A009235 -id:A009235 - cp's OEIS frontend

A330797 Evaluation of the Stirling cycle polynomials at -1/2 and normalized with (-2)^n.

1, 1, -1, 3, -15, 105, -945, 10395, -135135, 2027025, -34459425, 654729075, -13749310575, 316234143225, -7905853580625, 213458046676875, -6190283353629375, 191898783962510625, -6332659870762850625, 221643095476699771875, -8200794532637891559375, 319830986772877770815625

Offset: 0

Peter Luschny, Jan 06 2020

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G. C. Greubel, Table of n, a(n) for n = 0..400
Y. Alp and E. G. Kocer, Exponential Almost-Riordan Arrays, Results Math 79, 173 (2024). See page 25.

The equivalent for Stirling2 is A009235.
Cf. A001147, A101485, A103639, A130534.
Cf. A330799, A330800, A330802, A330803.

m:=30;
R:=PowerSeriesRing(Rationals(), m+2);
A330797:= func< n | Coefficient(R!(Laplace( Sqrt(1+2*x) )), n) >;
[A330797(n): n in [0..m]]; // G. C. Greubel, Sep 14 2023

a := n -> ((-2)^(n-1)*GAMMA(n-1/2))/sqrt(Pi): seq(a(n), n=1..9);
# Alternative:
arec := proc(n) option remember: if n = 0 then 1 else
(3 - 2*n)*arec(n-1) fi end: seq(arec(n), n=0..20);
# Or:
gf := (1+2*x)^(1/2); ser := series(gf, x, 24);
seq(n!*coeff(ser, x, n), n=0..20);

a[n_]:= (-2)^n*Sum[Abs[StirlingS1[n, k]]*(-1/2)^k, {k, 0, n}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Nov 19 2021 *)
Table[(-2)^(n-1)*Pochhammer[1/2, n-1], {n,0,30}] (* G. C. Greubel, Sep 14 2023 *)

def A330797(n): return (-2)^(n-1)*rising_factorial(1/2, n-1)
[A330797(n) for n in (0..20)]

a(n) = (-2)^n*Sum_{k=0..n} |Stirling1(n,k)|*(-1/2)^k.
a(n) = (-2)^(n-1)*RisingFactorial(1/2, n-1).
a(n) = ((-2)^(n-1)*Gamma(n - 1/2))/sqrt(Pi).
a(n) = n!*[x^n] (1+2*x)^(1/2).
D-finite with recurrence a(n) = (3 - 2*n)*a(n-1).
a(n) = (-1)^(n-1)*(2*n-3)!! = (-1)^(n-1)*A001147(n-1).
a(2*n) = -2^(2*n-1)*RisingFactorial(1/2, 2*n-1) = -A103639(n-1).
a(2*n+1) = 4^n*RisingFactorial(1/2, 2*n) = A101485(n).
a(n) ~ -((-2*n)^n/exp(n))/(sqrt(2)*n).
Sum_{n>=0} 1/a(n) = 2 - sqrt(Pi/(2*e))*erfi(1/sqrt(2)), where erfi is the imaginary error function. - Amiram Eldar, Jan 08 2023
O.g.f.: 1+x*2F0(1/2,1;;-2*x). - R. J. Mathar, Aug 10 2025

A350456 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 2x)) / (1 + 2x).

Original entry on oeis.org

1, 1, 1, -1, 1, -3, 17, -85, 385, -1767, 8929, -50633, 312705, -2036267, 13794417, -97295069, 717808897, -5549714767, 44868094145, -377741383697, 3298933836033, -29813463964115, 278462029910993, -2685972391332837, 26733375327601281, -274247228584531767

Offset: 0

Ilya Gutkovskiy, Feb 04 2022

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Shifts 2 places left under 2nd-order inverse binomial transform.

Cf. A007472, A009235, A010739, A051139, A351184, A351185, A351186, A351187.

nmax = 25; A[] = 0; Do[A[x] = 1 + x + x^2 A[x/(1 + 2 x)]/(1 + 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] (-2)^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 25}]

a(0) = a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * (-2)^k * a(n-k-2).

A308536 Expansion of e.g.f. exp(1 - exp(2*x)).

Original entry on oeis.org

1, -2, 0, 8, 16, -64, -576, -1152, 12800, 136704, 422912, -4464640, -72626176, -413966336, 1805123584, 64448004096, 651340611584, 1132294045696, -73000566390784, -1332193006190592, -10293724166750208, 56984418960539648, 3042980275005947904, 46913652420264329216

Offset: 0

Ilya Gutkovskiy, Jun 06 2019

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Cf. A000079, A000587, A009235, A055882, A213170.

nmax = 23; CoefficientList[Series[Exp[1 - Exp[2x]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = -Sum[2^k Binomial[n - 1, k - 1] a[n - k], {k, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]
Table[2^n BellB[n, -1], {n, 0, 23}]

O.g.f.: 1/(1 + 2*x/(1 - 2*x/(1 + 2*x/(1 - 4*x/(1 + 2*x/(1 - 6*x/(1 + 2*x/(1 - 8*x/(1 + ...))))))))), a continued fraction.
a(0) = 1; a(n) = -Sum_{k=1..n} 2^k*binomial(n-1,k-1)*a(n-k).
a(n) = exp(1) * 2^n * Sum_{k>=0} (-1)^k*k^n/k!.
a(n) = 2^n * A000587(n).

