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 A302059 G.f. A(x) satisfies: [x^n] A(x)^(3*n+2) / (x*A(x)^3)' = 0 for n>1. 5
 1, 1, 7, 223, 12876, 1041470, 106367932, 12976266030, 1828572078504, 291167594109079, 51587932089790950, 10054999390608521223, 2137102578158115474600, 491830012906134545686644, 121847472306799372796495932, 32335452582105155555135147936, 9152510003592208592171248479432, 2752729182740256406998201329266149 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS More generally, [x^n] G(x,k)^(k*(n+1)-1) / (x*G(x,k)^k)' = 0 is satisfied by an integer series G(x,k) when k is a fixed positive integer. LINKS Paul D. Hanna, Table of n, a(n) for n = 0..300 FORMULA G.f. A(x) satisfies: [x^n] A(x)^(3*n) / (A(x) + 3*x*A'(x)) = 0 for n>1. a(n) ~ c * 18^n * n! / n^(4/9), where c = 0.0140759690511059570797... - Vaclav Kotesovec, Oct 20 2020 EXAMPLE G.f.: A(x) = 1 + x + 7*x^2 + 223*x^3 + 12876*x^4 + 1041470*x^5 + 106367932*x^6 + 12976266030*x^7 + 1828572078504*x^8 + 291167594109079*x^9 + ... such that [x^n] A(x)^(3*n+2) / (x*A(x)^3)' = 0 for n>1. ILLUSTRATION OF DEFINITION. The table of coefficients in A(x)^(3*n+2) / (x*A(x)^3)' begins: n=0: [1, -4, -33, -1902, -149243, -15230208, -1876337625, -267941179110, ...]; n=1: [1, -1, -21, -1385, -118455, -12695769, -1615045187, -235896844893, ...]; n=2: [1, 2, 0, -760, -83692, -9927612, -1334925264, -201948321520, ...]; n=3: [1, 5, 30, 0, -44414, -6904812, -1034743200, -165997446000, ...]; n=4: [1, 8, 69, 922, 0, -3604176, -713169380, -127940381076, ...]; n=5: [1, 11, 117, 2033, 50252, 0, -368770482, -87667218244, ...]; n=6: [1, 14, 174, 3360, 107125, 3936174, 0, -45061548696, ...]; n=7: [1, 17, 240, 4930, 171483, 8235801, 394811962, 0, ...]; ... in which the main diagonal consists of all zeros after the initial terms, illustrating that [x^n] A(x)^(3*n+2) / (x*A(x)^3)' = 0 for n>1. RELATED SERIES. A(x)^3 = 1 + 3*x + 24*x^2 + 712*x^3 + 40134*x^4 + 3211848*x^5 + 326090932*x^6 + 39631822680*x^7 + 5570315156529*x^8 + ... (x*A(x)^3)' = 1 + 6*x + 72*x^2 + 2848*x^3 + 200670*x^4 + 19271088*x^5 + 2282636524*x^6 + 317054581440*x^7 + 50132836408761*x^8 + ... log(A(x)) = x + 13*x^2/2 + 649*x^3/3 + 50541*x^4/4 + 5136491*x^5/5 + 631363729*x^6/6 + 90027008835*x^7/7 + 14517298906509*x^8/8 + 2603034994183642*x^9/9 + ... PROG (PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A = concat(A, 0); A[#A] = Vec( Ser(A)^(3*#A-1)/(x*Ser(A)^3)' )[#A]); A[n+1]} for(n=0, 20, print1(a(n), ", ")) CROSSREFS Cf. A300994, A300627, A300995, A302060. Sequence in context: A140018 A202655 A009488 * A352312 A138247 A322709 Adjacent sequences:  A302056 A302057 A302058 * A302060 A302061 A302062 KEYWORD nonn AUTHOR Paul D. Hanna, Mar 31 2018 STATUS approved

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Last modified August 13 17:37 EDT 2022. Contains 356107 sequences. (Running on oeis4.)