A000246 -id:A000246 - cp's OEIS frontend

A365969 Expansion of e.g.f. exp( Sum_{k>=0} x^(5k+1) / (5k+1) ).

1, 1, 1, 1, 1, 1, 121, 841, 3361, 10081, 25201, 3684241, 50309281, 369738721, 1926648361, 7980936601, 1335634023361, 27705746752321, 302258931418081, 2283161710263841, 13419441405835201, 2498339829188508481, 70152448708746111961, 1025314852704395518441

Offset: 0

Seiichi Manyama, Sep 23 2023

nonn

Cf. A000246, A193437, A193438.

Cf. A333882, A365977.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=0, N\5, x^(5*k+1)/(5*k+1)))))

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..floor((n-1)/5)} a(n-5*k-1)/(n-5*k-1)!.

a(0) = a(1) = ... = a(4) = 1; a(n) = a(n-1) + 120 * binomial(n-1,5) * a(n-5).

A110491 Expansion of e.g.f.: sqrt(1+2x)/sqrt(1-2x).

Original entry on oeis.org

1, 2, 4, 24, 144, 1440, 14400, 201600, 2822400, 50803200, 914457600, 20118067200, 442597478400, 11507534438400, 299195895398400, 8975876861952000, 269276305858560000, 9155394399191040000, 311283409572495360000

Offset: 0

Paul Barry, Jul 22 2005

Row sums of exponential Riordan array [1, arctanh(2x)]. - Paul Barry, Apr 17 2008

Conjecture: {a(n-1), n>=1} is the T-transform of A093178, where T maps a sequence {b(n), n>=1} to the sequence {c(n)} defined by c(n) = det(M_n), where M_n is the n X n matrix with elements M_n(i,j) = b(2*j) for i>j and M_n(i,j) = b(i+j-1) for i<=j. - Lechoslaw Ratajczak, Aug 04 2021

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

Cf. A000246, A002867, A063886.

S:= series(sqrt(1+2*x)/sqrt(1-2*x),x,31):
seq(coeff(S,x,j)*j!,j=0..30); # Robert Israel, Jun 08 2016

With[{nn=20},CoefficientList[Series[Sqrt[1+2x]/Sqrt[1-2x],{x,0,nn}],x] Range[0,nn]!] (* or *) Join[{1},Table[2n!Binomial[n-1,Floor[(n-1)/2]], {n,20}]] (* Harvey P. Dale, Nov 11 2011 *)
Table[2^n Binomial[1/2,n] n! Hypergeometric2F1[1/2, -n, 3/2 - n, -1], {n, 0, 20}] (* Benedict W. J. Irwin, Jun 06 2016 *)

my(x='x+O('x^25)); Vec(serlaplace(sqrt(1+2*x)/sqrt(1-2*x))) \\ Michel Marcus, Aug 05 2021

E.g.f.: sqrt((1+2x)/(1-2x)); a(n)=2*n!*binomial(n-1, floor((n-1)/2))+0^n.
The sequence 0,1,0,2,0,4,... has e.g.f. arctanh(x). - Paul Barry, Apr 17 2008
D-finite with recurrence a(n) -2*a(n-1) -4*(n-1)*(n-2)*a(n-2)=0. - R. J. Mathar, Sep 20 2012
a(n) ~ 2^(n+1)*n^n/exp(n). - Vaclav Kotesovec, Sep 25 2013
a(n) = 2^n*binomial(1/2,n)*n!*2F1(1/2,-n;3/2-n;-1). - Benedict W. J. Irwin, Jun 06 2016
From Robert Israel, Jun 08 2016: (Start)
a(n) = n! * A063886(n).
E.g.f. satisfies 2*g(x)+(4*x^2-1)*g'(x) = 0, from which Mathar's recurrence follows. (End)
Sum_{n>=0} 1/a(n) = 1 + (StruveL(-1,1/2) + StruveL(0,1/2))*Pi/4, where StruveL is the modified Struve function. - Amiram Eldar, Aug 15 2025

A113774 Number of partitions of {1,...,n} into block sizes not a multiple of 3.

