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 A181066 Expansion of g.f.: exp( Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^3 *x^k ] *x^n/n ). 3
 1, 1, 2, 7, 31, 157, 865, 5051, 30774, 193669, 1250319, 8240232, 55239187, 375624781, 2585449450, 17982937876, 126222946496, 893073250063, 6363674671524, 45631735776036, 329065051395940, 2385126419825231 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Compare g.f. to a g.f. of the Catalan numbers (A000108): . exp( Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^2 *x^k ] *x^n/n ). LINKS G. C. Greubel, Table of n, a(n) for n = 0..500 EXAMPLE G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 31*x^4 + 157*x^5 + 865*x^6 +... The logarithm begins: log(A(x)) = x + 3*x^2/2 + 16*x^3/3 + 95*x^4/4 + 606*x^5/5 + 4032*x^6/6 +...+ A181067(n)*x^n/n +... which equals the series: log(A(x)) = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 +...)*x + (1 + 2^3*x + 3^3*x^2 + 4^3*x^3 + 5^3*x^4 + 6^3*x^5 + ...)*x^2/2 + (1 + 3^3*x + 6^3*x^2 + 10^3*x^3 + 15^3*x^4 + 21^3*x^5 + ...)*x^3/3 + (1 + 4^3*x + 10^3*x^2 + 20^3*x^3 + 35^3*x^4 + 56^3*x^5 + ...)*x^4/4 + (1 + 5^3*x + 15^3*x^2 + 35^3*x^3 + 70^3*x^4 + 126^3*x^5 + ...)*x^5/5 + (1 + 6^3*x + 21^3*x^2 + 56^3*x^3 + 126^3*x^4 + 252^3*x^5 + ...)*x^6/6 + (1 + 7^3*x + 28^3*x^2 + 84^3*x^3 + 210^3*x^4 + 462^3*x^5 + ...)*x^7/7 + ... MATHEMATICA With[{m=30}, CoefficientList[Series[Exp[Sum[Sum[Binomial[n+k-1, k]^3*x^k*x^n/n, {k, 0, m+2}], {n, m+1}]], {x, 0, m}], x]] (* G. C. Greubel, Apr 05 2021 *) PROG (PARI) {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, n, binomial(m+k-1, k)^3*x^k)*x^m/m)+x*O(x^n)), n)} (Magma) m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( Exp( (&+[ (&+[ Binomial(n+k-1, k)^3*x^(n+k)/n : k in [0..m+2]]): n in [1..m+1]]) ) )); // G. C. Greubel, Apr 05 2021 (Sage) m=30; def A181066_list(prec): P. = PowerSeriesRing(ZZ, prec) return P( exp( sum( sum( binomial(n+k-1, k)^3*x^(n+k)/n for k in (0..m+2) ) for n in (1..m+1)) ) ).list() A181066_list(m) # G. C. Greubel, Apr 05 2021 CROSSREFS Cf. A000108, A181067 (log), A181068 (variant). Sequence in context: A030823 A030873 A030913 * A325452 A030945 A088554 Adjacent sequences: A181063 A181064 A181065 * A181067 A181068 A181069 KEYWORD nonn AUTHOR Paul D. Hanna, Oct 03 2010 STATUS approved

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Last modified June 19 02:16 EDT 2024. Contains 373492 sequences. (Running on oeis4.)