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A300620 Table of row functions R(n,x) that satisfy: [x^k] exp( k * R(n,x) ) = k^n * [x^(k-1)] exp( k * R(n,x) ) for k>=1, n>=1, read by antidiagonals.

%I A300620 #8 Mar 27 2018 16:28:09
%S A300620 1,1,1,1,3,3,1,7,30,14,1,15,207,550,85,1,31,1290,14226,15375,621,1,63,
%T A300620 7803,340550,1852800,601398,5236,1,127,46830,8086594,215528250,
%U A300620 409408077,31299268,49680,1,255,280647,192663030,25359510515,280823532696,142286748933,2093655600,521721,1,511,1682130,4605331346,3013207159725,197431364485587,676005054191880,73448832515952,175312873125,5994155
%N A300620 Table of row functions R(n,x) that satisfy: [x^k] exp( k * R(n,x) ) = k^n * [x^(k-1)] exp( k * R(n,x) ) for k>=1, n>=1, read by antidiagonals.
%H A300620 Paul D. Hanna, <a href="/A300620/b300620.txt">Table of n, a(n) for n = 1..435 of rows 1..30 as a flattened table read by antidiagonals.</a>
%e A300620 This table of coefficients T(n,k) begins:
%e A300620 n=1: [1, 1, 3, 14, 85, 621, 5236, ...];
%e A300620 n=2: [1, 3, 30, 550, 15375, 601398, 31299268, ...];
%e A300620 n=3: [1, 7, 207, 14226, 1852800, 409408077, 142286748933, ...];
%e A300620 n=4: [1, 15, 1290, 340550, 215528250, 280823532696, 676005054191880, ...];
%e A300620 n=5: [1, 31, 7803, 8086594, 25359510515, 197431364485587, ...];
%e A300620 n=6: [1, 63, 46830, 192663030, 3013207159725, 140620832995924134, ...];
%e A300620 n=7: [1, 127, 280647, 4605331346, 359881205186350, 100749338488125315273, 82972785219971584775198767, ...]; ...
%e A300620 such that row functions R(n,x) = Sum_{k>=1} T(n,k)*x^k satisfy:
%e A300620 [x^k] exp( k * R(n,x) ) = k^n * [x^(k-1)] exp( k * R(n,x) ) for k>=1.
%e A300620 Row functions R(n,x) begin:
%e A300620 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 A300620 R(2,x) = x + 3*x^2 + 30*x^3 + 550*x^4 + 15375*x^5 + 601398*x^6 + 31299268*x^7 + ...
%e A300620 R(3,x) = x + 7*x^2 + 207*x^3 + 14226*x^4 + 1852800*x^5 + 409408077*x^6 + ...
%e A300620 R(4,x) = x + 15*x^2 + 1290*x^3 + 340550*x^4 + 215528250*x^5 + 280823532696*x^6 + ...
%e A300620 etc.
%o A300620 (PARI) {T(n,k) = my(A=[1]); for(i=1, k+1, A=concat(A, 0); V=Vec(Ser(A)^(#A-1)); A[#A] = ((#A-1)^n*V[#A-1] - V[#A])/(#A-1) ); polcoeff( log(Ser(A)), k)}
%o A300620 for(n=1, 8, for(k=1,8, print1(T(n,k), ", "));print(""))
%Y A300620 Cf. A088716 (row 1), A300617 (row 2), A300619 (row 3).
%K A300620 nonn,tabl
%O A300620 1,5
%A A300620 _Paul D. Hanna_, Mar 12 2018