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.
1, 1, 1, 1, 3, 3, 1, 7, 30, 14, 1, 15, 207, 550, 85, 1, 31, 1290, 14226, 15375, 621, 1, 63, 7803, 340550, 1852800, 601398, 5236, 1, 127, 46830, 8086594, 215528250, 409408077, 31299268, 49680, 1, 255, 280647, 192663030, 25359510515, 280823532696, 142286748933, 2093655600, 521721, 1, 511, 1682130, 4605331346, 3013207159725, 197431364485587, 676005054191880, 73448832515952, 175312873125, 5994155
Offset: 1
Examples
This table of coefficients T(n,k) begins:
n=1: [1, 1, 3, 14, 85, 621, 5236, ...];
n=2: [1, 3, 30, 550, 15375, 601398, 31299268, ...];
n=3: [1, 7, 207, 14226, 1852800, 409408077, 142286748933, ...];
n=4: [1, 15, 1290, 340550, 215528250, 280823532696, 676005054191880, ...];
n=5: [1, 31, 7803, 8086594, 25359510515, 197431364485587, ...];
n=6: [1, 63, 46830, 192663030, 3013207159725, 140620832995924134, ...];
n=7: [1, 127, 280647, 4605331346, 359881205186350, 100749338488125315273, 82972785219971584775198767, ...]; ...
such that row functions R(n,x) = Sum_{k>=1} T(n,k)*x^k satisfy:
[x^k] exp( k * R(n,x) ) = k^n * [x^(k-1)] exp( k * R(n,x) ) for k>=1.
Row functions R(n,x) begin:
R(1,x) = x + x^2 + 3*x^3 + 14*x^4 + 85*x^5 + 621*x^6 + 5236*x^7 + 49680*x^8 + ...
R(2,x) = x + 3*x^2 + 30*x^3 + 550*x^4 + 15375*x^5 + 601398*x^6 + 31299268*x^7 + ...
R(3,x) = x + 7*x^2 + 207*x^3 + 14226*x^4 + 1852800*x^5 + 409408077*x^6 + ...
R(4,x) = x + 15*x^2 + 1290*x^3 + 340550*x^4 + 215528250*x^5 + 280823532696*x^6 + ...
etc.
Links
Programs
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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)} for(n=1, 8, for(k=1,8, print1(T(n,k), ", "));print(""))