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 A359928 a(n) = coefficient of x^n*y^(n+1)/n! in (1/2) * log( Sum_{n>=0} (n^2 + 3*n*y + 2*y^2)^n * x^n/n! ). 4
 1, 6, 93, 2448, 92505, 4589568, 283008621, 20903023872, 1800986581521, 177455695795200, 19690717755237309, 2430478269127673856, 330392930155527272553, 49053029845102480576512, 7898602773992589665290125, 1371137549213022697047785472, 255275516636592894833768588961 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Paul D. Hanna, Table of n, a(n) for n = 1..200 FORMULA E.g.f. A(x) = Sum_{n>=1} a(n) * x^n/n! may be defined by the following. (1) a(n) = [x^n*y^(n+1)/n!] (1/2)*log( Sum_{n>=0} (n + y)^n*(n + 2*y)^n *x^n/n! ). (2) A(x) = lim_{N->oo} (1/N)*log( Sum_{n>=0} (N + n)^n*(N + 2*n)^n * (x/N)^n/n! ). a(n) ~ c * d^n * n! / n^(5/2), where d = 12.7029497597456784744445675253711147535742245945208995646083627... and c = 0.15440395598650604464793307483290467035754174771895993579108... - Vaclav Kotesovec, Mar 21 2024 EXAMPLE E.g.f.: A(x) = x + 6*x^2/2! + 93*x^3/3! + 2448*x^4/4! + 92505*x^5/5! + 4589568*x^6/6! + 283008621*x^7/7! + 20903023872*x^8/8! + 1800986581521*x^9/9! + 177455695795200*x^10/10! + ... Exponentiation yields the e.g.f. of A319147: exp(A(x)) = 1 + x + 7*x^2/2! + 112*x^3/3! + 2965*x^4/4! + 111856*x^5/5! + 5528419*x^6/6! + 339433984*x^7/7! + 24965493865*x^8/8! + 2142654088960*x^9/9! + ... + A319147(n)*x^n/n! + ... which equals the limit exp(A(x)) = lim_{N->oo} [ Sum_{n>=0} (N^2 + 3*N*n + 2*n^2)^n * (x/N)^n/n! ]^(1/N). RELATED SEQUENCES. a(n) is divisible by n where a(n)/n begins: [1, 3, 31, 612, 18501, 764928, 40429803, 2612877984, 200109620169, ...]. PROG (PARI) {a(n) = (1/2) * n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m + y)^m*(m + 2*y)^m *x^m/m! ) +x*O(x^n) ), n, x), n+1, y)} for(n=1, 30, print1(a(n), ", ")) CROSSREFS Cf. A359927, A319834, A318634. Sequence in context: A158121 A328427 A103212 * A033935 A218682 A078103 Adjacent sequences: A359925 A359926 A359927 * A359929 A359930 A359931 KEYWORD nonn AUTHOR Paul D. Hanna, Jan 20 2023 STATUS approved

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Last modified June 22 12:55 EDT 2024. Contains 373571 sequences. (Running on oeis4.)