A007889 -id:A007889 - cp's OEIS frontend

A202617 E.g.f. satisfies: A(x) = exp( x*(1 + A(x)^2)/2 ).

1, 1, 3, 19, 185, 2441, 40747, 823691, 19564785, 534145105, 16482667091, 567343245635, 21552042260905, 895664877901145, 40422799315249275, 1968883362773653051, 102942561775293158369, 5750760587905912310177, 341848844954020959953059, 21545207157567497255044979

Offset: 0

Paul D. Hanna, Dec 21 2011

nonn

Compare to e.g.f. F(x) of A007889: F(x) = exp(x*(1 + F(x))/2), where A007889(n) = number of intransitive (or alternating) trees: vertices are [0,n] and for no i

Related sequence: A058014(n) = number of labeled trees with n+1 nodes such that the degrees of all nodes, excluding the first node, are odd.

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 185*x^4/4! + 2441*x^5/5! +...
where log(A(x)) = x*(1 + A(x)^2)/2 and
log(A(x)) = x + 2*x^2/2! + 12*x^3/3! + 112*x^4/4! + 1440*x^5/5! + 23616*x^6/6! +...
A(x)^2 = 1 + 2*x + 8*x^2/2! + 56*x^3/3! + 576*x^4/4! + 7872*x^5/5! + 134656*x^6/6! +...

Seiichi Manyama, Table of n, a(n) for n = 0..372

Cf. A007889, A058014, A214225, A138860, A202357.

CoefficientList[Series[Sqrt[-ProductLog[-E^x*x]/x], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 10 2014 *)

a(n)=local(A=1+x); for(i=0, n, A=exp(x*(1+A^2)/2 +x*O(x^n))); n!*polcoeff(A, n)

/* Coefficients of A(x)^p are given by: */
{a(n, p=1)=(1/2^n)*sum(k=0, n, binomial(n, k)*p*(2*k+p)^(n-1))}

a(n)=n!*polcoeff(exp(sum(k=1,n,k^(k-1)*cosh(k*x +x*O(x^n))*x^k/k!) +x*O(x^n)),n)
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Nov 20 2012

E.g.f. A(x) equals the formal inverse of function 2*log(x)/(1+x^2).
E.g.f.: exp( Sum_{n>=1} n^(n-1) * cosh(n*x) * x^n / n! ). - Paul D. Hanna, Nov 20 2012
E.g.f.: exp(G(x)) where G(x) = x/(1 - tanh(G(x))) is the e.g.f. of A214225. - Paul D. Hanna, Nov 20 2012
E.g.f. satisfies: A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the e.g.f. of A058014.
a(n) = (1/2^n)*Sum_{k=0..n} C(n,k)*(2*k+1)^(n-1).
Powers of e.g.f.:
If A(x)^p = Sum_{n>=0} a(n,p)*x^n/n! then a(n,p) = (1/2^n)* Sum_{k=0..n} binomial(n,k)*p*(2*k+p)^(n-1).
a(n) ~ sqrt(1+c) * n^(n-1) / (2 * exp(n) * c^(n+1/2)), where c = LambertW(exp(-1)) = 0.278464542761... (see A202357). - Vaclav Kotesovec, Jan 10 2014
E.g.f.: sqrt(-LambertW(-x*exp(x))/x). - Vaclav Kotesovec, Jan 10 2014

A349719 E.g.f. satisfies: A(x) = exp( x * (1 + 1/A(x))/2 ).

Original entry on oeis.org

1, 1, 0, 1, -4, 26, -212, 2108, -24720, 334072, -5112544, 87396728, -1650607040, 34132685120, -767025716736, 18612106195456, -485013257865472, 13509071081429888, -400505695457942528, 12592502771190979712, -418524228123134068224

Offset: 0

Seiichi Manyama, Nov 27 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..411
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A007889, A202617, A283828, A349714, A349715, A349716, A349720, A349721.

