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A300625 Table of row functions R(n,x) that satisfy: [x^k] exp( k^n * R(n,x) ) = k^n * [x^(k-1)] exp( k^n * R(n,x) ) for k>=1, n>=1, read by antidiagonals.

%I A300625 #11 Mar 13 2018 17:33:33
%S A300625 1,1,1,1,2,3,1,4,27,14,1,8,243,736,85,1,16,2187,40448,30525,621,1,32,
%T A300625 19683,2351104,12519125,1715454,5236,1,64,177147,142475264,6153518125,
%U A300625 6111917748,123198985,49680,1,128,1594323,8856272896,3436799053125,31779658925496,4308276119854,10931897664,521721,1,256,14348907,558312194048,2049047412828125,212148041589128016,287364845865893467,4151360558858752,1172808994833,5994155
%N A300625 Table of row functions R(n,x) that satisfy: [x^k] exp( k^n * R(n,x) ) = k^n * [x^(k-1)] exp( k^n * R(n,x) ) for k>=1, n>=1, read by antidiagonals.
%H A300625 Paul D. Hanna, <a href="/A300625/b300625.txt">Table of n, a(n) for n = 1..435 of rows 1..30 as a flattened table read by antidiagonals.</a>
%e A300625 This table of coefficients T(n,k) begins:
%e A300625 n=1: [1, 1, 3, 14, 85, 621, 5236, 49680, ...];
%e A300625 n=2: [1, 2, 27, 736, 30525, 1715454, 123198985, 10931897664, ...];
%e A300625 n=3: [1, 4, 243, 40448, 12519125, 6111917748, 4308276119854, ..];
%e A300625 n=4: [1, 8, 2187, 2351104, 6153518125, 31779658925496, ...];
%e A300625 n=5: [1, 16, 19683, 142475264, 3436799053125, 212148041589128016, ...];
%e A300625 n=6: [1, 32, 177147, 8856272896, 2049047412828125, 1569837215111038900704, ...];
%e A300625 n=7: [1, 64, 1594323, 558312194048, 1256793474918203125, 12020665333382306853887808, ...]; ...
%e A300625 such that row functions R(n,x) = Sum_{k>=1} T(n,k)*x^k satisfy:
%e A300625 [x^k] exp( k^n * R(n,x) ) = k^n * [x^(k-1)] exp( k^n * R(n,x) ) for k>=1.
%e A300625 Row functions R(n,x) begin:
%e A300625 R(1,x) = x + x^2 + 3*x^3 + 14*x^4 + 85*x^5 + 621*x^6 + 5236*x^7 + 49680*x^8 + ...
%e A300625 R(2,x) = x + 2*x^2 + 27*x^3 + 736*x^4 + 30525*x^5 + 1715454*x^6 + ...
%e A300625 R(3,x) = x + 4*x^2 + 243*x^3 + 40448*x^4 + 12519125*x^5 + 6111917748*x^6 + ...
%e A300625 R(4,x) = x + 8*x^2 + 2187*x^3 + 2351104*x^4 + 6153518125*x^5 + ...
%e A300625 etc.
%o A300625 (PARI) {T(n, k) = my(A=[1]); for(i=1, k+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)^n)); A[#A] = ((#A-1)^n * V[#A-1] - V[#A])/(#A-1)^n ); polcoeff( log(Ser(A)), k)}
%o A300625 /* Print as a table of row functions: */
%o A300625 for(n=1, 8, for(k=1, 8, print1(T(n, k), ", ")); print(""))
%o A300625 /* Print as a flattened triangle: */
%o A300625 for(n=1, 12, for(k=1, n-1, print1(T(n-k, k), ", ")); )
%Y A300625 Cf. A088716 (row 1), A300591 (row 2), A300595 (row 3), A300597 (row 4).
%K A300625 nonn,tabl
%O A300625 1,5
%A A300625 _Paul D. Hanna_, Mar 12 2018