A000806 -id:A000806 - cp's OEIS frontend

A006198 Number of partitions into pairs.

1, 1, 6, 41, 365, 3984, 51499, 769159, 13031514, 246925295, 5173842311, 118776068256, 2964697094281, 79937923931761, 2315462770608870, 71705109685449689, 2364107330976587909, 82676528225908987824, 3056806370495613000259, 119137361202296994159415

Offset: 1

N. J. A. Sloane

nonn

a(n) is the subset of the set of unordered pairings of the first 2n integers (A001147) forbidding pairs of the form (i,i+1) for all i in [2,n-1]. There are many other selections of forbidden pairs giving the same count. - Olivier Gérard, Feb 08 2011

			G.f. = x + x^2 + 6*x^3 + 41*x^4 + 365*x^5 + 3984*x^6 + 51499*x^7 + ...

G. Kreweras and Y. Poupard, Sur les partitions en paires d'un ensemble fini totalement ordonné, Publications de l'Institut de Statistique de l'Université de Paris, 23 (1978), 57-74.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Gheorghe Coserea, Table of n, a(n) for n = 1..201
Simon Plouffe, Approximations of generating functions and a few conjectures, arXiv:0911.4975 [math.NT], 2009.

a[ n_] := With[ {m = Abs[n] - 1}, If[ m < 0, 0, Sign[n] Hypergeometric1F1[-m, -2 m - 1, -2] (2 m + 1)!!]]; (* Michael Somos, Jan 27 2014 *)
a[ n_] := With[ {m = Abs[n] - 1}, If[ m < 0, 0, Sign[n] Sum[ (-1)^k (2 m + 1 - k)! / (2^(m - k) k! (m - k)!), {k, 0, m}]]]; (* Michael Somos, Jan 27 2014 *)
a[ n_] := With[ {m = Abs[n] - 1}, If[ m < 0, 0, Sign[n] Numerator @ FromContinuedFraction[ Table[(-1)^Quotient[k, 2] If[ OddQ[k], k, 1], {k, 2 m + 1}]]]]; (* Michael Somos, Jan 27 2014 *)
Rest[CoefficientList[Series[E^(-1 + Sqrt[1 - 2*x])*(-1 + 1/Sqrt[1 - 2*x]), {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Nov 29 2015 *)
Table[(2 n - 1)!! Hypergeometric1F1[1 - n, 1 - 2 n, -2], {n, 20}] (* Eric W. Weisstein, Nov 14 2018 *)

{a(n) = sign(n) * if( n==0, 0, contfracpnqn( vector( 2*abs(n) -1, k, (-1)^(k\2) * if( k%2, k, 1))) [1,1]) }; /* Michael Somos, Jan 27 2014 */

{a(n) = sign(n) * sum( k=0, n=abs(n)-1, (-1)^k * (2*n + 1 - k)! / (2^(n - k) * k! * (n - k)!) ) }; /* Michael Somos, Jan 27 2014 */

x = 'x+O('x^33); Vec(serlaplace(((2 - 2*x - (1 - 2*x)^(1/2)) / (1-2*x)^(3/2)) * exp((1-2*x)^(1/2) - 1))) \\ Gheorghe Coserea, Aug 05 2015

a(n) = |A000806(n-1)|+|A000806(n)|. G.f.: Sum_{n>=0} A001147(n)*(x/(1+x)^2)^n. - Vladeta Jovovic, Jun 27 2007
Recurrence: (4*n^2-8*n+1)*a(n-1) + (2*n-1)*a(n-2) + (3-2*n)*a(n) = 0. - Vaclav Kotesovec, Oct 05 2012
G.f.: T(0) - 1, where T(k) = 1 - (k+1)*x/( (k+1)*x - (1+x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 03 2013
a(-n) = -a(n) for all n in Z. - Michael Somos, Jan 27 2014
a(n+1) = Sum_{k=0..n} (-1)^k * (2n+1-k)! / (2^(n-k) * k! * (n-k)!) if n>=0. - Michael Somos, Jan 27 2014
0 = a(n) * (a(n+2) + a(n+3)) + a(n+1) * (-a(n+1) -3*a(n+2) -4*a(n+3) + a(n+4)) + a(n+2) * (-3*a(n+3) + a(n+4)) + a(n+3) * (-a(n+3)) for all n in Z. - Michael Somos, Jan 27 2014
E.g.f. (for offset 0): ((2 - 2*x - (1 - 2*x)^(1/2)) / (1-2*x)^(3/2)) * exp((1-2*x)^(1/2) - 1) (formula due to B. Salvy, see Plouffe link). - Gheorghe Coserea, Aug 05 2015
E.g.f. (for offset 1): exp(sqrt(1-2*x)-1) * (1/sqrt(1-2*x)-1). - Vaclav Kotesovec, Nov 29 2015
a(n) ~ 2^(n+1/2)*n^n/exp(n+1). - Vaclav Kotesovec, Nov 29 2015

