A202828 -id:A202828 - cp's OEIS frontend

A202829 Expansion of e.g.f.: exp(4x/(1-3x)) / sqrt(1-9*x^2).

1, 4, 49, 676, 13225, 293764, 7890481, 236359876, 8052729169, 300797402500, 12388985000401, 551925653637604, 26614517015830969, 1373655853915667716, 75803216516463190225, 4440662493517062816004, 275697752917311709134241, 18052104090118575573856516

Offset: 0

Paul D. Hanna, Dec 25 2011

nonn

			E.g.f.: A(x) = 1 + 4*x + 49*x^2/2! + 676*x^3/3! + 13225*x^4/4! + 293764*x^5/5! + ...
were A(x) = 1 + 2^2*x + 7^2*x^2/2! + 26^2*x^3/3! + 115^2*x^4/4! + 542^2*x^5/5! + ... + A202830(n)^2*x^n/n! + ...

G. C. Greubel, Table of n, a(n) for n = 0..350

Cf. A202830, A202827, A202828, A202831, A202833, A202835, A202836.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( Exp(4*x/(1-3*x))/Sqrt(1-9*x^2) ))); // G. C. Greubel, Jun 21 2022

With[{nn=20},CoefficientList[Series[Exp[(4x)/(1-3x)]/Sqrt[1-9x^2],{x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Mar 09 2012 *)

{a(n)=n!*polcoeff(exp(4*x/(1-3*x)+x*O(x^n))/sqrt(1-9*x^2+x*O(x^n)),n)}

{a(n)=sum(k=0,n\2,2^(n-3*k)*3^k*n!/((n-2*k)!*k!))^2}

def A202829_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(4*x/(1-3*x))/sqrt(1-9*x^2) ).egf_to_ogf().list()
A202829_list(40) # G. C. Greubel, Jun 21 2022

a(n) = A202830(n)^2, where the e.g.f. of A202830 is exp(2*x + 3*x^2/2).
a(n) = ( Sum_{k=0..[n/2]} 2^(n-3*k)*3^k * n!/((n-2*k)!*k!) )^2.
a(n) ~ n^n*exp(4*sqrt(n/3)-2/3-n)*3^n/2. - Vaclav Kotesovec, May 23 2013
D-finite with recurrence: a(n) = (3*n+1)*a(n-1) + 3*(n-1)*(3*n+1)*a(n-2) - 27*(n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013

A202831 Expansion of e.g.f.: exp(4x/(1-5x)) / sqrt(1-25*x^2).

Original entry on oeis.org

1, 4, 81, 1444, 44521, 1397124, 58354321, 2574344644, 136043683281, 7657406908804, 489836445798001, 33351743794661604, 2504378700538997881, 199445618093659242244, 17189578072429077875121, 1564487078400498014277124, 152146464623361858013314721

Offset: 0

Paul D. Hanna, Dec 25 2011

nonn

			E.g.f.: 1 + 4*x + 81*x^2/2! + 1444*x^3/3! + 44521*x^4/4! + 1397124*x^5/5! + ...
where A(x) = 1 + 2^2*x + 9^2*x^2/2! + 38^2*x^3/3! + 211^2*x^4/4! + 1182^2*x^5/5! + ... + A202832(n)^2*x^n/n! + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A202832, A202827, A202828, A202829, A202833, A202835, A202836.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(4*x/(1-5*x))/Sqrt(1-25*x^2) ))); // G. C. Greubel, Jun 21 2022

CoefficientList[Series[Exp[4*x/(1-5*x)]/Sqrt[1-25*x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, May 23 2013 *)

{a(n)=n!*polcoeff(exp(4*x/(1-5*x)+x*O(x^n))/sqrt(1-25*x^2+x*O(x^n)),n)}

{a(n)=n!^2*polcoeff(exp(2*x+5*x^2/2+x*O(x^n)),n)^2}

{a(n)=sum(k=0,n\2,2^(n-3*k)*5^k*n!/((n-2*k)!*k!))^2}

def A202831_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(4*x/(1-5*x))/sqrt(1-25*x^2) ).egf_to_ogf().list()
A202831_list(40) # G. C. Greubel, Jun 21 2022

a(n) = A202832(n)^2, where the e.g.f. of A202832 is exp(2*x + 5*x^2/2).
a(n) = ( Sum_{k=0..[n/2]} 2^(n-3*k)*5^k * n!/((n-2*k)!*k!) )^2.
a(n) ~ n^n*exp(4*sqrt(n/5)-2/5-n)*5^n/2. - Vaclav Kotesovec, May 23 2013
D-finite with recurrence: a(n) = (5*n-1)*a(n-1) + 5*(n-1)*(5*n-1)*a(n-2) - 125*(n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013

