A274390 Table of coefficients in the iterations of Euler's tree function (A000169), as read by antidiagonals.

1, 1, 0, 1, 2, 0, 1, 4, 9, 0, 1, 6, 30, 64, 0, 1, 8, 63, 332, 625, 0, 1, 10, 108, 948, 4880, 7776, 0, 1, 12, 165, 2056, 18645, 89742, 117649, 0, 1, 14, 234, 3800, 50680, 454158, 1986124, 2097152, 0, 1, 16, 315, 6324, 112625, 1537524, 13221075, 51471800, 43046721, 0, 1, 18, 408, 9772, 219000, 4090980, 55494712, 448434136, 1530489744, 1000000000, 0, 1, 20, 513, 14288, 387205, 9266706, 176238685, 2325685632, 17386204761, 51395228090, 25937424601, 0, 1, 22, 630, 20016, 637520, 18704322, 463975764, 8793850560, 111107380464, 759123121050, 1924687118684, 743008370688, 0, 1, 24, 759, 27100, 993105, 34617288, 1067280319, 26858490392, 499217336145, 5964692819140, 36882981687519, 79553145323940, 23298085122481, 0

Offset

0,5

Comments

See table A274391 for the coefficients in exp( T^n(x) ), n>=0, where T^n(x) is the e.g.f. of the n-th row of this table.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1034 of rows 0..45 of the flattened table.

Formula

Let T^n(x) denote the n-th iteration of Euler's tree function T(x), then the coefficients in T^n(x) form the n-th row of this table, and the functions satisfy:

(1) T^n(x) = x * exp( Sum_{i=1..n} T^i(x) ).

(2) T^n(x) = T^(n-1)(x) * exp( T^n(x) ).

(3) T^n(x) = T^(n+1)( x/exp(x) ).

Example

This table begins:
1,  0,   0,     0,       0,        0,          0,            0, ...;
1,  2,   9,    64,     625,     7776,     117649,      2097152, ...;
1,  4,  30,   332,    4880,    89742,    1986124,     51471800, ...;
1,  6,  63,   948,   18645,   454158,   13221075,    448434136, ...;
1,  8, 108,  2056,   50680,  1537524,   55494712,   2325685632, ...;
1, 10, 165,  3800,  112625,  4090980,  176238685,   8793850560, ...;
1, 12, 234,  6324,  219000,  9266706,  463975764,  26858490392, ...;
1, 14, 315,  9772,  387205, 18704322, 1067280319,  70311813880, ...;
1, 16, 408, 14288,  637520, 34617288, 2217367600, 163802295616, ...;
1, 18, 513, 20016,  993105, 59879304, 4254311817, 348285415872, ...;
1, 20, 630, 27100, 1480000, 98110710, 7656893020, 688058734520, ...;
...
where the e.g.f.s of the rows are iterations of T(x) and begin:
T^0(x) = x;
T^1(x) = 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^2(x) = 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^3(x) = 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^4(x) = 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! +...;
...
where T^n(x)/exp( T^n(x) ) = T^n( x/exp(x) ) = T^(n-1)(x).
Also we have
T(x) = x*exp( T(x) );
T^2(x) = x*exp( T(x) + T^2(x) );
T^3(x) = x*exp( T(x) + T^2(x) + T^3(x) );
T^4(x) = x*exp( T(x) + T^2(x) + T^3(x) + T^4(x) ); ...

Prog

(PARI) {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 square array */
for(n=0, 10, for(k=1, 10, print1(T(n, k), ", ")); print(""))
/* Print this table as a flattened array */
for(n=0, 12, for(k=1, n, print1(T(n-k, k), ", ")); )

Crossrefs

Cf. A274391, A000169, A207833, A227278; diagonals: A274389, A274392.

Cf. A274570 (transforms diagonals).

Cf. A274740 (same table, but read differently).

Sequence in context: A173003 A335461 A294411 * A244128 A016584 A293961

Adjacent sequences: A274387 A274388 A274389 * A274391 A274392 A274393

Keyword

nonn,tabl

Author

Paul D. Hanna, Jun 19 2016

Status

approved