A357588 -id:A357588 - cp's OEIS frontend

A006195 Reversion of Jacobi theta_3.

1, -2, 8, -40, 222, -1316, 8160, -52272, 343220, -2297682, 15623760, -107611608, 749209832, -5264005060, 37277153920, -265788870480, 1906489923022, -13747860118724, 99606357848920, -724732875917064, 5293303253527704, -38795196044205056

Offset: 0

N. J. A. Sloane

sign

J.-G. Penaud, Arbres et Animaux, Ph.D. Dissertation, Université Bordeaux I, 1990, cover page and p. 76. (Annotated scanned copy)
Index entries for reversions of series

Cf. A000122, A109085, A181315.

# Using function CompInv from A357588.
CompInv(22, n -> if n = 1 then 1 elif issqr(n-1) then 2 else 0 fi); # Peter Luschny, Oct 05 2022

nmax = 30; A[] = 0; Do[A[x] = Product[(1 + x^k*A[x]^k)/(1 - x^k*A[x]^k), {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * (-1)^Range[0, nmax] (* Vaclav Kotesovec, Sep 27 2023 *)
(* Calculation of constant d: *) 1/r /. FindRoot[{QPochhammer[-1, r*s] == 2*s*QPochhammer[r*s], (2* QPochhammer[r*s]*(-Log[r*s] + Log[1 - r*s] + QPolyGamma[0, 1, r*s])) / Log[r*s] + r*(Derivative[0, 1][QPochhammer][-1, r*s] - 2*s*Derivative[0, 1][QPochhammer][r*s, r*s]) == 0}, {r, 1/8}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Sep 27 2023 *)

N=66; x='x+O('x^N); /* that many terms */
Vec(serreverse(x*sum(n=-N,N,x^(n^2)))) /* show terms */ /* Joerg Arndt, May 25 2011 */

REVERT(A000122).
From Vaclav Kotesovec, Sep 27 2023: (Start)
G.f. A(x) satisfies A(x) = Product_{k>=1} (1 + (-x)^k*A(x)^k)/(1 - (-x)^k*A(x)^k).
a(n) ~ (-1)^n * c * d^n / n^(3/2), where d = 7.86298339570590526151934790995382716105758424871057843176888470144337... and c = 0.617020565581840591336246430220953133238702598666548444780767269...
(End)

Signs corrected by N. J. A. Sloane, Dec 24 2001

A059372 Revert transform of factorials n! (n >= 1).

Original entry on oeis.org

1, -2, 2, -4, -4, -48, -336, -2928, -28144, -298528, -3454432, -43286528, -583835648, -8433987584, -129941213184, -2127349165824, -36889047574272, -675548628690432, -13030733384956416, -264111424634864640

Offset: 1

N. J. A. Sloane, Jan 28 2001

First diagonal of triangle in A059370.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 171, #34.

Vaclav Kotesovec, Table of n, a(n) for n = 1..400 (first 100 terms from T. D. Noe)
M. H. Albert, M. D. Atkinson and M. Klazar, The Enumeration of Simple Permutations, J. Integer Seqs., Vol. 6, 2003.
Eli Bagno, Estrella Eisenberg, Shulamit Reches, and Moriah Sigron, Blockwise simple permutations, arXiv:2303.13115 [math.CO], 2023.
Emeric Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, arXiv:math/0407326 [math.CO], 2004; J. Num. Theory 117 (2006), 191-215.
Index entries for reversions of series

Cf. A000142, A059370.

# From Transforms, see the footer of the page.
REVERT([seq(k!, k=1..20)]); # Peter Luschny, May 01 2021
# Using function CompInv from A357588.
CompInv(10, n -> factorial(n)); # Peter Luschny, Oct 09 2022

nmax = 20; t[n_, k_] := t[n, k] = Sum[(m + 1)!*t[n - m - 1, k - 1], {m, 0, n - k}]; t[n_, 1] = n!; t[n_, n_] = 1; tnk = Table[t[n, k], {n, 1, nmax}, {k, 1, nmax}]; Inverse[tnk][[All, 1]] (* Jean-François Alcover, Jul 13 2016 *)

a(n) ~ -exp(-2) * n! * (1 - 4/n + 2/n^2 - 34/(3*n^3) - 296/(3*n^4) - 4818/(5*n^5) - 508532/(45*n^6)). - Vaclav Kotesovec, Aug 04 2015

G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} k! * A(x)^k. - Ilya Gutkovskiy, Apr 22 2020

More terms from Vladeta Jovovic, Mar 05 2001

Definition refined by Georg Fischer, May 01 2021

A007311 Reversion of o.g.f. for Bell numbers (A000110) omitting a(0)=1.

Original entry on oeis.org

1, -2, 3, -5, 7, -14, 11, -66, -127, -992, -5029, -30899, -193321, -1285300, -8942561, -65113125, -494605857, -3911658640, -32145949441, -274036507173, -2419502677445, -22093077575496, -208364964369913, -2027216779571754, -20323053380033763, -209715614081160850

Offset: 1

N. J. A. Sloane, Mira Bernstein

sign

As the definition says, this entry deliberately omits the zero-th term 1. - N. J. A. Sloane, Jun 16 2021

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Index entries for reversions of series

Cf. A000110.

read transforms; A := series(exp(exp(x)-1),x,60); SERIESTOLISTMULT(%); subsop(1=NULL,%); REVERT(%);
# Alternative, using function CompInv from A357588:
CompInv(26, n -> combinat:-bell(n)); # Peter Luschny, Oct 05 2022

a(n)=if(n<1,0,polcoeff(serreverse(-1+serlaplace(exp(exp(x+x*O(x^n))-1))),n))

G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} Bell(k) * A(x)^k. - Ilya Gutkovskiy, Apr 22 2020

Signs corrected Dec 24 2001

A050397 Reversion of sequence of involutions (A000085).

