A166897
a(n) = Sum_{k=0..[n/2]} C(n-k,k)^3*n/(n-k), n>=1.
Original entry on oeis.org
1, 3, 13, 39, 126, 477, 1765, 6495, 24709, 95128, 367368, 1431453, 5620343, 22170543, 87858813, 349708431, 1397003136, 5598513261, 22502171771, 90681323364, 366299212873, 1482827487650, 6014529069540, 24439715146941
Offset: 1
L.g.f.: L(x) = x + 3*x^2/2 + 13*x^3/3 + 39*x^4/4 + 126*x^5/5 + 477*x^6/6 +...
exp(L(x)) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 45*x^5 + 142*x^6 + 459*x^7 +...+ A166896(n)*x^n/n +...
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Table[Sum[Binomial[n-k,k]^3 n/(n-k),{k,0,Floor[n/2]}],{n,30}] (* Harvey P. Dale, Mar 05 2013 *)
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a(n)=sum(k=0,n\2,binomial(n-k,k)^3*n/(n-k))
A166898
G.f.: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^4 * x^k] * x^n/n ), an integer series in x.
Original entry on oeis.org
1, 1, 2, 10, 38, 137, 646, 3241, 15623, 79439, 427562, 2317396, 12715372, 71543343, 408543758, 2353591560, 13717994046, 80827739181, 480016288156, 2871701561720, 17304832805996, 104933348346951, 639814473417775
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 38*x^4 + 137*x^5 + 646*x^6 + 3241*x^7 +...
log(A(x)) = x + 3*x^2/2 + 25*x^3/3 + 111*x^4/4 + 456*x^5/5 + 2697*x^6/6 + 15961*x^7/7 +...+ A166899(n)*x^n/n +...
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{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m, k)^4*x^k)*x^m/m)+x*O(x^n)), n)}
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{a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m\2, binomial(m-k, k)^4*m/(m-k))*x^m/m)+x*O(x^n)), n)}
A167539
a(n) = Sum_{k=0..[n/2]} C(n-k,k)^2 * n/(n-k), n>=1.
Original entry on oeis.org
1, 3, 7, 15, 36, 87, 211, 519, 1285, 3198, 7998, 20079, 50571, 127725, 323367, 820407, 2085306, 5309169, 13537045, 34561890, 88347091, 226079208, 579110262, 1484766015, 3809948461, 9783998877, 25143452881, 64658016249, 166375274790
Offset: 1
L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 15*x^4/4 + 36*x^5/5 + 87*x^6/6 +...
exp(L(x)) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 45*x^5 + 142*x^6 + 459*x^7 +...+ A004148(n+1)*x^n/n +...
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Table[Sum[(Binomial[n - k, k]^2)*(n/(n - k)), {k, 0, n/2}], {n, 1, 100}] (* G. C. Greubel, Jun 15 2016 *)
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{a(n) = sum(k=0,n\2, binomial(n-k,k)^2 * n/(n-k))}
for(n=1,30,print1(a(n),", "))
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{a(n) = n * polcoeff( log( (1 - x - x^2 - sqrt((1+x+x^2)*(1-3*x+x^2) +x^6*O(x^n) )) / (2*x^3) ), n)}
for(n=1,30,print1(a(n),", "))
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{a(n) = sum(k=0,n-1,sum(j=0,k, binomial(n-k+j,n-k)*n/(n-k+j) * binomial(n-k,k-j)*binomial(k-j,j)))}
for(n=1,30,print1(a(n),", "))
Showing 1-3 of 3 results.