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A357693
Expansion of e.g.f. cos( sqrt(2) * log(1+x) ).
1
1, 0, -2, 6, -18, 60, -216, 756, -1620, -14256, 349272, -5452920, 78885576, -1143659088, 17074183104, -265437239760, 4316991698448, -73572489226368, 1314108286270560, -24584195654596512, 481215937895868384, -9843358555320333120, 210128893733994567552
OFFSET
0,3
LINKS
Eric Weisstein's World of Mathematics, Pochhammer Symbol.
FORMULA
a(n) = Sum_{k=0..floor(n/2)} (-2)^k * Stirling1(n,2*k).
a(n) = (-1)^n * ( (sqrt(2) * i)_n + (-sqrt(2) * i)_n )/2, where (x)_n is the Pochhammer symbol and i is the imaginary unit.
a(0) = 1, a(1) = 0; a(n) = -(2*n-3) * a(n-1) - (n^2-4*n+6) * a(n-2).
MATHEMATICA
With[{nn=30}, CoefficientList[Series[Cos[Sqrt[2]Log[1+x]], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Nov 04 2024 *)
PROG
(PARI) my(N=30, x='x+O('x^N)); apply(round, Vec(serlaplace(cos(sqrt(2)*log(1+x)))))
(PARI) a(n) = sum(k=0, n\2, (-2)^k*stirling(n, 2*k, 1));
(PARI) a(n) = (-1)^n*round((prod(k=0, n-1, sqrt(2)*I+k)+prod(k=0, n-1, -sqrt(2)*I+k)))/2;
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; v[2]=0; for(i=2, n, v[i+1]=-(2*i-3)*v[i]-(i^2-4*i+6)*v[i-1]); v;
CROSSREFS
Column k=2 of A357720.
Cf. A357725.
Sequence in context: A191280 A150046 A009458 * A150047 A150048 A363511
KEYWORD
sign
AUTHOR
Seiichi Manyama, Oct 10 2022
STATUS
approved