A361290 -id:A361290 - cp's OEIS frontend

A360766 a(0) = 0; a(n) = ( (n + sqrt(n))^n - (n - sqrt(n))^n )/(2 * sqrt(n)).

0, 1, 4, 30, 320, 4400, 73872, 1462552, 33325056, 858283776, 24641000000, 779935205984, 26972930949120, 1011642325897216, 40890444454377728, 1771640957790000000, 81896889467638120448, 4022826671022707900416, 209224123984489179202560

Offset: 0

Seiichi Manyama, Mar 11 2023

Main diagonal of A361290.
Cf. A007070, A030192, A093145, A154237.
Cf. A084062.

a(n) = polcoeff(lift(Mod('x, ('x-n)^2-n)^n), 1); \\ Kevin Ryde, Mar 16 2023

a(n) = Sum_{k=0..floor((n-1)/2)} n^(n-1-k) * binomial(n,2*k+1).
a(n) = [x^n] x/(1 - 2*n*x + (n-1)*n*x^2).
a(n) = n! * [x^n]  exp(n * x) * sinh(sqrt(n) * x) / sqrt(n) for n > 0.

A361293 a(n) = 20 * a(n-1) - 90 * a(n-2) for n>1, with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 20, 310, 4400, 60100, 806000, 10711000, 141680000, 1869610000, 24641000000, 324555100000, 4273412000000, 56258281000000, 740558540000000, 9747925510000000, 128308241600000000, 1688851536100000000, 22229288978000000000, 292589141311000000000

Offset: 0

Seiichi Manyama, Mar 12 2023

Index entries for linear recurrences with constant coefficients, signature (20,-90).

Column k=10 of A361290.

Cf. A289414.

LinearRecurrence[{20,-90},{0,1},20] (* Harvey P. Dale, Dec 16 2023 *)

a(n) = polcoef(lift(Mod('x, ('x-10)^2-10)^n), 1);

my(N=20, x='x+O('x^N)); concat (0,  Vec(x/(1-20*x+90*x^2)))

my(N=20, x='x+O('x^N)); concat (0, apply(round, Vec(serlaplace(exp(10*x)*sinh(sqrt(10)*x)/sqrt(10)))))

a(n) = ( (10 + sqrt(10))^n - (10 - sqrt(10))^n )/(2 * sqrt(10)).
a(n) = Sum_{k=0..floor((n-1)/2)} 10^(n-1-k) * binomial(n,2*k+1).
G.f.: x/(1 - 20 * x + 90 * x^2).
E.g.f.: exp(10 * x) * sinh(sqrt(10) * x) / sqrt(10).

cp's OEIS Frontend

A360766 a(0) = 0; a(n) = ( (n + sqrt(n))^n - (n - sqrt(n))^n )/(2 * sqrt(n)).

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A361293 a(n) = 20 * a(n-1) - 90 * a(n-2) for n>1, with a(0)=0, a(1)=1.

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