A202836 Expansion of e.g.f.: exp(9x/(1-4x)) / sqrt(1-16*x^2).

Original entry on oeis.org

1, 9, 169, 3969, 119025, 4173849, 169754841, 7764958161, 395853630561, 22158814509225, 1352182116776841, 89167147951863969, 6319166996322943569, 478498255838869322169, 38549853656690487255225, 3290600595687160597292529, 296613603422471046790496961

Offset: 0

Paul D. Hanna, Dec 25 2011

nonn

			E.g.f.: A(x) = 1 + 9*x + 169*x^2/2! + 3969*x^3/3! + 119025*x^4/4! + ...
where A(x) = 1 + 3^2*x + 13^2*x^2/2! + 63^2*x^3/3! + 345^2*x^4/4! + 2043^2*x^5/5! + ... + A202837(n)^2*x^n/n! + ...

G. C. Greubel, Table of n, a(n) for n = 0..350

Cf. A202837, A202827, A202828, A202829, A202831, A202833, A202835.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(9*x/(1-4*x))/Sqrt(1-16*x^2) ))); // G. C. Greubel, Jun 22 2022

CoefficientList[Series[Exp[9*x/(1-4*x)]/Sqrt[1-16*x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, May 23 2013 *)

{a(n)=n!*polcoeff(exp(9*x/(1-4*x)+x*O(x^n))/sqrt(1-16*x^2+x*O(x^n)),n)}

{a(n)=n!^2*polcoeff(exp(3*x+2*x^2+x*O(x^n)),n)^2}

{a(n)=sum(k=0,n\2,3^(n-2*k)*2^k*n!/((n-2*k)!*k!))^2}

def A202836_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(9*x/(1-4*x))/sqrt(1-16*x^2) ).egf_to_ogf().list()
A202836_list(40) # G. C. Greubel, Jun 22 2022

a(n) = A202837(n)^2, where the e.g.f. of A202837 is exp(3*x + 2*x^2).
a(n) ~ n^n*exp(3*sqrt(n)-9/8-n)*2^(2*n-1) * (1+33/(32*sqrt(n))). - Vaclav Kotesovec, May 23 2013
D-finite with recurrence: a(n) = (4*n+5)*a(n-1) + 4*(n-1)*(4*n+5)*a(n-2) - 64*(n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013

A202827 Expansion of e.g.f.: exp(4*x/(1-x)) / sqrt(1-x^2).

Original entry on oeis.org

1, 4, 25, 196, 1849, 20164, 249001, 3422500, 51739249, 851822596, 15155825881, 289527934084, 5906625426025, 128089110981316, 2940882813228649, 71239270847432164, 1815115761586307041, 48511703775281296900, 1356708799439194070809, 39615996090901693902916

Offset: 0

Paul D. Hanna, Dec 25 2011

nonn

			E.g.f.: A(x) = 1 + 4*x + 25*x^2/2! + 196*x^3/3! + 1849*x^4/4! +...
where A(x) = 1 + 2^2*x + 5^2*x^2/2! + 14^2*x^3/3! + 43^2*x^4/4! +...+ A005425(n)^2*x^n/n! +...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A005425, A202828, A202829, A202831, A202833, A202835, A202836.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(4*x/(1-x))/Sqrt(1-x^2) ))); // G. C. Greubel, Jun 21 2022

With[{nn=20},CoefficientList[Series[Exp[(4x)/(1-x)]/Sqrt[1-x^2], {x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Dec 31 2011 *)