a[n_] := (1/2^n) * Sum[If[k == n == 1, 1, (-k + 1)^(n - 1)] * Binomial[n, k], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, Nov 27 2021 *)

a(n) = sum(k=0, n, (-k+1)^(n-1)*binomial(n, k))/2^n;

my(N=40, x='x+O('x^N)); Vec(serlaplace((x/2)/lambertw(x/2*exp(-x/2))))

my(N=40, x='x+O('x^N)); Vec(2*sum(k=0, N, (-k+1)^(k-1)*x^k/(2-(-k+1)*x)^(k+1)))

a(n) = (1/2^n) * Sum_{k=0..n} (-k+1)^(n-1) * binomial(n,k).
E.g.f.: (x/2)/LambertW( x/2 * exp(-x/2) ).
G.f.: 2 * Sum_{k>=0} (-k+1)^(k-1) * x^k/(2 - (-k+1)*x)^(k+1).
a(n) ~ -(-1)^n * sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (2^n * exp(n) * LambertW(exp(-1))^(n-1)). - Vaclav Kotesovec, Dec 05 2021

A138860 E.g.f. satisfies: A(x) = exp( x*(A(x) + A(x)^2)/2 ).

Original entry on oeis.org

1, 1, 4, 31, 364, 5766, 115300, 2788724, 79197040, 2583928360, 95256535936, 3916137470664, 177651980724160, 8815348234689920, 474993826614917632, 27619367979975064576, 1723821221240101984000, 114948301218300412117632

Offset: 0

Paul D. Hanna, Apr 01 2008, Apr 02 2008, Apr 03 2008

nonn

The related sequence A007889 enumerates the number of intransitive (or alternating) trees.

a(n+1) is the number of incomplete ternary trees on n labeled vertices in which each left child has a larger label than its parent and each middle child has a smaller label than its parent. - Brian Drake, Jul 28 2008

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 31*x^3/3! + 364*x^4/4! + 5766*x^5/5! + ...

Seiichi Manyama, Table of n, a(n) for n = 0..363

Cf. A007889, A088789, A058014, A036778, A138903.

Cf. A138764.

Table[1/2^n * Sum[Binomial[n,k]*(n+k+1)^(n-1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 15 2013 *)

a(n)=(1/2^n)*sum(k=0,n,binomial(n,k)*(n+k+1)^(n-1))

/* Series Reversion: */
a(n)=local(X=x+x*O(x^n));n!*polcoeff(exp(serreverse(2*x/(exp(X)+exp(2*X)) )),n)

/* Coefficients of A(x)^p are given by: */
{a(n,p=1)=(1/2^n)*sum(k=0,n,binomial(n,k)*p*(n+k+p)^(n-1))}

a(n) = (1/2^n)* Sum_{k=0..n} binomial(n,k)*(n+k+1)^(n-1) - Vladeta Jovovic, Mar 31 2008.
E.g.f. satisfies: A( 2*x/( exp(x) + exp(2*x) ) ) = exp(x).
E.g.f.: A(x) = inverse function of 2*log(x)/(x + x^2).
E.g.f.: A(x) = exp( Series_Reversion[ 2*x/(exp(x) + exp(2*x)) ] ).
E.g.f.: A(x) = G(x/2) where G(x) = e.g.f. of A138764.
More generally, if A(x) = Sum_{n>=0} a(n)*x^n/n! = exp( x*[A(x) + A(x)^m]/2 ) then a(n) = (1/2^n)* Sum_{k=0..n} binomial(n,k)*(n+(m-1)*k+1)^(n-1) and if B(x) = Sum_{n>=0} b(n)*x^n/n! = log(A(x)) then b(n) = (1/2^n)* Sum_{k=0..n} binomial(n,k)*(n+(m-1)*k)^(n-1). - Paul D. Hanna and Vladeta Jovovic, Apr 02 2008
Powers of e.g.f.: If A(x)^p = Sum_{n>=0} a(n,p)*x^n/n! then
. a(n,p) = (1/2^n)* Sum_{k=0..n} binomial(n,k)*p*(n+k+p)^(n-1).
Given e.g.f. A(x), let B(x) = e.g.f. of A007889, then
. A(x) = B(x*A(x)) = (1/x)*Series_Reversion(x/B(x)) and
. B(x) = A(x/B(x)) = x/Series_Reversion(x*A(x)).
a(n) ~ n^(n-1)*(1+r)^n*r^(n+1)/(sqrt(1+3*r)*(1-r)^(2*n+1)*exp(n)*2^n), where r = 0.6472709258412625... is the root of the equation (r/(1-r))^(1+r) = e. - Vaclav Kotesovec, Jun 15 2013

A349714 E.g.f. satisfies: A(x) = exp( x * (1 + A(x)^3)/2 ).

