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A274390
Table of coefficients in the iterations of Euler's tree function (A000169), as read by antidiagonals.
10
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.
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. A274570 (transforms diagonals).
Cf. A274740 (same table, but read differently).
Sequence in context: A173003 A335461 A294411 * A244128 A016584 A293961
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jun 19 2016
STATUS
approved