{a(n)=n!*polcoeff(exp(4*x/(1-x)+x*O(x^n))/sqrt(1-x^2+x*O(x^n)),n)}

{a(n)=sum(k=0,n\2,2^(n-3*k)*n!/((n-2*k)!*k!))^2}

def A202827_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(4*x/(1-x))/sqrt(1-x^2) ).egf_to_ogf().list()
A202827_list(40) # G. C. Greubel, Jun 21 2022

a(n) = A005425(n)^2, where the e.g.f. of A005425 is exp(2*x + x^2/2).
a(n) = ( Sum_{k=0..[n/2]} 2^(n-3*k)*n!/((n-2*k)!*k!) )^2. [From formula by Huajun Huang in A005425]
a(n) ~ n^n*exp(4*sqrt(n)-2-n)/2 * (1+5/(3*sqrt(n))). - Vaclav Kotesovec, May 23 2013
D-finite with recurrence: a(n) = (n+3)*a(n-1) +(n-1)*(n+3)*a(n-2) - (n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013

A202833 Expansion of e.g.f.: exp(9*x/(1-x)) / sqrt(1-x^2).

Original entry on oeis.org

1, 9, 100, 1296, 19044, 311364, 5588496, 108993600, 2291345424, 51585311376, 1236953249856, 31447331115264, 844332494760000, 23859653712215616, 707522071322329344, 21958125453144843264, 711555574637600891136, 24025060090437573945600

Offset: 0

Paul D. Hanna, Dec 25 2011

nonn

			E.g.f.: A(x) = 1 + 9*x + 100*x^2/2! + 1296*x^3/3! + 19044*x^4/4! + ...
where A(x) = 1 + 3^2*x + 10^2*x^2/2! + 36^2*x^3/3! + 138^2*x^4/4! + ... + A202834(n)^2*x^n/n! + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A202834, A202827, A202828, A202829, A202831, A202835, A202836.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( Exp(9*x/(1-x))/Sqrt(1-x^2) ))); // G. C. Greubel, Jun 22 2022

With[{nn=20},CoefficientList[Series[Exp[((9x)/(1-x))]/Sqrt[1-x^2], {x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Apr 23 2012 *)

{a(n)=n!*polcoeff(exp(9*x/(1-x)+x*O(x^n))/sqrt(1-x^2+x*O(x^n)),n)}

{a(n)=n!^2*polcoeff(exp(3*x+x^2/2+x*O(x^n)),n)^2}

def A202833_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(9*x/(1-x))/sqrt(1-x^2) ).egf_to_ogf().list()
A202833_list(40) # G. C. Greubel, Jun 22 2022

a(n) = A202834(n)^2, where the e.g.f. of A202834 is exp(3*x + x^2/2).
a(n) = ( Sum_{k=0..[n/2]} 3^(n-2*k)/2^k * n!/((n-2*k)!*k!) )^2.
a(n) ~ n^n*exp(6*sqrt(n)-9/2-n)/2 * (1+15/(4*sqrt(n))). - Vaclav Kotesovec, May 23 2013
D-finite with recurrence: a(n) = (n+8)*a(n-1) + (n-1)*(n+8)*a(n-2) - (n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013

A202835 Expansion of e.g.f.: exp(9x/(1-2x)) / sqrt(1-4*x^2).

Original entry on oeis.org

1, 9, 121, 2025, 40401, 927369, 24000201, 689220009, 21710549025, 743187098889, 27441452694681, 1086166287819369, 45846179189949681, 2054407698719865225, 97357866191666622441, 4862830945258077841449, 255239441235423753980481, 14040944744510973314880009

Offset: 0

Paul D. Hanna, Dec 25 2011

nonn

			E.g.f.: A(x) = 1 + 9*x + 121*x^2/2! + 2025*x^3/3! + 40401*x^4/4! +...
where A(x) = 1 + 3^2*x + 11^2*x^2/2! + 45^2*x^3/3! + 201^2*x^4/4! + 963^2*x^5/5! +...+ A083886(n)^2*x^n/n! +...