Original entry on oeis.org

1, 1, 2, 4, 11, 32, 112, 415, 1732, 7678, 37115, 190016, 1039546, 5996083, 36528196, 233492044, 1564012751, 10940385668, 79762304116, 604791685063, 4760047233424, 38825234812882, 327641201731475, 2856835856307428, 25702896025566886, 238331921722835203

Offset: 0

Vladeta Jovovic, Jan 19 2006

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A003724, A115276, A000246, A102736, A088009, A001590.

nmax := 30: B := add(op(1+(i mod 3),[0,1,1])*x^i/i!,i=0..nmax) : egf := 0 : for i from 0 to nmax do egf := convert(egf+taylor(B^i,x=0,nmax+1)/i!,polynom) : od: for i from 0 to nmax do printf("%d ", i!*coeftayl(egf,x=0,i)) ; od: # R. J. Mathar, Feb 06 2008
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(`if`(
      irem(j, 3)=0, 0, binomial(n-1, j-1)*a(n-j)), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 17 2015

a=Sum[x^(3i)/(3i)!,{i,1,20}]; Range[0, 20]! CoefficientList[Series[Exp[Exp[x] - 1 - a], {x, 0, 20}], x] (* Geoffrey Critzer, Jan 02 2011 *)

E.g.f.: exp(B(x)), where B(x) is e.g.f. of A011655.

More terms from R. J. Mathar, Feb 06 2008

A275385 Number of labeled functional digraphs on n nodes with only odd sized cycles and such that every vertex is at a distance of at most 1 from a cycle.

Original entry on oeis.org

1, 1, 3, 12, 73, 580, 5601, 63994, 844929, 12647016, 211616065, 3914510446, 79320037281, 1747219469164, 41569414869633, 1062343684252530, 29023112392093441, 844101839207139280, 26038508978625589377, 849150487829425227094, 29189561873274715264545

Offset: 0

Geoffrey Critzer, Jul 25 2016

nonn

Equivalently, these are the functions counted by A116956 with the additional constraint that every element is mapped to a recurrent element. A recurrent element is an element on a cycle in the functional digraph.

Alois P. Heinz, Table of n, a(n) for n = 0..411

Cf. A000246, A006153, A009444, A116956, A216401.

b:= proc(n) option remember; `if`(n=0, 1, add(`if`(j::odd,
       (j-1)!*b(n-j)*binomial(n-1, j-1), 0), j=1..n))
    end:
a:= n-> add(b(j)*j^(n-j)*binomial(n, j), j=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 25 2016

nn = 20; Range[0, nn]! CoefficientList[Series[Sqrt[(1 + z*Exp[z])/(1 - z*Exp[z])], {z, 0, nn}], z]

default(seriesprecision, 30);
S=sqrt((1 + x*exp(x))/(1 - x*exp(x)));
v=Vec(S); for(n=2,#v-1,v[n+1]*=n!); v \\ Charles R Greathouse IV, Jul 29 2016

E.g.f.: sqrt((1 + z*exp(z))/(1 - z*exp(z))).
Exponential transform of A216401.
a(n) ~ 2 * n^n / (sqrt(1+LambertW(1)) * LambertW(1)^n * exp(n)). - Vaclav Kotesovec, Jun 26 2022

A316666 Number of simple relaxed compacted binary trees of right height at most one with no sequences on level 1 and no final sequences on level 0.

