A166990 G.f.: A(x) = exp( Sum_{n>=1} A000172(n)*x^n/n ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.
1, 2, 7, 30, 147, 786, 4472, 26644, 164477, 1044258, 6782484, 44887236, 301782361, 2056250570, 14172792355, 98667874038, 692948001906, 4904403499992, 34951124337300, 250617829087656, 1807055528439771, 13095146839953030
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + 2*x + 7*x^2 + 30*x^3 + 147*x^4 + 786*x^5 + 4472*x^6 +... log(A(x)) = 2*x + 10*x^2/2 + 56*x^3/3 + 346*x^4/4 + 2252*x^5/5 + 15184*x^6/6 + 104960*x^7/7 +...+ A000172(n)*x^n/n +...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..500
Programs
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Mathematica
a[n_] := Sum[(Binomial[n, k])^3, {k, 0, n}]; f[x_] := Sum[a[n]*x^n/n, {n, 1, 75}]; CoefficientList[Series[Exp[f[x]], {x, 0, 50}], x] (* G. C. Greubel, May 30 2016 *) nmax = 30; Clear[a]; franel = RecurrenceTable[{n^2*a[n] == (7*n^2 - 7*n + 2)*a[n-1] + 8*(n-1)^2*a[n-2], a[1] == 2, a[2] == 10}, a, {n, 1, nmax}]; $RecursionLimit -> Infinity; a[n_] := a[n] = If[n == 0, 1, Sum[franel[[k]]*a[n-k], {k, 1, n}]/n]; Table[a[n], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 27 2024 *) -
PARI
{a(n)=polcoeff(exp(sum(m=1,n,sum(k=0,m,binomial(m,k)^3)*x^m/m)+x*O(x^n)),n)}
Formula
Self-convolution of A166991.
a(n) ~ c * 8^n / n^2, where c = 0.58462945... - Vaclav Kotesovec, Nov 27 2017, updated Oct 29 2024
Comments