Revision History for A323206
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing entries 1-10
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#49 by Peter Luschny at Wed Feb 27 13:23:04 EST 2019
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#48 by Peter Luschny at Wed Feb 27 13:22:51 EST 2019
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| FORMULA
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A(n, k) = [x^(k+1)] cf(n, x) where cf(n, x) = K_{i>=1} c(i)/b(i) in the notation of Gauß with b(i) = 1, c(1) = 1, c(2) = -x and c(i) = -n*x for i > 2.
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| MATHEMATICA
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nomnum[k_, mn_] := If[k < 2, 1, If[k == 2, -x, -mn x]];
cf[n_, len_] := ContinuedFractionK[nomnum[k, n], 1, {k, len + 2}];
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| STATUS
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approved
editing
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#47 by Peter Luschny at Wed Feb 27 10:49:38 EST 2019
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#46 by Peter Luschny at Wed Feb 27 10:49:33 EST 2019
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| MATHEMATICA
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(* Alternative *): *)
(* Continued fraction: *)
nom[k_, m_] := If[k < 2, 1, If[k == 2, -x, -m x]];
cf[n_, len_] := ContinuedFractionK[nom[k, n], 1, {k, len + 2}];
Arow[n_, len_] := Rest[CoefficientList[Series[cf[n, len], {x, 0, len}], x]];
For[n = 0, n < 9, n++, Print[Arow[n, 8]]]
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| STATUS
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approved
editing
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#45 by Peter Luschny at Wed Feb 27 06:38:55 EST 2019
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#44 by Peter Luschny at Wed Feb 27 06:37:50 EST 2019
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| PROG
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{A(n, k) = polcoeff((1/x)*serreverse(x*((1+(n-1)*(-x))/((1-(-x))^2) + )+x*O(x^k))), k)}
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#43 by Peter Luschny at Wed Feb 27 06:37:22 EST 2019
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| PROG
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(PARI)
{A(n, k) = polcoeff((1/x)*serreverse(x*((1+(n-1)*(-x))/((1-(-x))^2) + x*O(x^k))), k)}
for(n=0, 8, for(k=0, 8, print1(A(n, k), ", ")); print())
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| STATUS
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approved
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#42 by Peter Luschny at Wed Feb 27 05:38:52 EST 2019
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#41 by Peter Luschny at Wed Feb 27 05:37:15 EST 2019
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| FORMULA
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A(k, n, k) = Sum_{j=0..nk} (binomial(2*nk-j, nk) - binomial(2*nk-j, nk+1))*k^(n^(k-j).
A(k, n, k) = Sum_{j=0..nk} binomial(nk + j, nk)*(1 - j/(nk + 1))*kn^j (cf. A009766).
A(k, n, k) = 1 + Sum_{j=0..nk-1} ((1+j)*binomial(2*nk-j, nk+1)/(nk-j))*k^(n^(k-j).
A(k, n, k) ~ ((4*n)^k/(Pi^(1/2)*k^(3/2)))*(1+1/(2*n-1))^2.
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approved
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Discussion
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| Wed Feb 27
| 05:38
| Peter Luschny: Changed name of variables (k,n) -> (n,k) in some formulas in order to make things more consistent.
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#40 by Peter Luschny at Wed Feb 27 04:47:05 EST 2019
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