A274390 -id:A274390 - cp's OEIS frontend

A274570 Triangle, read by rows, that transforms diagonals in the array A274390 of coefficients in successive iterations of Euler's tree function (A000169).

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

Offset: 0

Paul D. Hanna, Jun 28 2016

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).

			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;
  ...

Paul D. Hanna, Table of n, a(n) for n = 0..860, of rows 0..40 of flattened triangle.

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

{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(""))

A274389 Main diagonal of rectangular array A274390 of coefficients in the iterations of Euler's tree function (A000169).

Original entry on oeis.org

1, 2, 30, 948, 50680, 4090980, 463975764, 70311813880, 13718193268896, 3348658563980040, 999698412743754460, 358297471515195652308, 151813934699349280088328, 75064081768759279536110316, 42833194538353991390132088540, 27937122503026656234469859408880, 20653210428143999114034181337343616, 17178393944175652034128269331788145680, 15970217696130529428248774113884778921452, 16497536217367322285994072192399435877530380, 18836957575278690757486149667782477659475272520

Offset: 0

Paul D. Hanna, Jun 24 2016

nonn

Paul D. Hanna, Table of n, a(n) for n = 0..100

Cf. A274390, A000169, A274392.

{ITERATE(F,n,k) = my(G=x +x*O(x^k)); for(i=1,n,G=subst(G,x,F));G}
{A274390(n,k) = my(TREE = serreverse(x*exp(-x +x*O(x^k)))); k!*polcoeff(ITERATE(TREE,n,k),k)}
/* Print A274390 */
for(n=0,10,for(k=1,10,print1(A274390(n,k),", "));print("..."))
/* Print this sequence, as the main diagonal of A274390 */
for(n=0,20,print1(A274390(n,n+1),", "))

A274392 A diagonal of rectangular array A274390 of coefficients in the iterations of Euler's tree function (A000169).

Original entry on oeis.org

1, 4, 63, 2056, 112625, 9266706, 1067280319, 163802295616, 32300931452769, 7956776354536450, 2394142654816299431, 863996246301971667600, 368314015001746325448313, 183100281424495288847092386, 104989565698848905178879275775, 68778360046311927838608116567296, 51049027217135211093037275781929857, 42614907995326324626989103964953188610, 39750079580111447237206552931429888023399, 41188867531604111691413161924808444678694800, 47163303540183246052916530453746351377795346681

Offset: 1

Paul D. Hanna, Jun 24 2016

nonn

Paul D. Hanna, Table of n, a(n) for n = 1..100

Cf. A274390, A000169, A274389.

{ITERATE(F,n,k) = my(G=x +x*O(x^k)); for(i=1,n,G=subst(G,x,F));G}
{A274390(n,k) = my(TREE = serreverse(x*exp(-x +x*O(x^k)))); k!*polcoeff(ITERATE(TREE,n,k),k)}
/* Print A274390 */
for(n=0,10,for(k=1,10,print1(A274390(n,k),", "));print("..."))
/* Print this sequence, as a diagonal of A274390 */
for(n=1,20,print1(A274390(n,n),", "))

A274391 Table of coefficients in functions that satisfy W_n(x) = W_{n-1}(x)^W_n(x), with W_0(x) = exp(x), as read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 16, 1, 1, 1, 7, 43, 125, 1, 1, 1, 9, 82, 525, 1296, 1, 1, 1, 11, 133, 1345, 8321, 16807, 1, 1, 1, 13, 196, 2729, 28396, 162463, 262144, 1, 1, 1, 15, 271, 4821, 71721, 734149, 3774513, 4782969, 1, 1, 1, 17, 358, 7765, 151376, 2300485, 22485898, 101808185, 100000000, 1, 1, 1, 19, 457, 11705, 283321, 5787931, 87194689, 796769201, 3129525793, 2357947691, 1, 1, 1, 21, 568, 16785, 486396, 12567187, 261066156, 3815719969, 32084546824, 108063152091, 61917364224, 1, 1, 1, 23, 691, 23149, 782321, 24539593, 656778529, 13577077401, 189440927857, 1447917011461, 4143297446729, 1792160394037, 1, 1, 1, 25, 826, 30941, 1195696, 44223529, 1457297878, 39536713209, 800175234736, 10525328121221, 72411962077126, 174723134310277, 56693912375296, 1

Offset: 0

Paul D. Hanna, Jun 19 2016

The e.g.f. of each row is an infinite exponential tetration of the e.g.f. of the prior row: W_{n+1}(x) = W_n(x)^W_n(x)^W_n(x)^..., starting with exp(x) as the e.g.f. of row zero. All of these row functions may be expressed in terms of the LambertW(x) function.

