A392214 G.f. A(x) satisfies [x^n] A(x)^prime(n) = prime(n)^n for n >= 1.

1, 1, 2, 15, 206, 11144, 207797, 18714429, 494557540, 61317318250, 12455879194406, 410753791638467, 153648885183805408, 15880925795681080591, 896689682126823560788, 194260367501836970466125, 61230354518890254685729774, 17617164299664315219199900199, 1059970583614353180199012019329, 626501197254599686781125463416382

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..500

Example

G.f.: A(x) = 1 + x + 2*x^2 + 15*x^3 + 206*x^4 + 11144*x^5 + 207797*x^6 + 18714429*x^7 + 494557540*x^8 + 61317318250*x^9 + 12455879194406*x^10 + ...
Below we illustrate the property [x^n] A(x)^prime(n) = prime(n)^n for n >= 0; here prime(0) is taken to be 1.
The table of coefficients of x^k in A(x)^prime(n) begins
  n = 0: [1,  1,   2,   15,   206,  11144,  207797, ...];
  n = 1: [1,  2,   5,   34,   446,  22760,  438931, ...];
  n = 2: [1,  3,   9,   58,   726,  34905,  694208, ...];
  n = 3: [1,  5,  20,  125,  1435,  61051, 1280845, ...];
  n = 4: [1,  7,  35,  224,  2401,  90216, 1976023, ...];
  n = 5: [1, 11,  77,  550,  5456, 161051, 3735094, ...];
  n = 6: [1, 13, 104,  793,  7761, 204997, 4826809, ...]; ...
in which the main diagonal equals A307539:
[1, 2, 9, 125, 2401, 161051, 4826809, ..., prime(n)^n, ...].

Prog

(PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A = concat(A, 0); m = #A-1;
A[#A] = prime(m)^(m-1) - polcoef(Ser(A)^prime(m), m)/prime(m) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A392213, A381353, A307539 (prime(n)^n).

Sequence in context: A020565 A377861 A282521 * A099718 A143881 A380762

Adjacent sequences: A392211 A392212 A392213 * A392215 A392216 A392217

Keyword

nonn

Author

Paul D. Hanna, Feb 03 2026

Status

approved