A369941 Expansion of e.g.f. exp(x^2/(2 * sqrt(1-2*x))).

1, 0, 1, 3, 21, 180, 1950, 25200, 378105, 6452460, 123337620, 2609352900, 60515126595, 1526350635810, 41593384352520, 1217650295862000, 38109761157092625, 1269767021491168200, 44871119612057138400, 1676214082334522530800

Offset

0,4

Links

Table of n, a(n) for n=0..19.

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-3*k) * binomial(n-1-3*k/2,n-2*k)/k!.

From Vaclav Kotesovec, Feb 20 2024: (Start)

Recurrence (for n>9): 4*(n-9)*(n-2)*a(n) = 12*(n-1)*(2*n^2 - 23*n + 48)*a(n-1) - 48*(n-2)*(n-1)*(n^2 - 12*n + 30)*a(n-2) + 16*(n-8)*(n-3)*(n-2)*(n-1)*(2*n - 9)*a(n-3) + 4*(n-9)*(n-3)*(n-2)*(n-1)*a(n-4) - 6*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 17)*a(n-5) + 9*(n-8)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6).

a(n) ~ 2^(n - 5/6) * exp(3*2^(-8/3)*n^(1/3) - n) * n^(n - 1/3) / sqrt(3) * (1 - 7*2^(2/3)/(128*n^(1/3))). (End)

Mathematica

Flatten[{{1, 0, 1, 3}, RecurrenceTable[{-9 (-8 + n) (-5 + n) (-4 + n) (-3 + n) (-2 + n) (-1 + n) a[-6 + n] + 6 (-4 + n) (-3 + n) (-2 + n) (-1 + n) (-17 + 2 n) a[-5 + n] - 4 (-9 + n) (-3 + n) (-2 + n) (-1 + n) a[-4 + n] - 16 (-8 + n) (-3 + n) (-2 + n) (-1 + n) (-9 + 2 n) a[-3 + n] + 48 (-2 + n) (-1 + n) (30 - 12 n + n^2) a[-2 + n] - 12 (-1 + n) (48 - 23 n + 2 n^2) a[-1 + n] + 4 (-9 + n) (-2 + n) a[n] == 0, a[4] == 21, a[5] == 180, a[6] == 1950, a[7] == 25200, a[8] == 378105, a[9] == 6452460}, a, {n, 4, 20}]}] (* Vaclav Kotesovec, Feb 19 2024 *)

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x^2/(2*sqrt(1-2*x)))))

Crossrefs

Cf. A362204.

Sequence in context: A195105 A285272 A295541 * A192946 A216171 A054879

Adjacent sequences: A369938 A369939 A369940 * A369942 A369943 A369944

Keyword

nonn

Author

Seiichi Manyama, Feb 06 2024

Status

approved