Original entry on oeis.org

1, -2, 4, -10, 30, -104, 392, -1568, 6520, -27976, 122944, -551680, 2518912, -11684000, 54957216, -261897024, 1263216192, -6164172608, 30416619200, -151750104800, 765364073120, -3902783995520, 20123276097920

Offset: 1

Christian G. Bower, Nov 15 1999

sign

Andrew Howroyd, Table of n, a(n) for n = 1..200
N. J. A. Sloane, Transforms
Index entries for reversions of series

Cf. A000085, A050398.

# Using function CompInv from A357588.
CompInv(23, n -> simplify(hypergeom([-n/2, (1-n)/2], [], 2))); # Peter Luschny, Oct 05 2022

seq(n)=Vec(serreverse(serlaplace(-1 + exp(x+x^2/2 + O(x*x^n))))) \\ Andrew Howroyd, May 06 2023

A007316 Reversion of g.f. for Euler numbers A000111(n-1).

Original entry on oeis.org

1, -1, 1, -2, 3, -9, 9, -71, -96, -1325, -6843, -54922, -417975, -3586117, -32531983, -316599861, -3274076017, -35914014266, -416386323306, -5088908019824, -65392831090975, -881473287321301, -12437647407521019, -183345613125389337

Offset: 1

N. J. A. Sloane, Mira Bernstein

sign

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Index entries for reversions of series

Cf. A000111.

# Using function CompInv from A357588.
CompInv(24, n -> if n=1 then 1 else 2^(n-1)*abs(euler(n-1, 1/2) + euler(n-1, 1)) fi); # Peter Luschny, Oct 05 2022

Extended with signs by Christian G. Bower, Feb 15 1999

Signs and a(23) and a(24) corrected by Sean A. Irvine, Dec 08 2017

A092413 Coefficient of x^n in solution of x = y + y^2 + y^4 + y^8 + ...

Original entry on oeis.org

1, -1, 2, -6, 20, -70, 256, -970, 3772, -14960, 60280, -246090, 1015700, -4231216, 17767456, -75126078, 319588340, -1366846548, 5873832384, -25350152100, 109828012448, -477486940848, 2082520454864, -9109146150050, 39950535931956

Offset: 1

Ralf Stephan, Mar 22 2004

sign

Reversion of Fredholm-Rueppel sequence (A036987) shifted right.

Cf. A049140.

# Using function CompInv from A357588.
CompInv(25, n -> if 2^ilog2(n) = n then 1 else 0 fi); # Peter Luschny, Oct 05 2022

PARI
```
serreverse(sum(k=0,8,x^(2^k))+O(x^257))
```

A249512 Expansion of 1/(1-xsqrt(4x^2+1)-2*x^2).

Original entry on oeis.org

1, 1, 3, 7, 15, 33, 75, 169, 375, 835, 1875, 4203, 9375, 20931, 46875, 104919, 234375, 523737, 1171875, 2621545, 5859375, 13098001, 29296875, 65523597, 146484375, 327500413, 732421875, 1637918089

Offset: 0

Vladimir Kruchinin, Oct 31 2014

nonn

# Using function CompInv from A357588.
1, CompInv(27, n -> simplify(GegenbauerC(n-1, 1-n, 3/2))); # Peter Luschny, Oct 05 2022

CoefficientList[Series[1/(1-x*Sqrt[4*x^2+1]-2*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 31 2014 *)

a(n) := if n=0 then 1  else sum(k*4^(n-k)*binomial(n/2,n-k),k,1,n)/n;

def a(n):
    if is_odd(n):
        return simplify((4^(n-1)*binomial(n/2, n-1)*hypergeometric([2, 1-n], [2-n/2], -1/4))/n)
    return 3*5^(n//2-1) if n>0 else 1
[a(n) for n in (0..27)] # Peter Luschny, Oct 31 2014

a(n) = sum(k = 1..n, k*4^(n-k)*binomial(n/2,n-k))/n, a(0)=1.
a(n) ~ 3 * 5^(n/2-1). - Vaclav Kotesovec, Oct 31 2014
a(n) = 3 * 5^(n/2-1) if n is even and n>0 else a(n) = ((4^(n-1)* binomial(n/2, n-1)*hypergeometric([2, 1-n],[2-n/2], -1/4))/n). - Peter Luschny, Oct 31 2014
D-finite with recurrence: (-n+1)*a(n) +(-n+2)*a(n-1) +(n+11)*a(n-2) +(n+10)*a(n-3) +20*(n-4)*a(n-4) +20*(n-5)*a(n-5)=0. - R. J. Mathar, Jan 25 2020

cp's OEIS Frontend

A006195 Reversion of Jacobi theta_3.

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A059372 Revert transform of factorials n! (n >= 1).

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A007311 Reversion of o.g.f. for Bell numbers (A000110) omitting a(0)=1.

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A050397 Reversion of sequence of involutions (A000085).

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A007316 Reversion of g.f. for Euler numbers A000111(n-1).

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Maple

Extensions

A092413 Coefficient of x^n in solution of x = y + y^2 + y^4 + y^8 + ...

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Author

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Comments

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Maple

PARI

A249512 Expansion of 1/(1-x*sqrt(4*x^2+1)-2*x^2).

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Maple

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Maxima

Sage

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A249512 Expansion of 1/(1-xsqrt(4x^2+1)-2*x^2).