A334190 a(n) = exp(1/2) * Sum_{k>=0} (2k + 1)^n / ((-2)^k k!).

Original entry on oeis.org

1, 0, -2, -4, 4, 64, 248, 48, -6512, -51200, -171296, 830400, 17870400, 144684032, 441316224, -5976726784, -119879356160, -1123892297728, -3962230563328, 70410917051392, 1686366492509184, 19578100126072832, 101728414306826240, -1258662784047370240, -42727186269262737408

Offset: 0

Ilya Gutkovskiy, Apr 18 2020

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Column k=2 of A334192.

Cf. A007405, A009235, A293037, A308536, A334191.

nmax = 24; CoefficientList[Series[1/(1 - x) Sum[(-x/(1 - x))^k/Product[(1 - 2 j x/(1 - x)), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 24; CoefficientList[Series[Exp[x + (1 - Exp[2 x])/2], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] * 2^k * BellB[k, -1/2], {k, 0, n}], {n, 0, 24}] (* Vaclav Kotesovec, Apr 18 2020 *)

G.f.: (1/(1 - x)) * Sum_{k>=0} (-x/(1 - x))^k / Product_{j=1..k} (1 - 2*j*x/(1 - x)).

E.g.f.: exp(x + (1 - exp(2*x)) / 2).

A367743 Expansion of e.g.f. exp(1 - x - exp(2*x)).

Original entry on oeis.org

1, -3, 5, 1, -7, -75, -99, 1241, 10161, 18989, -332299, -3857551, -14440151, 141168997, 2807256333, 20182451657, -42073176479, -2999363709091, -38439478980891, -161835672017439, 3439471815545177, 87228227501354517, 937579822282327421, 216540362854403513, -198501712690150659055

Offset: 0

Ilya Gutkovskiy, Nov 29 2023

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Cf. A000587, A004211, A009235, A109747, A124311, A126390, A308536, A308645, A367744.

nmax = 24; CoefficientList[Series[Exp[1 - x - Exp[2 x]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -a[n - 1] - Sum[Binomial[n - 1, k - 1] 2^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 24}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 2^k BellB[k, -1], {k, 0, n}], {n, 0, 24}]

a(n) = exp(1) * Sum_{k>=0} (-1)^k * (2*k-1)^n / k!.
a(0) = 1; a(n) = -a(n-1) - Sum_{k=1..n} binomial(n-1,k-1) * 2^k * a(n-k).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * 2^k * A000587(k).

A305708 Expansion of e.g.f. exp(cos(x)/exp(x) - 1).

Original entry on oeis.org

1, -1, 1, 1, -11, 43, -83, -275, 3833, -21561, 51369, 375593, -5860147, 40452371, -101676235, -1409619211, 23912208945, -189650997937, 454996127889, 11250036170129, -204691511497499, 1799897065507003, -3741969787709699, -164548323889940675, 3183842522596250537, -30356999697044585833

Offset: 0

Ilya Gutkovskiy, Jun 08 2018

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			exp(cos(x)/exp(x) - 1) = 1 - x + x^2/2! + x^3/3! - 11*x^4/4! + 43*x^5/5! - 83*x^6/6! - 275*x^7/7! + ...

Cf. A004211, A009116, A009216, A009235, A009255, A009276, A146559, A155585.

a:=series(exp(cos(x)/exp(x)-1),x=0,26): seq(n!*coeff(a,x,n),n=0..25); # Paolo P. Lava, Mar 26 2019

nmax = 25; CoefficientList[Series[Exp[Cos[x]/Exp[x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Re[(-1 - I)^k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 25}]

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A330797 Evaluation of the Stirling cycle polynomials at -1/2 and normalized with (-2)^n.

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A350456 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 2*x)) / (1 + 2*x).

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A308536 Expansion of e.g.f. exp(1 - exp(2*x)).

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A334190 a(n) = exp(1/2) * Sum_{k>=0} (2*k + 1)^n / ((-2)^k * k!).

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A367743 Expansion of e.g.f. exp(1 - x - exp(2*x)).

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A305708 Expansion of e.g.f. exp(cos(x)/exp(x) - 1).

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Maple

Mathematica

A350456 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 2x)) / (1 + 2x).

A334190 a(n) = exp(1/2) * Sum_{k>=0} (2k + 1)^n / ((-2)^k k!).