			This table begins:
1, 1,  1,   1,     1,       1,         1,          1,            1, ...;
1, 1,  3,  16,   125,    1296,     16807,     262144,      4782969, ...;
1, 1,  5,  43,   525,    8321,    162463,    3774513,    101808185, ...;
1, 1,  7,  82,  1345,   28396,    734149,   22485898,    796769201, ...;
1, 1,  9, 133,  2729,   71721,   2300485,   87194689,   3815719969, ...;
1, 1, 11, 196,  4821,  151376,   5787931,  261066156,  13577077401, ...;
1, 1, 13, 271,  7765,  283321,  12567187,  656778529,  39536713209, ...;
1, 1, 15, 358, 11705,  486396,  24539593, 1457297878,  99609347825, ...;
1, 1, 17, 457, 16785,  782321,  44223529, 2940281793, 224869459201, ...;
1, 1, 19, 568, 23149, 1195696,  74840815, 5506111864, 465734919289, ...;
1, 1, 21, 691, 30941, 1754001, 120403111, 9709554961, 899836571001, ...;
...
in which the e.g.f. of row n equals W_n(x) = exp( T^n(x) ), where T^n(x) is the n-th iteration of the Euler tree function T(x).
The row functions begin:
W_0(x) = 1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! + x^6/6! +...;
W_1(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! + 16807*x^6/6! + +...+ (n+1)^(n-1)*x^n/n! +...;
W_2(x) = 1 + x + 5*x^2/2! + 43*x^3/3! + 525*x^4/4! + 8321*x^5/5! + 162463*x^6/6! + +...+ A227176(n)*x^n/n! +...;
W_3(x) = 1 + x + 7*x^2/2! + 82*x^3/3! + 1345*x^4/4! + 28396*x^5/5! + 734149*x^6/6! +...+ A268653(n)*x^n/n! +...;
W_4(x) = 1 + x + 9*x^2/2! + 133*x^3/3! + 2729*x^4/4! + 71721*x^5/5! + 2300485*x^6/6! +...+ A268654(n)*x^n/n! +...;
W_5(x) = 1 + x + 11*x^2/2! + 196*x^3/3! + 4821*x^4/4! + 151376*x^5/5! + 5787931*x^6/6! +...;
W_6(x) = 1 + x + 13*x^2/2! + 271*x^3/3! + 7765*x^4/4! + 283321*x^5/5! + 12567187*x^6/6! +...;
...
and satisfy:
(0) W_0(x) = exp(x),
(1) W_1(x) = exp(x)^W_1(x) = exp(T(x)) = LambertW(-x)/(-x),
(2) W_2(x) = W_1(x)^W_2(x) = exp(T(T(x))),
(3) W_3(x) = W_2(x)^W_3(x) = exp(T(T(T(x)))),
(4) W_4(x) = W_3(x)^W_4(x) = exp(T(T(T(T(x))))),
...
Euler's tree function T(x), and its iterates begin:
T(x) = x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 625*x^5/5! + 7776*x^6/6! + 117649*x^7/7! + 2097152*x^8/8! +...+ n^(n-1)*x^n/n! +...
T(T(x)) = x + 4*x^2/2! + 30*x^3/3! + 332*x^4/4! + 4880*x^5/5! + 89742*x^6/6! + 1986124*x^7/7! + 51471800*x^8/8! +...+ A207833(n)*x^n/n! +...
T(T(T(x))) = x + 6*x^2/2! + 63*x^3/3! + 948*x^4/4! + 18645*x^5/5! + 454158*x^6/6! + 13221075*x^7/7! + 448434136*x^8/8! +...+ A227278(n)*x^n/n! +...
T(T(T(T(x)))) = x + 8*x^2/2! + 108*x^3/3! + 2056*x^4/4! + 50680*x^5/5! + 1537524*x^6/6! + 55494712*x^7/7! + 2325685632*x^8/8! +...
...
Note that the e.g.f. of the n-th row function, W_n(x), also equals the ratio of two iterates of the Euler tree function: W_n(x) = T^n(x) / T^(n-1)(x).
See A274390 for the table of coefficients in these iterated tree functions.