G. C. Greubel, Table of n, a(n) for n = 0..350

Cf. A083886, A202827, A202828, A202829, A202831, A202833, A202836.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(9*x/(1-2*x))/Sqrt(1-4*x^2) ))); // G. C. Greubel, Jun 21 2022

CoefficientList[Series[Exp[9*x/(1-2*x)]/Sqrt[1-4*x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, May 23 2013 *)

{a(n)=n!*polcoeff(exp(9*x/(1-2*x)+x*O(x^n))/sqrt(1-4*x^2+x*O(x^n)),n)}

{a(n)=n!^2*polcoeff(exp(3*x+x^2+x*O(x^n)),n)^2}

{a(n)=sum(k=0,n\2,3^(n-2*k)*n!/((n-2*k)!*k!))^2}

def A202835_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(9*x/(1-2*x))/sqrt(1-4*x^2) ).egf_to_ogf().list()
A202835_list(40) # G. C. Greubel, Jun 21 2022

a(n) = A083886(n)^2, where the e.g.f. of A083886 is exp(3*x + x^2).
a(n) = ( Sum_{k=0..[n/2]} 3^(n-2*k) * n!/((n-2*k)!*k!) )^2.
a(n) ~ n^n*exp(3*sqrt(2*n)-9/4-n)*2^(n-1). - Vaclav Kotesovec, May 23 2013
D-finite with recurrence: a(n) = (2*n+7)*a(n-1) + 2*(n-1)*(2*n+7)*a(n-2) - 8*(n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013

A123071 Bishops on a 2n+1 X 2n+1 board (see Robinson paper for details).

Original entry on oeis.org

1, 2, 4, 12, 36, 120, 400, 1520, 5776, 23712, 97344, 431808, 1915456, 9012608, 42406144, 210988800, 1049760000, 5475340800, 28558296064, 155672726528, 848579961856, 4810614454272, 27271456395264, 160376430784512, 943132599095296, 5735299537018880

Offset: 0

N. J. A. Sloane, Sep 28 2006

nonn

R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). [The sequence S(2k+1) eq(24) p. 210.]
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)

Cf. A000898, A005635, A135401 (B(n)), A202828.

For Maple program see A005635.
# alternative
# this is A000898, replicated as 1,1,2,2,6,6,20,20,76,76,...
B := proc(n)
    if n=0 or n= -2 then
        1 ;
    elif type (n,'odd') then
        procname(n-1) ;
    else
        2*procname(n-2)+(n-2)*procname(n-4) ;
    end if;
end proc:
A123071 := proc(n)
    B(n)*B(n+1) ;
end proc:
seq(A123071(n),n=0..20) ; # R. J. Mathar, Apr 02 2017

B[n_] := B[n] = Which[n == 0 || n == -2, 1, OddQ[n], B[n-1], True, 2*B[n-2] + (n-2)*B[n-4]];
a[n_] := B[n]*B[n+1];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jul 23 2022, after R. J. Mathar *)

Conjecture: 2*a(n) +a(n-1) -2*n*a(n-2) +(-n-10)*a(n-3) -2*(n-2)*(n+2)*a(n-4) +(-n^2-2*n+23)*a(n-5) +2*(n-5)*(n^2-7*n+11)*a(n-6) +(n-6)*(n-5)^2*a(n-7)=0. - R. J. Mathar, Apr 02 2017

A202878 Expansion of e.g.f.: exp(16*x/(1-x)) / sqrt(1-x^2).

Original entry on oeis.org

1, 16, 289, 5776, 126025, 2972176, 75186241, 2027520784, 57988974481, 1751546371600, 55668326576641, 1855807478279056, 64713593898036889, 2354701531657512976, 89209297718289390625, 3512141211682081889296, 143435878498076017059361

Offset: 0

Paul D. Hanna, Dec 25 2011

nonn

			E.g.f.: A(x) = 1 + 16*x + 289*x^2/2! + 5776*x^3/3! + 126025*x^4/4! + ...
where A(x) = 1 + 4^2*x + 17^2*x^2/2! + 76^2*x^3/3! + 355^2*x^4/4! + 1724^2*x^5/5! + ... + A202879(n)^2*x^n/n! + ...