Original entry on oeis.org

1, 1, 4, 37, 532, 10426, 259300, 7823908, 277713904, 11339452792, 523621438336, 26982030104536, 1534947906550528, 95550736737542464, 6460746383585984512, 471533064029919744256, 36946948091091750496000, 3093472887944746070621056

Offset: 0

Seiichi Manyama, Nov 26 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..352
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A007889, A202617, A349715, A349716, A349719, A349720, A349721.

a[n_] := (1/2^n) * Sum[(3*k + 1)^(n - 1) * Binomial[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Nov 27 2021 *)

a(n) = sum(k=0, n, (3*k+1)^(n-1)*binomial(n, k))/2^n;

my(N=20, x='x+O('x^N)); Vec(serlaplace((-lambertw(-3*x/2*exp(3*x/2))/(3*x/2))^(1/3)))

my(N=20, x='x+O('x^N)); Vec(2*sum(k=0, N, (3*k+1)^(k-1)*x^k/(2-(3*k+1)*x)^(k+1)))

a(n) = (1/2^n) * Sum_{k=0..n} (3*k+1)^(n-1) * binomial(n,k).
E.g.f.: ( -LambertW( -3*x/2 * exp(3*x/2) )/(3*x/2) )^(1/3).
G.f.: 2 * Sum_{k>=0} (3*k+1)^(k-1) * x^k/(2 - (3*k+1)*x)^(k+1).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 3^(n-1) * n^(n-1) / (LambertW(exp(-1))^(n + 1/3) * 2^n * exp(n)). - Vaclav Kotesovec, Nov 26 2021

A349716 E.g.f. satisfies: A(x) = exp( x * (1 + A(x)^5)/2 ).

Original entry on oeis.org

1, 1, 6, 91, 2156, 69926, 2884576, 144555356, 8529135216, 579220982056, 44503081624976, 3816776859516776, 361462121953291456, 37464997600663289216, 4218485281787859411456, 512762346462142021355776, 66919363061333997572830976, 9332997074366800051673277056

Offset: 0

Seiichi Manyama, Nov 26 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..330
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A007889, A202617, A349714, A349715, A349719, A349720, A349721.

a[n_] := (1/2^n) * Sum[(5*k + 1)^(n - 1) * Binomial[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Nov 27 2021 *)

a(n) = sum(k=0, n, (5*k+1)^(n-1)*binomial(n, k))/2^n;

my(N=20, x='x+O('x^N)); Vec(serlaplace((-lambertw(-5*x/2*exp(5*x/2))/(5*x/2))^(1/5)))

my(N=20, x='x+O('x^N)); Vec(2*sum(k=0, N, (5*k+1)^(k-1)*x^k/(2-(5*k+1)*x)^(k+1)))

a(n) = (1/2^n) * Sum_{k=0..n} (5*k+1)^(n-1) * binomial(n,k).
E.g.f.: ( -LambertW( -5*x/2 * exp(5*x/2) )/(5*x/2) )^(1/5).
G.f.: 2 * Sum_{k>=0} (5*k+1)^(k-1) * x^k/(2 - (5*k+1)*x)^(k+1).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 5^(n-1) * n^(n-1) / (LambertW(exp(-1))^(n + 1/5) * 2^n * exp(n)). - Vaclav Kotesovec, Nov 26 2021

A349720 E.g.f. satisfies: A(x) = exp( x * (1 + 1/A(x)^2)/2 ).

Original entry on oeis.org

1, 1, -1, 7, -63, 801, -13025, 258343, -6048511, 163276417, -4992740289, 170571634311, -6439161507647, 266180947507489, -11958385377911713, 580151397382158631, -30227616424300542975, 1683438461080186841601, -99796591057813372007297

Offset: 0

Seiichi Manyama, Nov 27 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..372
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A007889, A202617, A349714, A349715, A349716, A349719, A349721.

