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 A243425 G.f. A(x) satisfies: coefficient of x^n in A(x)^(2*n) equals A005260(n) = Sum_{k=0..n} C(n,k)^4. 1
 1, 1, 3, 9, 60, 417, 3430, 29927, 278316, 2693437, 26976407, 277394148, 2916106328, 31220964707, 339508802940, 3741551907530, 41714692453164, 469827584596185, 5339334757945439, 61165396353689573, 705720529604453193, 8195208178337460065, 95724512701573485819, 1124070800784913396731 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..600 FORMULA G.f.: sqrt( x / Series_Reversion( x*exp( Sum_{n>=1} A005260(n)*x^n/n ) ) ), where A005260(n) = Sum_{k=0..n} C(n,k)^4. a(n) ~ c * d^n / n^(5/2), where d= 13.142352254618115022093263384837224..., c = 0.051491668112404252102416729094836... . - Vaclav Kotesovec, Jun 05 2014 EXAMPLE G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 60*x^4 + 417*x^5 + 3430*x^6 +... Form a table of coefficients in A(x)^(2*n) for n>=0, which begins: [1,  0,   0,    0,     0,      0,      0,       0,        0, ...]; [1,  2,   7,   24,   147,   1008,   8135,   70296,   648172, ...]; [1,  4,  18,   76,   439,   2940,  22936,  194300,  1761411, ...]; [1,  6,  33,  164,   960,   6378,  48526,  403440,  3598050, ...]; [1,  8,  52,  296,  1810,  12128,  90972,  744656,  6542519, ...]; [1, 10,  75,  480,  3105,  21252, 158845, 1286240, 11157705, ...]; [1, 12, 102,  724,  4977,  35100, 263844, 2125020, 18253680, ...]; [1, 14, 133, 1036,  7574,  55342, 421484, 3395016, 28975933, ...]; [1, 16, 168, 1424, 11060,  84000, 651848, 5277696, 44916498, ...]; ... then the main diagonal forms A005260(n) = Sum_{k=0..n} C(n,k)^4. PROG (PARI) {a(n)=polcoeff(sqrt(x/serreverse(x*exp(sum(m=1, n+1, sum(k=0, m, binomial(m, k)^4)*x^m/m +x^2*O(x^n))))), n)} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A242903. Sequence in context: A340389 A140812 A202210 * A018513 A143761 A269961 Adjacent sequences:  A243422 A243423 A243424 * A243426 A243427 A243428 KEYWORD nonn AUTHOR Paul D. Hanna, Jun 04 2014 STATUS approved

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Last modified May 17 23:07 EDT 2022. Contains 353779 sequences. (Running on oeis4.)