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A343276 a(n) = n! * [x^n] -x*(x + 1)*exp(x)/(x - 1)^3. 1
0, 1, 10, 81, 652, 5545, 50886, 506905, 5480056, 64116657, 808856290, 10959016321, 158851484100, 2454385635481, 40285778016862, 700261611998985, 12853532939027056, 248482678808005345, 5047002269952482106, 107466341437781300017, 2394019421567804960380 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..20.

FORMULA

a(n) = Sum_{k=0..n} rf(n - k + 1, k)*k^2, where rf is the rising factorial.

a(n) = (2 + n*(n + 2))*a(n - 1)/(n - 1) - (n + 1)*a(n - 2) for n >= 3.

A002775(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(k).

MAPLE

egf := -x*(x + 1)*exp(x)/(x - 1)^3: ser := series(egf, x, 32):

seq(n!*coeff(ser, x, n), n = 0..20);

MATHEMATICA

a[n_] := Sum[Pochhammer[n - k + 1, k]*k^2, {k, 0, n}];

Table[a[n], {n, 0, 20}]

PROG

(SageMath)

def a(n): return sum(rising_factorial(n - k + 1, k)*k^2 for k in (0..n))

print([a(n) for n in (0..20)])

(Python)

def a():

a, b, n = 0, 1, 2

yield 0

while True:

yield b

a, b = b, -(n + 1)*a + ((2 + n*(n + 2))*b)//(n - 1)

n += 1

A343276 = a(); print([next(A343276) for _ in range(21)])

CROSSREFS

Cf. A002775, A093964.

Sequence in context: A095004 A037541 A037485 * A350503 A277205 A308862

Adjacent sequences: A343273 A343274 A343275 * A343277 A343278 A343279

KEYWORD

nonn

AUTHOR

Peter Luschny, Apr 20 2021

STATUS

approved

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Last modified December 1 22:12 EST 2022. Contains 358484 sequences. (Running on oeis4.)