a[n_] := (1/2^n) * Sum[(-2*k + 1)^(n - 1) * Binomial[n, k], {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, Nov 27 2021 *)

a(n) = sum(k=0, n, (-2*k+1)^(n-1)*binomial(n, k))/2^n;

my(N=20, x='x+O('x^N)); Vec(serlaplace((x/lambertw(x*exp(-x)))^(1/2)))

my(N=20, x='x+O('x^N)); Vec(2*sum(k=0, N, (-2*k+1)^(k-1)*x^k/(2-(-2*k+1)*x)^(k+1)))

a(n) = (1/2^n) * Sum_{k=0..n} (-2*k+1)^(n-1) * binomial(n,k).
E.g.f.: ( x/LambertW( x * exp(-x) ) )^(1/2).
G.f.: 2 * Sum_{k>=0} (-2*k+1)^(k-1) * x^k/(2 - (-2*k+1)*x)^(k+1).
a(n) ~ -(-1)^n * sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (2 * exp(n) * LambertW(exp(-1))^(n - 1/2)). - Vaclav Kotesovec, Dec 05 2021

A349715 E.g.f. satisfies: A(x) = exp( x * (1 + A(x)^4)/2 ).

Original entry on oeis.org

1, 1, 5, 61, 1161, 30201, 998413, 40077493, 1893550865, 102951388657, 6331847746581, 434653328279853, 32944254978940825, 2732662648183661545, 246228744062320481309, 23949858491053731087781, 2501088964314980938821153, 279111248034686114681365473

Offset: 0

Seiichi Manyama, Nov 26 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..339
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A007889, A202617, A349714, A349716, A349719, A349720, A349721.

a[n_] := (1/2^n) * Sum[(4*k + 1)^(n - 1) * Binomial[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Nov 27 2021 *)

a(n) = sum(k=0, n, (4*k+1)^(n-1)*binomial(n, k))/2^n;

my(N=20, x='x+O('x^N)); Vec(serlaplace((-lambertw(-2*x*exp(2*x))/(2*x))^(1/4)))

my(N=20, x='x+O('x^N)); Vec(2*sum(k=0, N, (4*k+1)^(k-1)*x^k/(2-(4*k+1)*x)^(k+1)))

a(n) = (1/2^n) * Sum_{k=0..n} (4*k+1)^(n-1) * binomial(n,k).
E.g.f.: ( -LambertW( -2*x * exp(2*x) )/(2*x) )^(1/4).
G.f.: 2 * Sum_{k>=0} (4*k+1)^(k-1) * x^k/(2 - (4*k+1)*x)^(k+1).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 2^(n-2) * n^(n-1) / (LambertW(exp(-1))^(n + 1/4) * exp(n)). - Vaclav Kotesovec, Nov 26 2021

A349721 E.g.f. satisfies: A(x) = exp( x * (1 + 1/A(x)^3)/2 ).

Original entry on oeis.org

1, 1, -2, 19, -260, 4966, -121328, 3613996, -127035920, 5147600680, -236245559984, 12112405259560, -686148484748480, 42560312499982720, -2868921992458611200, 208828244778853125376, -16324500711130356582656, 1363986660232205656646272

Offset: 0

Seiichi Manyama, Nov 27 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..352
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A007889, A202617, A349714, A349715, A349716, A349719, A349720.

a[n_] := (1/2^n) * Sum[(-3*k + 1)^(n - 1) * Binomial[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Nov 27 2021 *)

a(n) = sum(k=0, n, (-3*k+1)^(n-1)*binomial(n, k))/2^n;

my(N=20, x='x+O('x^N)); Vec(serlaplace(((3*x/2)/lambertw(3*x/2*exp(-3*x/2)))^(1/3)))

my(N=20, x='x+O('x^N)); Vec(2*sum(k=0, N, (-3*k+1)^(k-1)*x^k/(2-(-3*k+1)*x)^(k+1)))

a(n) = (1/2^n) * Sum_{k=0..n} (-3*k+1)^(n-1) * binomial(n,k).
E.g.f.: ( (3*x/2)/LambertW( 3*x/2 * exp(-3*x/2) ) )^(1/3).
G.f.: 2 * Sum_{k>=0} (-3*k+1)^(k-1) * x^k/(2 - (-3*k+1)*x)^(k+1).
a(n) ~ -(-1)^n * sqrt(1 + LambertW(exp(-1))) * 3^(n-1) * n^(n-1) / (2^n * exp(n) * LambertW(exp(-1))^(n - 1/3)). - Vaclav Kotesovec, Dec 05 2021

A100526 Number of local binary search trees (i.e., labeled binary trees such that every left child has a smaller label than its parent and every right child has a larger label than its parent) with n vertices such that the root has only one child.