G. C. Greubel, Table of n, a(n) for n = 0..350

Cf. A202879, A202827, A202828, A202829, A202831, A202835, A202836, A202837.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(16*x/(1-x))/Sqrt(1-x^2) ))); // G. C. Greubel, Jun 22 2022

CoefficientList[Series[Exp[16*x/(1-x)]/Sqrt[1-x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, May 23 2013 *)

{a(n)=n!*polcoeff(exp(16*x/(1-x)+x*O(x^n))/sqrt(1-x^2+x*O(x^n)),n)}

{a(n)=n!^2*polcoeff(exp(4*x+x^2/2+x*O(x^n)),n)^2}

def A202878_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(16*x/(1-x))/sqrt(1-x^2) ).egf_to_ogf().list()
A202878_list(40) # G. C. Greubel, Jun 22 2022

a(n) = A202879(n)^2, where the e.g.f. of A202879 is exp(4*x + x^2/2).
a(n) = ( Sum_{k=0..floor(n/2)} 4^(n-2*k)/2^k * n!/((n-2*k)!*k!) )^2.
a(n) ~ n^n*exp(8*sqrt(n)-8-n)/2 * (1+22/(3*sqrt(n))). - Vaclav Kotesovec, May 23 2013
D-finite with recurrence: a(n) = (n+15)*a(n-1) + (n-1)*(n+15)*a(n-2) - (n-1)*(n-2)^2*a(n-3). - Vaclav Kotesovec, May 23 2013

A387317 Number of good involutions of all nontrivial linear quandles of order n.

Original entry on oeis.org

1, 4, 1, 2, 1, 44, 1, 2, 1, 414, 1, 2, 31, 5784, 1, 2, 1, 97358, 237, 2, 1, 1917064, 1, 2, 1, 42406158, 1

Offset: 3

Luc Ta, Aug 26 2025

A linear quandle is a pair (Z/nZ, k) where k is a unit in Z/nZ, viewed as an Alexander quandle under the operation a(b) := ka + (1-k)b. A linear quandle is trivial if and only if k = 1.

A good involution f of a quandle Q is an involution that commutes with all inner automorphisms and satisfies the identity f(y)(x) = y^-1(x). The pair (Q,f) is called a symmetric quandle.

Seiichi Kamada, Quandles with good involutions, their homologies and knot invariants, Intelligence of Low Dimensional Topology 2006, World Scientific Publishing Co. Pte. Ltd., 2007, 101-108.

Lực Ta, Good involutions of twisted conjugation subquandles and Alexander quandles, arXiv:2508.16772 [math.GT], 2025. See Table 2.
Lực Ta, Symmetric-Linear-Quandles, GitHub, 2025.
Index entries for sequences related to quandles and racks

Cf. A060594, A202828, A386233, A386234.

GAP
```
See Ta, GitHub link
```

If A060594(n) = 2, then a(n) = 1 if n is odd, a(n) = 4 if n = 4, and a(n) = 2 otherwise. See Ta, Ex. 5.8 and Prop. 5.9.

For all n >= 1, we have a(4n) >= A202828(n), with equality if and only if n = 1. See Ta, Thm. 5.11.

Some terms corrected by Luc Ta, Sep 03 2025

cp's OEIS Frontend

A202829 Expansion of e.g.f.: exp(4*x/(1-3*x)) / sqrt(1-9*x^2).

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A202831 Expansion of e.g.f.: exp(4*x/(1-5*x)) / sqrt(1-25*x^2).

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A202836 Expansion of e.g.f.: exp(9*x/(1-4*x)) / sqrt(1-16*x^2).

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A202827 Expansion of e.g.f.: exp(4*x/(1-x)) / sqrt(1-x^2).

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A202833 Expansion of e.g.f.: exp(9*x/(1-x)) / sqrt(1-x^2).

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A202835 Expansion of e.g.f.: exp(9*x/(1-2*x)) / sqrt(1-4*x^2).

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A202829 Expansion of e.g.f.: exp(4x/(1-3x)) / sqrt(1-9*x^2).

A202831 Expansion of e.g.f.: exp(4x/(1-5x)) / sqrt(1-25*x^2).

A202836 Expansion of e.g.f.: exp(9x/(1-4x)) / sqrt(1-16*x^2).

A202835 Expansion of e.g.f.: exp(9x/(1-2x)) / sqrt(1-4*x^2).