A101683 Write exp(sqrt(1+x)-1) = Sum c(n) x^n/n!; then a(n) = numerator of c(n).

Original entry on oeis.org

1, 1, 0, 1, -5, 9, -329, 3655, -11961, 721315, -12310199, 29326887, -4939227215, 113836841041, -356357531655, 77087063678521, -2238375706930349, 17366683494629835, -2294640596998068569, 80381887628910919255

Offset: 0

Ralf Stephan, Dec 13 2004

Odd part of A000806.

			exp(sqrt(1+x)-1) = 1+(1/2)*x+(1/48)*x^3-(5/384)*x^4+(3/320)*x^5-(329/46080)*x^6+(731/129024)*x^7-(1329/286720)*x^8+... - From _N. J. A. Sloane_, Aug 29 2012

Robert Israel, Table of n, a(n) for n = 0..404

Denominators are 2^A101684(n).

c[0]:= 1: c[1]:= 1/2:
for n from 2 to 100 do c[n]:= (c[n-2]-(4*n-6)*c[n-1])/4 od:
seq(numer(c[n]),n=0..100); # Robert Israel, Nov 30 2023

With[{nn=20},Numerator[CoefficientList[Series[Exp[Sqrt[1+x]-1],{x,0,nn}],x]Range[0,nn]!]] (* Harvey P. Dale, Aug 29 2012 *)

my(x='x+O('x^30)); apply(numerator, Vec(serlaplace(exp(sqrt(1+x)-1)))) \\ Michel Marcus, Nov 30 2023

Numerator of c(n) satisfying c(n) = (c(n-2) - (4*n-6)*c(n-1))/4, c(0) = 1, c(1) = 1/2. - Robert Israel, Nov 30 2023

Definition clarified by N. J. A. Sloane, Aug 29 2012

A104548 Triangle read by rows giving coefficients of Bessel polynomial p_n(x).

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 3, 3, 0, 1, 6, 15, 15, 0, 1, 10, 45, 105, 105, 0, 1, 15, 105, 420, 945, 945, 0, 1, 21, 210, 1260, 4725, 10395, 10395, 0, 1, 28, 378, 3150, 17325, 62370, 135135, 135135, 0, 1, 36, 630, 6930, 51975, 270270, 945945, 2027025, 2027025, 0

Offset: 0

Eric W. Weisstein, Mar 14 2005

			Bessel polynomials begin with:
      x;
      x +     x^2;
    3*x +   3*x^2 +    x^3;
   15*x +  15*x^2 +  6*x^3 +    x^4;
  105*x + 105*x^2 + 45*x^3 + 10*x^4 + x^5;
  ...
Triangle of coefficients begins as:
  0;
  1,  0;
  1,  1    0;
  1,  3,   3     0;
  1,  6,  15,   15      0;
  1, 10,  45,  105,   105      0;
  1, 15, 105,  420,   945,   945       0;
  1, 21, 210, 1260,  4725, 10395,  10395       0;
  1, 28, 378, 3150, 17325, 62370, 135135, 135135    0;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Eric Weisstein's World of Mathematics, Bessel Polynomial

Essentially the same as A001498 (the main entry).

Cf. A000085, A000806, A001464, A001515.