Original entry on oeis.org

1, 2, 6, 28, 180, 1476, 14728, 173216, 2346480, 35981200, 616111056, 11652662880, 241259095168, 5427319729664, 131818482923520, 3437894427590656, 95825936705566464, 2842834581982211328, 89435890422890433280, 2974081497762693670400, 104234511362034627442176

Offset: 1

Emeric Deutsch, Nov 24 2004

nonn

			a(3)=6 because we have 3L2L1, 2L1R3, 3L1R2, 1R2R3, 1R3L2, 2R3L1 (Li means left child labeled i, RI means right child labeled i).
E.g.f.: A(x) = x + 2*x^2/2! + 6*x^3/3! + 28*x^4/4! + 180*x^5/5! + 1476*x^6/6! +...
Given G(x) such that G( sqrt( G(x^2*exp(-x)) ) ) = x, where
G(x) = x + 1/2*x^2 + 1/8*x^3 + 1/12*x^4 + 77/384*x^5 + 23/120*x^6 + 2077/46080*x^7 + 179/5040*x^8 + 239525/2064384*x^9 +...+ A273952(n)*x^n/(2^(n-1)*(n-1)!) +...
then A(x) = G( sqrt( G(x^2*exp(x)) ) ). - _Paul D. Hanna_, Jun 06 2016

G. C. Greubel, Table of n, a(n) for n = 1..400
C. Chauve, S. Dulucq and A. Rechnitzer, Enumerating alternating trees, J. Combin. Theory Ser. A 94 (2001), 142-151.
A. Postnikov, Intransitive Trees, J. Combin. Theory Ser. A 79 (1997), 360-366.

Cf. A007889, A273952.

[2^(1-n)*(&+[ k^(n-1)*Binomial(n, k): k in [1..n]]): n in [1..40]]; // G. C. Greubel, Mar 27 2023

seq((1/2^(n-1))*add(k^(n-1)*binomial(n,k),k=1..n),n=1..22);

Rest[CoefficientList[Series[-2*LambertW[-x*E^(x/2)/2], {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jun 23 2016 *)

/* Egf A(x) = G(sqrt(G(x^2*exp(x)))) if G(sqrt(G(x^2*exp(-x)))) = x */
{a(n) = my(G=x); for(i=1,n, G = serreverse( sqrt( subst(G,x, x^2*exp(-x +O(x^n))) ) )); A = subst(G,x,sqrt(subst(G,x,x^2*exp(x +O(x^n))))); n!*polcoeff(A,n)}
for(n=1,30,print1(a(n),", ")) \\ Paul D. Hanna, Jun 06 2016

def A100526(n): return 2^(1-n)*sum( k^(n-1)*binomial(n, k) for k in range(1,n+1) )
[A100526(n) for n in range(1,40)] # G. C. Greubel, Mar 27 2023

a(n) = (1/2^(n-1))*Sum_{k=1..n} k^(n-1)*binomial(n, k).
a(n) = n*A007889(n-1).
E.g.f.: -2*LambertW(-x*exp(x/2)/2). - Paul D. Hanna, Jun 07 2016, after Vladeta Jovovic's formula in A038049
E.g.f.: G( sqrt( G(x^2*exp(x)) ) ), where G( sqrt( G(x^2*exp(-x)) ) ) = x, and G(x) is the e.g.f. of A273952. - Paul D. Hanna, Jun 06 2016
a(n) ~ sqrt(1 + LambertW(exp(-1)))*n^(n-1) / (2^(n-1) * exp(n) * LambertW(exp(-1))^n). - Vaclav Kotesovec, Jun 23 2016

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cp's OEIS Frontend

A349562 Number of labeled rooted forests with 2-colored leaves.

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A202617 E.g.f. satisfies: A(x) = exp( x*(1 + A(x)^2)/2 ).

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A349719 E.g.f. satisfies: A(x) = exp( x * (1 + 1/A(x))/2 ).

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A138860 E.g.f. satisfies: A(x) = exp( x*(A(x) + A(x)^2)/2 ).

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A349714 E.g.f. satisfies: A(x) = exp( x * (1 + A(x)^3)/2 ).

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A349716 E.g.f. satisfies: A(x) = exp( x * (1 + A(x)^5)/2 ).

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A349720 E.g.f. satisfies: A(x) = exp( x * (1 + 1/A(x)^2)/2 ).

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