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A181411
a(n) = Sum_{k=0..n} C(n,k)*sigma(n+k) for n>=1.
2
4, 18, 55, 150, 379, 915, 2146, 4934, 11080, 24833, 54476, 119091, 259432, 556700, 1195135, 2561094, 5428597, 11488866, 24350993, 51296325, 107427025, 225330244, 472762497, 985966379, 2049357779, 4267962522, 8887535983, 18431783744
OFFSET
1,1
FORMULA
Equals the logarithmic derivative of A181410.
Conjecture: a(n) ~ c * n * 2^n, where c = Pi^2/4 = A091476. - Vaclav Kotesovec, Oct 05 2020
EXAMPLE
L.g.f.: L(x) = 4*x + 18*x^2/2 + 55*x^3/3 + 150*x^4/4 + 379*x^5/5 +...
Exponentiation yields the g.f. of A181410:
exp(L(x)) = 1 + 4*x + 17*x^2 + 65*x^3 + 234*x^4 + 804*x^5 +...
The initial terms begin:
a(1) = 1*1 + 1*3 = 4;
a(2) = 1*3 + 2*4 + 1*7 = 18;
a(3) = 1*4 + 3*7 + 3*6 + 1*12 = 55;
a(4) = 1*7 + 4*6 + 6*12 + 4*8 + 1*15 = 150; ...
MATHEMATICA
Table[Sum[Binomial[n, k] * DivisorSigma[1, n+k], {k, 0, n}], {n, 1, 30}] (* Vaclav Kotesovec, Oct 05 2020 *)
PROG
(PARI) {a(n)=sum(k=0, n, binomial(m, k)*sigma(n+k))}
CROSSREFS
Cf. A181410.
Sequence in context: A212250 A229788 A242206 * A238915 A212680 A027286
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 19 2010
STATUS
approved