A104548:= func< n,k | k eq n select 0 else Binomial(n-1,k)*Factorial(n+k-1)/(2^k*Factorial(n-1)) >;
[A104548(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Jan 02 2023

T[n_, k_]:= If[k==n, 0, Binomial[n-1,k]*(n+k-1)!/(2^k*(n-1)!)];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 02 2023 *)

def A104548(n,k): return 0 if (k==n) else binomial(n-1,k)*factorial(n+k-1)/(2^k*factorial(n-1))
flatten([[A104548(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 02 2023

From G. C. Greubel, Jan 02 2023: (Start)
T(n, k) = binomial(n-1,k)*(n+k-1)!/(2^k*(n-1)!), with T(n, n) = 0.
Sum_{k=0..n} T(n, k) = A001515(n-1).
Sum_{k=0..n} (-1)^k*T(n, k) = A000806(n-1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A000085(n-1).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A001464(n-1). (End)

T(0, 0) = 0 prepended by G. C. Greubel, Jan 02 2023

A271215 Number of loop-free assembly graphs with n rigid vertices.

Original entry on oeis.org

1, 0, 1, 4, 24, 184, 1911, 24252, 362199, 6162080, 117342912, 2469791336, 56919388745, 1425435420600, 38543562608825, 1119188034056244, 34733368101580440, 1147320305439301344, 40190943859500501151, 1488212241729974297796, 58080468361734193793551

Offset: 0

Jonathan Burns, Apr 13 2016

nonn

Number of chord diagrams (equivalent up to reflection) that do not contain any simple chords, e.g., 121332 contains the simple chord 33.

			For n=0 the a(0)=1 solution is { ∅ }.
For n=1, a(1)=0 since the only assembly graph with one rigid vertex is the loop 11.
For n=2, the a(2)=1 solution is { 1212 }.
For n=3, the a(3)=4 solutions are { 121323, 123123, 123231, 123132 }.

J. Burns, Counting a Class of Signed Permutations and Chord Diagrams related to DNA Rearrangement, Preprint.

Kristin DeSplinter, Satyan L. Devadoss, Jordan Readyhough, and Bryce Wimberly, Unfolding cubes: nets, packings, partitions, chords, arXiv:2007.13266 [math.CO], 2020. See Table 1 p. 15.

Cf. A000806, A132101, A271218.

(Table[Sum[Binomial[n,i]*(2*n-i)!/2^(n-i)*(-1)^(i)/n!,{i,0,n}],{n,0,20}]+RecurrenceTable[{a[n]==2a[n-1]+(2n-3)a[n-2]-(2n-5)a[n-3]+2a[n-4]-a[n-5],a[0]==1,a[1]==0,a[2]==1,a[3]==3,a[4]==12},a[n],{n,0,20}])/2

f(n) = sum(k=0, n, (2*n-k)! / (k! * (n-k)!) * (-1/2)^(n-k) ); \\ A000806
lista(nn) = {my(va = vector(nn)); va[1] = 1; va[2] = 0; va[3] = 1; va[4] = 3; va[5] = 12; for (n=5, nn-1, va[n+1] = 2*va[n] + (2*n-3)*va[n-1] - (2*n-5)*va[n-2] + 2*va[n-3] - va[n-4];); vector(nn-1, n, (va[n] + abs(f(n-1)))/2);} \\ Michel Marcus, Jul 28 2020

a(n) ~ (2n/e)^n / (e * sqrt(2)).
a(n) = (|A000806(n)| + A271218(n)) / 2.
a(n)/A132101(n) ~ 1/e.

A380307 Expansion of e.g.f. exp( (1+5*x)^(1/5) - 1 ).

Original entry on oeis.org

1, 1, -3, 25, -335, 6177, -144947, 4128937, -138327615, 5327738497, -231899041475, 11255588133945, -602683483719503, 35288931375293857, -2242963870471014963, 153791777744471484745, -11314787069889491407103, 889087243145447511507969, -74312052321224600661026051

Offset: 0

Seiichi Manyama, Jan 20 2025

sign

Seiichi Manyama, Table of n, a(n) for n = 0..355

Cf. A000806, A380208.