Original entry on oeis.org

1, 0, 1, 3, 15, 87, 597, 4701, 41787, 413691, 4512993, 53779833, 695000919, 9680369943, 144560191149, 2303928046437, 39031251610227, 700394126116851, 13270625547477177, 264748979672169681, 5547121478845459983, 121784530649198053263, 2795749225338111831429, 66981491857058929294653

Offset: 0

Michael Wallner, Jul 10 2018

nonn

A relaxed compacted binary tree of size n is a directed acyclic graph consisting of a binary tree with n internal nodes, one leaf, and at most n pointers. It is constructed from a binary tree of size n, where the first leaf in a post-order traversal is kept and all other leaves are replaced by pointers. These links may point to any node that has already been visited by the post-order traversal. It is called simple if for nodes with two pointers both point to the same node. The right height is the maximal number of right-edges (or right children) on all paths from the root to any leaf after deleting all pointers. See the Wallner link.

a(n) is one of two "basis" sequences for sequences of the form a(0)=a, a(1)=b, a(n) = n*a(n-1) + (n-1)*a(n-2), the second basis sequence being A096654 (with 0 appended as a(0)). The sum of these sequences is listed as A000255. - Gary Detlefs, Dec 11 2018

G. C. Greubel, Table of n, a(n) for n = 0..448
Antoine Genitrini, Bernhard Gittenberger, Manuel Kauers and Michael Wallner, Asymptotic Enumeration of Compacted Binary Trees, arXiv:1703.10031 [math.CO], 2017.
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017.

Cf. A000032, A000246, A001879, A051577, A213527, A288950, A288952, A288953 (subclasses of relaxed compacted binary trees of right height at most one, see the Wallner link).
Cf. A000166, A000255, A000262, A052852, A123023, A130905, A176408, A201203 (variants of simple relaxed compacted binary trees of right height at most one, see the Wallner link).
Cf. A000255, A096654.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (3*Exp(-x) + x-2)/(1-x)^2 )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Dec 12 2018

aseq := n-> 3*round((n+2)*n!/exp(1))-(n+2)*n!: bseq := n-> (n+2)*n!- 2* round((n+2)*n!/exp(1)): s := (a,b,n)-> a*aseq(n) + b*bseq( n): seq(s(1,0,n),n = 0..20);  # Gary Detlefs, Dec 11 2018

terms = 24;
CoefficientList[(3E^-z+z-2)/(1-z)^2 + O[z]^terms, z] Range[0, terms-1]! (* Jean-François Alcover, Sep 14 2018 *)

Vec(serlaplace((3*exp(-x + O(x^25)) + x - 2)/(1 - x)^2)) \\ Andrew Howroyd, Jul 10 2018

E.g.f.: (3*exp(-z)+z-2)/(1-z)^2.
a(n) ~ (3*exp(-1) - 1) * n * n!. - Vaclav Kotesovec, Jul 12 2018
a(n) = 3*round((n+2)*n!/e) - (n+2)*n!. - Gary Detlefs, Dec 11 2018
From Seiichi Manyama, Apr 25 2025: (Start)
a(n) = 3 * A000255(n) - n! - (n+1)!.
a(0) = 1, a(1) = 0; a(n) = n*a(n-1) + (n-1)*a(n-2). (End)

A346943 a(n) = a(n-1) + n(n+1)a(n-2) with a(0)=1, a(1)=1.

Original entry on oeis.org

1, 1, 7, 19, 159, 729, 7407, 48231, 581535, 4922325, 68891175, 718638075, 11465661375, 142257791025, 2550046679775, 36691916525775, 730304613424575, 11958031070311725, 261722208861516375, 4805774015579971875, 114729101737416849375, 2334996696935363855625

Offset: 0

Vaclav Kotesovec, Aug 08 2021, following a suggestion from John M. Campbell

From Peter Bala, Dec 09 2024: (Start)

b(n) := A000246(n+2) = (n+2)!/2^(n+1) * binomial(n+1, floor((n+1)/2)) satisfies the same second-order recurrence as a(n) with the initial conditions b(0) = 1 and b(1) = 3. This leads to the finite continued fraction a(n)/b(n) = 1/(1 + 2/(1 + 6/(1 + ... + n*(n+1)/1). Letting n tend to infinity gives the continued fraction representation 1/(1 + 2/(1 + 6/(1 + ... + n*(n+1)/(1 + ...) = Pi/2 - 1, due to Euler - see paragraph 31, p. 48. (End)

Leonhard Euler, E123: De fractionibus continuis observationes, originally published in Commentarii academiae scientiarum Petropolitanae 11, 1750, pp. 32-81, reprinted in Opera Omnia: Series 1, Volume 14, 291-349.