Paul D. Hanna, Table of n, a(n) for n = 0..1080, of rows 0..45 of the flattened table.

Cf. A274390, A000272, A227176, A268653, A268654; diagonals: A274387, A274388.

Cf. A274741 (same table, but read differently).

{ITERATE(F,n,k) = my(G=x +x*O(x^k)); for(i=1,n,G=subst(G,x,F));G}
{T(n,k) = my(TREE = serreverse(x*exp(-x +x*O(x^k)))); k!*polcoeff(exp(ITERATE(TREE,n,k)),k)}
/* Print this table as a square array */
for(n=0,10,for(k=0,10,print1(T(n,k),", "));print(""))
/* Print this table as a flattened array */
for(n=0,12,for(k=0,n,print1(T(n-k,k),", "));)

Let W_n(x) denote the e.g.f. of the n-th row function of this table, and T^n(x) the n-th iteration of Euler's tree function T(x) (cf. A274390), then
(1) W_n(x) = exp( T^n(x) ).
(2) W_n(x) = T^n(x) / T^(n-1)(x).
(3) W_n(x) = W_{n+1}( x/exp(x) ).
(4) W_n(x) = W_n( x/exp(x) )^W_n(x).

A274571 Column 0 of triangle A274570.

Original entry on oeis.org

1, 1, 7, 127, 4377, 245481, 20391523, 2354116899, 360734454993, 70865037282673, 17367953099244051, 5195706189463681299, 1863485648739527246569, 789370246456579516316121, 389929989984983783920348611, 222178771969671609391720410691, 144648509476124539709343154760897, 106712830948451223242311469280356609, 88557950557114913966623605248882438755, 82132537612235618834557329353828430430755

Offset: 0

Paul D. Hanna, Jun 28 2016

nonn

Triangle A274570 transforms diagonals in the array A274390 of coefficients of successive iterations of Euler's tree function (A000169).

Paul D. Hanna, Table of n, a(n) for n = 0..40

Cf. A274570, A274572, A274573, A274574.

{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 triangle A274570: */
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
/* Print this sequence, which is column 0 */
for(n=0,20,print1(T(n,0),", "))

A274572 Column 1 of triangle A274570.

Original entry on oeis.org

1, 2, 20, 470, 19912, 1326382, 127677580, 16767030632, 2880746218304, 627213971899610, 168767535712912684, 54994347890521005100, 21342142821229037730064, 9726400286221416303901358, 5143644030714149522751534524, 3124088412968372614077895431788, 2159818183532141245447039295746240, 1686295004858842334963772859214802354, 1476540037893212558044217633785452773068, 1440964034588041764141738802548853847618732

Offset: 0

Paul D. Hanna, Jun 28 2016

nonn

Triangle A274570 transforms diagonals in the array A274390 of coefficients of successive iterations of Euler's tree function (A000169).

Paul D. Hanna, Table of n, a(n) for n = 0..39

Cf. A274570, A274571, A274573, A274574.

{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 triangle A274570: */
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
/* Print this sequence, which is column 1 */
for(n=0,20,print1(T(n+1,1),", "))

A274573 Column 2 of triangle A274570.

Original entry on oeis.org

1, 3, 39, 1125, 56505, 4354923, 476265591, 70056231213, 13329387478113, 3184105899176739, 932720103991595919, 328710383679927689157, 137188135970104212440721, 66909975066823379752742835, 37706189062081696231083478647, 24312515006613477431766856702797, 17784145956730483348850500758855969, 14647274671009402833580157237962722147, 13492886769176913829412675003231182928079

Offset: 0

Paul D. Hanna, Jun 28 2016

nonn

Triangle A274570 transforms diagonals in the array A274390 of coefficients of successive iterations of Euler's tree function (A000169).

Paul D. Hanna, Table of n, a(n) for n = 0..38

Cf. A274570, A274571, A274572, A274574.

{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 triangle A274570: */
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
/* Print this sequence, which is column 2 */
for(n=0,20,print1(T(n+2,2),", "))

A274574 Central terms of triangle A274570.

