D * [1, 6/2!, 108/3!, 3800/4!, 219000/5!, 18704322/6!, ...]~ =

%e [1, 8/2!, 165/3!, 6324/4!, 387205/5!, 34617288/6!, ...];

%e ...

%e where array A274390 consists of coefficients in the iterations of Euler's tree function (A000169), and begins:

%e 1, 0, 0, 0, 0, 0, 0, ...;

%e 1, 2, 9, 64, 625, 7776, 117649, ...;

%e 1, 4, 30, 332, 4880, 89742, 1986124, ...;

%e 1, 6, 63, 948, 18645, 454158, 13221075, ...;

%e 1, 8, 108, 2056, 50680, 1537524, 55494712, ...;

%e 1, 10, 165, 3800, 112625, 4090980, 176238685, ...;

%e 1, 12, 234, 6324, 219000, 9266706, 463975764, ...;

%e 1, 14, 315, 9772, 387205, 18704322, 1067280319, ...;

%e 1, 16, 408, 14288, 637520, 34617288, 2217367600, ...;

%e ...

%e Note that this triangle also transforms the diagonals of table A274391 into each other, if we omit column 0 from those diagonals.

%e After truncating column 0, table A274391 begins:

%e 1, 1, 1, 1, 1, 1, 1, ...;

%e 1, 3, 16, 125, 1296, 16807, 262144, ...;

%e 1, 5, 43, 525, 8321, 162463, 3774513, ...;

%e 1, 7, 82, 1345, 28396, 734149, 22485898, ...;

%e 1, 9, 133, 2729, 71721, 2300485, 87194689, ...;

%e 1, 11, 196, 4821, 151376, 5787931, 261066156, ...;

%e 1, 13, 271, 7765, 283321, 12567187, 656778529, ...;

%e 1, 15, 358, 11705, 486396, 24539593, 1457297878, ...;

%e ...

%e 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).

%e For example:

%e D * [1, 3/2!, 43/3!, 1345/4!, 71721/5!, 5787931/6!, ...]~ =

%e [1, 5/2!, 82/3!, 2729/4!, 151376/5!, 12567187/6!, ...];

%e D * [1, 5/2!, 82/3!, 2729/4!, 151376/5!, 12567187/6!, ...] =

%e [1, 7/2!, 133/3!, 4821/4!, 283321/5!, 24539593/6!, ...];

%e D * [1, 7/2!, 133/3!, 4821/4!, 283321/5!, 24539593/6!, ... =

%e [1, 9/2!, 196/3!, 7765/4!, 486396/5!, 44223529/6!, ...];

%e ...

%e The matrix inverse of triangle D, as shown with elements [D^-1][n,k] * (n-k)!, begins:

%e 1;

%e -1, 1;

%e -3, -2, 1;

%e -40, -8, -3, 1;

%e -1155, -140, -15, -4, 1;

%e -57696, -5040, -324, -24, -5, 1;

%e -4417175, -302092, -13923, -616, -35, -6, 1;

%e -479964528, -26990720, -970848, -30720, -1040, -48, -7, 1;

%e -70186001319, -3352727646, -98952435, -2439864, -58995, -1620, -63, -8, 1;

%e -13284014648320, -551688200000, -13810202640, -279099200, -5254000, -102960, -2380, -80, -9, 1;

%e -3158467118697099, -116039984093000, -2522473482375, -43202840076, -666167975, -10157796, -167475, -3344, -99, -10, 1;

%e ...

%e The matrix square of triangle D, as shown with elements [D^2][n,k] * (n-k)!, begins:

%e 1;

%e 2, 1;

%e 18, 4, 1;

%e 377, 52, 6, 1;

%e 14304, 1414, 102, 8, 1;

%e 859977, 65904, 3411, 168, 10, 1;

%e 75306424, 4699274, 188496, 6668, 250, 12, 1;

%e 9061819643, 476161840, 15542811, 426144, 11485, 348, 14, 1;

%e 1435831150784, 65093379838, 1788015528, 39885108, 833280, 18162, 462, 16, 1;

%e 289948340816657, 11551390491440, 273593165397, 5134299808, 87266525, 1474704, 26999, 592, 18, 1;

%e ...

%o (PARI) {T(n, k)=local(F=x,

%o LW=serreverse(x*exp(-x+x*O(x^(n+2)))), M, N, P, m=max(n, k));

%o M=matrix(m+3, m+3, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, LW)); polcoeff(F, c));

%o N=matrix(m+1, m+1, r, c, M[r, c]);

%o P=matrix(m+1, m+1, r, c, M[r+1, c]);

%o (n-k)!*(P~*N~^-1)[n+1, k+1]}

%o /* Print this triangle: */

%o for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

%Y Cf. A274390, A274571, A274572, A274573, A274574.

%K nonn,tabl

%O 0,4

%A _Paul D. Hanna_, Jun 28 2016

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Last modified November 13 02:59 EST 2019. Contains 329085 sequences. (Running on oeis4.)