Cf. A000246, A000932, A024167.

RecurrenceTable[{a[n] == a[n-1] + n*(n+1)*a[n-2], a[0]==1, a[1]==1}, a, {n,0,20}]
nmax = 20; CoefficientList[Series[(-2 + Pi + 2*Pi*x + 4*Sqrt[1 - x^2] + 2*x*(-2 + Sqrt[1 - x^2]) - 4*(1 + 2*x) * ArcSin[Sqrt[1 - x]/Sqrt[2]]) / (2*(1 - x)^(5/2) * (1 + x)^(3/2)), {x, 0, nmax}], x] * Range[0, nmax]!

a(n) ~ n! * (Pi - 2) * n^(3/2) / sqrt(2*Pi).
a(n) ~ (Pi - 2) * n^(n+2) / exp(n).
E.g.f. A(x) satisfies the differential equation -6*A(x) - (6*x + 1)*A'(x) + (1 - x^2)*A''(x) = 0, A(0)=1, A'(0)=1.
E.g.f.: (-2 + Pi + 2*Pi*x + 4*sqrt(1-x^2) + 2*x*(-2+sqrt(1-x^2)) - 4*(1+2*x) * arcsin(sqrt(1-x)/sqrt(2))) / (2*(1-x)^(5/2) * (1+x)^(3/2)).

A346371 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( Sum_{n>=0} x^(2n+1) / (2n+1)^2 ).

Original entry on oeis.org

1, 1, 2, 10, 88, 1496, 34256, 1305872, 57804160, 3960382848, 288097804032, 31177032137472, 3374496463248384, 530644850402565120, 79955455534325999616, 17241179374803330287616, 3448609425518084068048896, 977269122457749276877750272, 250420488297020919542581493760

Offset: 0

Ilya Gutkovskiy, Jul 14 2021

nonn

Cf. A000246, A074707.

nmax = 18; CoefficientList[Series[Exp[Sum[x^(2 k + 1)/(2 k + 1)^2, {k, 0, Infinity}]], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = (1/n) Sum[(Binomial[n, 2 k + 1] (2 k + 1)!)^2 a[n - 2 k - 1]/(2 k + 1), {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 18}]

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

cp's OEIS Frontend

A365969 Expansion of e.g.f. exp( Sum_{k>=0} x^(5*k+1) / (5*k+1) ).

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A110491 Expansion of e.g.f.: sqrt(1+2x)/sqrt(1-2x).

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A113774 Number of partitions of {1,...,n} into block sizes not a multiple of 3.

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A275385 Number of labeled functional digraphs on n nodes with only odd sized cycles and such that every vertex is at a distance of at most 1 from a cycle.

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A316666 Number of simple relaxed compacted binary trees of right height at most one with no sequences on level 1 and no final sequences on level 0.

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A346943 a(n) = a(n-1) + n*(n+1)*a(n-2) with a(0)=1, a(1)=1.

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A346371 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( Sum_{n>=0} x^(2*n+1) / (2*n+1)^2 ).

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A365969 Expansion of e.g.f. exp( Sum_{k>=0} x^(5k+1) / (5k+1) ).

A346943 a(n) = a(n-1) + n(n+1)a(n-2) with a(0)=1, a(1)=1.

A346371 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( Sum_{n>=0} x^(2n+1) / (2n+1)^2 ).