Original entry on oeis.org

1, 2, 39, 2188, 247465, 47290506, 13732594855, 5645761143968, 3124313624563281, 2240929551882269890, 2023001689428835457551, 2245340983227461222262600, 3005921392102922941037743561, 4777188534537036418050441999458, 8892651921874938986718539648539335, 19167346139929272962512547586833106016, 47363669252787891219004826832547428944065, 133017373943189884985366059167726505579520322, 421334607602498277189468756234637306051458074191, 1495034827615578030423476599123008111000477082402040, 5906697677063490360959940664316005473632429506949424681

Offset: 0

Paul D. Hanna, Jun 28 2016

nonn

Triangle A274570 transforms diagonals in the array A274390 of coefficients of successive iterations of Euler's tree function (A000169).

Cf. A274570, A274571, A274572, A274573.

{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 triangle : */
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
/* Print this sequence, which is central terms */
for(n=0,20,print1(T(2*n,n),", "))

A274740 Table of coefficients in the iterations of Euler's tree function (A000169), as read by antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 9, 4, 1, 0, 64, 30, 6, 1, 0, 625, 332, 63, 8, 1, 0, 7776, 4880, 948, 108, 10, 1, 0, 117649, 89742, 18645, 2056, 165, 12, 1, 0, 2097152, 1986124, 454158, 50680, 3800, 234, 14, 1, 0, 43046721, 51471800, 13221075, 1537524, 112625, 6324, 315, 16, 1, 0, 1000000000, 1530489744, 448434136, 55494712, 4090980, 219000, 9772, 408, 18, 1, 0, 25937424601, 51395228090, 17386204761, 2325685632, 176238685, 9266706, 387205, 14288, 513, 20, 1

Offset: 0

Paul D. Hanna, Jul 04 2016

See examples and formulas at A274390, which is the main entry for this table.

This entry is the same as table A274390, but read by antidiagonals from top down.

			See examples at A274390, which is the main entry for this table.
This table 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, ...;
...
This table may also be written as a triangle:
1;
0, 1;
0, 2, 1;
0, 9, 4, 1;
0, 64, 30, 6, 1;
0, 625, 332, 63, 8, 1;
0, 7776, 4880, 948, 108, 10, 1;
0, 117649, 89742, 18645, 2056, 165, 12, 1;
0, 2097152, 1986124, 454158, 50680, 3800, 234, 14, 1;
0, 43046721, 51471800, 13221075, 1537524, 112625, 6324, 315, 16, 1;
0, 1000000000, 1530489744, 448434136, 55494712, 4090980, 219000, 9772, 408, 18, 1, 0;
...

Cf. A274390.

{ITERATE(F, n, k) = my(G=x +x*O(x^k)); for(i=1, n, G=subst(G, x, F)); G}
{T(n, k) = my(TREE = serreverse(x*exp(-x +x*O(x^k)))); k!*polcoeff(ITERATE(TREE, n, k), k)}
/* Print this table as a rectangular array */
for(n=0, 10, for(k=1, 10, print1(T(n, k), ", ")); print(""))
/* Print this table as a triangle */
for(n=1, 12, for(k=0, n-1, print1(T(k, n-k), ", "));print("") )
/* Print this table as a flattened array */
for(n=0, 12, for(k=0, n-1, print1(T(k, n-k), ", ")); )

See formulas at A274390, which is the main entry for this table.

cp's OEIS Frontend

A274570 Triangle, read by rows, that transforms diagonals in the array A274390 of coefficients in successive iterations of Euler's tree function (A000169).

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A274389 Main diagonal of rectangular array A274390 of coefficients in the iterations of Euler's tree function (A000169).

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A274392 A diagonal of rectangular array A274390 of coefficients in the iterations of Euler's tree function (A000169).

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A274391 Table of coefficients in functions that satisfy W_n(x) = W_{n-1}(x)^W_n(x), with W_0(x) = exp(x), as read by antidiagonals.

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Formula

A274571 Column 0 of triangle A274570.

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A274572 Column 1 of triangle A274570.

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A274573 Column 2 of triangle A274570.

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A274574 Central terms of triangle A274570.

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A274740 Table of coefficients in the iterations of Euler's tree function (A000169), as read by antidiagonals.

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