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 A320668 O.g.f. A(x) satisfies: [x^n] exp(-n^3*A(x)) / (1 - n*x)^(n^2) = 0, for n > 0. 2
 1, 1, 3, 48, 1125, 74844, 4538576, 571979264, 61768818081, 11756208796500, 1930305045364047, 501690433505046336, 114627985830970025544, 38401761759325497631504, 11530876917646339177773375, 4792821920208552461683208192, 1816651428077402993910096849969, 911361374568809242258003199407404 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS It is remarkable that this sequence should consist entirely of integers. LINKS EXAMPLE O.g.f.: A(x) = x + x^2 + 3*x^3 + 48*x^4 + 1125*x^5 + 74844*x^6 + 4538576*x^7 + 571979264*x^8 + 61768818081*x^9 + 11756208796500*x^10 + ... ILLUSTRATION OF DEFINITION. The table of coefficients of x^k/k! in exp(-n^3*A(x)) / (1 - n*x)^(n^2) begins: n=1: [1, 0, -1, -16, -1143, -134816, -53867825, ...]; n=2: [1, 0, 0, -80, -8832, -1076928, -431006720, ...]; n=3: [1, 0, 27, 0, -24543, -3592512, -1464710445, ...]; n=4: [1, 0, 128, 896, 0, -7099904, -3495833600, ...]; n=5: [1, 0, 375, 4000, 371625, 0, -6020725625, ...]; n=6: [1, 0, 864, 11664, 2270592, 78335424, 0, ...]; n=7: [1, 0, 1715, 27440, 9134433, 444056032, 73395100555, 0, ...]; ... in which the coefficient of x^n in row n forms a diagonal of zeros. PROG (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(-m^3*x*Ser(A))/(1-m*x +x^2*O(x^m))^(m^2))[m+1]/m^3 ); A[n]} for(n=1, 20, print1(a(n), ", ")) CROSSREFS Cf. A320418, A320669, A319939. Sequence in context: A218382 A195635 A203427 * A319732 A199012 A304208 Adjacent sequences:  A320665 A320666 A320667 * A320669 A320670 A320671 KEYWORD nonn AUTHOR Paul D. Hanna, Oct 19 2018 STATUS approved

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Last modified December 1 11:07 EST 2021. Contains 349429 sequences. (Running on oeis4.)