Cf. A000110, A028575.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((1+5*x)^(1/5)-1)))

a(n) = Sum_{k=0..n} 5^(n-k) * Stirling1(n,k) * Bell(k).
a(n) = (1/e) * 5^n * n! * Sum_{k>=0} binomial(k/5,n)/k!.
a(0) = 1; a(n) = Sum_{k=1..n} (Product_{j=0..k-1} (-5*j+1)) * binomial(n-1,k-1) * a(n-k).

A006147 Reversion of Ramanujan numbers.

Original entry on oeis.org

1, 24, 900, 40352, 1994322, 104816880, 5747466920, 325077729600, 18826860841119, 1110900168420264, 66547088543789532, 4036419643768799328, 247405021070280491110, 15299980644645295780560, 953473460271200881693560, 59817824263728235912396224

Offset: 1

N. J. A. Sloane

nonn

			x + 24*x^2 + 900*x^3 + 40352*x^4 + 1994322*x^5 + 104816880*x^6 + ...

Amiram Eldar, Table of n, a(n) for n = 1..100
G. Kreweras and Y. Poupard, Sur les partitions en paires d'un ensemble fini totalement ordonné, Publications de l'Institut de Statistique de l'Université de Paris, 23 (1978), 57-74. (Annotated scanned copy)
Index entries for reversions of series.

Cf. A000594.

Rest[CoefficientList[InverseSeries[Series[q * QPochhammer[q, q]^24, {q, 0, 16}]], q]] (* Amiram Eldar, Jan 06 2025 *)

REVERT(A000594).

Signs corrected Dec 24 2001

A373175 Expansion of e.g.f. exp(sqrt(2x+1)-1)/(2-sqrt(2x+1))^2.

Original entry on oeis.org

1, 3, 8, 25, 87, 386, 1663, 11313, 39560, 717067, -2408199, 128675438, -2009225567, 53624676795, -1282589050168, 35660396328721, -1032462831852297, 32302377782200418, -1070227545188815745, 37651172275242136857, -1398665563931458389304, 54757245858874447661683

Offset: 0

Stefano Spezia, May 26 2024

sign

Y. Alp and E. G. Kocer, Exponential Almost-Riordan Arrays, Results Math 79, 173 (2024). See page 24.

Cf. A000806, A330797, A373176.

a[n_]:=n!SeriesCoefficient[Exp[Sqrt[2x+1]-1]/(2-Sqrt[2x+1])^2,{x,0,n}]; Array[a,22,0]

my(x = 'x+O('x^30)); Vec(serlaplace(exp(sqrt(2*x+1)-1)/(2-sqrt(2*x+1))^2)) \\ Michel Marcus, May 27 2024

A373176 Expansion of e.g.f. 2exp(sqrt(2x+1)-1)/(2-sqrt(2*x+1))^3.

Original entry on oeis.org

2, 8, 30, 122, 548, 2802, 15638, 100760, 661242, 5519558, 36021212, 495019758, 944742290, 96695115272, -1151063332242, 46492769525882, -1177828529162332, 39211350154011570, -1272035779868081338, 45289997660347946648, -1679496857400789295638, 65976928289858329056518

Offset: 0

Stefano Spezia, May 26 2024

sign

Y. Alp and E. G. Kocer, Exponential Almost-Riordan Arrays, Results Math 79, 173 (2024). See page 24.

Cf. A000806, A330797, A373175.

a[n_]:=n!SeriesCoefficient[2Exp[Sqrt[2x+1]-1]/(2-Sqrt[2x+1])^3,{x,0,n}]; Array[a,22,0]

A272261 Number of one-to-one functions f from [n] to [2n] where f(x) may not be equal to x or to 2n+1-x.

Original entry on oeis.org

1, 0, 4, 40, 576, 10528, 233920, 6124032, 184656640, 6302821888, 240245858304, 10115537336320, 466275700903936, 23354247194542080, 1262994451308888064, 73347095164693676032, 4552571878016243466240, 300763132329730843475968, 21071629550593224017182720

Offset: 0

Marko Riedel, Apr 23 2016

nonn

Marko R. Riedel, Brian M. Scott et al. Enumerating a type of function by inclusion-exclusion

Cf. A000806.

a := n -> add(binomial(n,q)*(-1)^q*2^q*binomial(2*n-q,n-q)*(n-q)!, q=0..n): seq(a(n), n=0..20);
seq(simplify(KummerU(-n, -2*n, -2)), n = 0..18); # Peter Luschny, May 10 2022

Table[CoefficientList[Series[E^(-1 + Sqrt[1 - 4 x])/Sqrt[1 - 4 x], {x, 0, 20}],x][[n]] (n - 1)!, {n, 1, 20}] (* Benedict W. J. Irwin, Jul 14 2016 *)

a(n) = Sum_{q=0..n} C(n,q) (-1)^q 2^q C(2n-q,n-q) (n-q)!.
a(n) = abs(A000806(n)) * 2^n.
E.g.f.: exp(-1+sqrt(1-4*x))/sqrt(1-4*x). - Benedict W. J. Irwin, Jul 14 2016
a(n) ~ 2^(2*n+1/2) * n^n / exp(n+1). - Vaclav Kotesovec, Jul 16 2016
Conjecture: Alternating sign g.f. is Sum_{k>=0} HermiteH[k,sqrt(x)]x^(k/2). - Benedict W. J. Irwin, Nov 30 2016
Conjecture D-finite with recurrence: a(n) + 2*(-2*n+1)*a(n-1) - 4*a(n-2)=0. - R. J. Mathar, Jan 27 2020
a(n) = KummerU(-n, -2*n, -2). - Peter Luschny, May 10 2022

A336653 First differences of A271215.

Original entry on oeis.org

-1, 1, 3, 20, 160, 1727, 22341, 337947, 5799881, 111180832, 2352448424, 54449597409, 1368516031855, 37118127188225, 1080644471447419, 33614180067524196, 1112586937337720904, 39043623554061199807, 1448021297870473796645, 56592256120004219495755, 2324706946641972649074513

Offset: 1

Michel Marcus, Jul 28 2020

sign

a(n) is the number of epsilon-paths of the n-cube for n>=2.

Kristin DeSplinter, Satyan L. Devadoss, Jordan Readyhough, and Bryce Wimberly, Unfolding cubes: nets, packings, partitions, chords, arXiv:2007.13266 [math.CO], 2020. See Table 1 p. 15.

Cf. A000806, A271215.

f(n) = sum(k=0, n, (2*n-k)! / (k! * (n-k)!) * (-1/2)^(n-k) ); \\ A000806
lista(nn) = {my(va = vector(nn)); va[1] = 1; va[2] = 0; va[3] = 1; va[4] = 3; va[5] = 12; for (n=5, nn-1, va[n+1] = 2*va[n] + (2*n-3)*va[n-1] - (2*n-5)*va[n-2] + 2*va[n-3] - va[n-4];); my(w=vector(nn-1, n, (va[n] + abs(f(n-1)))/2)); vector(#w-1, k, w[k+1] - w[k]);} \\ Michel Marcus, Jul 28 2020

a(n) = A271215(n) - A271215(n-1).

cp's OEIS Frontend

A006198 Number of partitions into pairs.

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A101683 Write exp(sqrt(1+x)-1) = Sum c(n) x^n/n!; then a(n) = numerator of c(n).

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A104548 Triangle read by rows giving coefficients of Bessel polynomial p_n(x).

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Magma

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A271215 Number of loop-free assembly graphs with n rigid vertices.

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A380307 Expansion of e.g.f. exp( (1+5*x)^(1/5) - 1 ).

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A006147 Reversion of Ramanujan numbers.

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Mathematica

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A373175 Expansion of e.g.f. exp(sqrt(2*x+1)-1)/(2-sqrt(2*x+1))^2.

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A373175 Expansion of e.g.f. exp(sqrt(2x+1)-1)/(2-sqrt(2x+1))^2.

A373176 Expansion of e.g.f. 2exp(sqrt(2x+1)-1)/(2-sqrt(2*x+1))^3.