A249938 -id:A249938 - cp's OEIS frontend

A249939 E.g.f.: 1/(5 - 4*cosh(x)).

1, 4, 100, 6244, 727780, 136330084, 37455423460, 14188457293924, 7087539575975140, 4514046217675793764, 3570250394992512270820, 3433125893070920512725604, 3944372161432193963534198500, 5336301013125557989981503385444, 8396749419933421378024498580446180

Offset: 0

Paul D. Hanna, Nov 19 2014

nonn

a(n) = 4*A242858(2*n) for n>0.
a(n) = A249940(n)/3.
a(n) == 4 (mod 96) for n>0.

			E.g.f.: E(x) = 1 + 4*x^2/2! + 100*x^4/4! + 6244*x^6/6! + 727780*x^8/8! +...
where E(x) = 1/(5 - 4*cosh(x)) = -exp(x) / (2 - 5*exp(x) + 2*exp(2*x)).
ALTERNATE GENERATING FUNCTION.
E.g.f.: A(x) = 1 + 4*x + 100*x^2/2! + 6244*x^3/3! + 727780*x^4/4! +...
where 3*A(x) = 1 + 2*exp(x)/2 + 2*exp(4*x)/2^2 + 2*exp(9*x)/2^3 + 2*exp(16*x)/2^4 + 2*exp(25*x)/2^5 + 2*exp(36*x)/2^6 + 2*exp(49*x)/2^7 +...

Seiichi Manyama, Table of n, a(n) for n = 0..200

Cf. A249938, A249940, A247082, A250914, A250915, A242858.

Cf. A210676, A210657, A028296, A094088, A210672, A210674.

/* E.g.f.: 1/(5 - 4*cosh(x)) */
{a(n) = local(X=x+O(x^(2*n+1))); (2*n)!*polcoeff( 1/(5 - 4*cosh(X)), 2*n)}
for(n=0, 20, print1(a(n), ", "))

/* Formula for a(n): */
{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = if(n==0, 1, sum(k=1, (2*n+1)\3, 2*(3*k-1)! * Stirling2(2*n+1, 3*k)))}
for(n=0, 20, print1(a(n), ", "))

/* Formula for a(n): */
{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = if(n==0, 1, (4/3)*sum(k=0, 2*n, k! * Stirling2(2*n, k) ))}
for(n=0, 20, print1(a(n), ", "))

E.g.f.: 1/3 + (2/3)*Sum_{n>=1} exp(n^2*x) / 2^n  =  Sum_{n>=0} a(n)*x^n/n!.
a(n) = (4/3) * Sum_{k=0..2*n} k! * Stirling2(2*n, k) for n>0 with a(0)=1.
a(n) = Sum_{k=1..[(2*n+1)/3]} 2 * (3*k-1)! * Stirling2(2*n+1, 3*k) for n>0 with a(0)=3, after Vladimir Kruchinin in A242858.

A247082 E.g.f.: (8 - 7cosh(x)) / (13 - 12cosh(x)).

Original entry on oeis.org

1, 5, 365, 66605, 22687565, 12420052205, 9972186170765, 11039636939221805, 16116066766061589965, 29996702068513925975405, 69334618695849722499185165, 194843145588759580915489113005, 654210085817395711127396030796365, 2586566313303319454399746941903834605, 11894287668430209899882926599828701863565

Offset: 0

Paul D. Hanna, Nov 28 2014

nonn

The number of 3-level labeled linear rooted trees with 2*n leaves.
A bisection of A050351.
a(n) == 5 (mod 360) for n>0.

			E.g.f.: E(x) = 1 + 5*x^2/2! + 365*x^4/4! + 66605*x^6/6! + 22687565*x^8/8! +...
where E(x) = (8 - 7*cosh(x)) / (13 - 12*cosh(x)), or, equivalently,
E(x) = (7 - 16*exp(x) + 7*exp(2*x)) / (12 - 26*exp(x) + 12*exp(2*x)).
ALTERNATE GENERATING FUNCTION.
E.g.f.: A(x) = 1 + 5*x + 365*x^2/2! + 66605*x^3/3! + 22687565*x^4/4! +...
where
6*A(x) = 4 + exp(x)*(2/3) + exp(4*x)*(2/3)^2 + exp(9*x)*(2/3)^3 + exp(16*x)*(2/3)^4 + exp(25*x)*(2/3)^5 + exp(36*x)*(2/3)^6 + exp(49*x)*(2/3)^7 +...

Seiichi Manyama, Table of n, a(n) for n = 0..195

Cf. A249938, A249939, A249940, A250914, A250915, A050351.

nmax=20; Table[(CoefficientList[Series[(8-7*Cosh[x]) / (13-12*Cosh[x]), {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[n]],{n,1,2*nmax+2,2}] (* Vaclav Kotesovec, Nov 29 2014 *)

/* E.g.f.: (8 - 7*cosh(x)) / (13 - 12*cosh(x)): */
{a(n) = local(X=x+O(x^(2*n+1))); (2*n)!*polcoeff( (8 - 7*cosh(X)) / (13 - 12*cosh(X)) , 2*n)}
for(n=0, 20, print1(a(n), ", "))

/* Formula for a(n): */
{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = if(n==0, 1, sum(k=0, 2*n, 2^(k-1) * k! * Stirling2(2*n, k) ))}
for(n=0, 20, print1(a(n), ", "))

/* As the Sum of an Infinite Series: */
\p60 \\ set precision
Vec(serlaplace(1/2+1/6*sum(n=0,2000,exp(n^2*x)*(2/3)^n*1.)))

E.g.f.: 1/2 + (1/6)*Sum_{n>=0} exp(n^2*x) * (2/3)^n  =  Sum_{n>=0} a(n)*x^n/n!.
a(n) = Sum_{k=0..2*n} 2^(k-1) * k! * Stirling2(2*n, k) for n>0 with a(0)=1. [After Vladeta Jovovic in A050351]
a(n) ~ (2*n)! / (6 * (log(3/2))^(2*n+1)). - Vaclav Kotesovec, Nov 29 2014

A250914 E.g.f.: (18 - 17cosh(x)) / (25 - 24cosh(x)).

Original entry on oeis.org

1, 7, 1015, 367927, 248956855, 270732878647, 431806658432695, 949587798053709367, 2753726282896986372535, 10181613308681289633868087, 46749244630988859672950920375, 260970234691672017384493753162807, 1740621952318191255997909826897420215, 13670746044282245244660044262911331401527

Offset: 0

Paul D. Hanna, Nov 28 2014

nonn

The number of 4-level labeled linear rooted trees with 2*n leaves.
A bisection of A050352.
a(n) == 7 (mod 1008) for n>0.

			E.g.f.: E(x) = 1 + 7*x^2/2! + 1015*x^4/4! + 367927*x^6/6! + 248956855*x^8/8! +...
where E(x) = (18 - 17*cosh(x)) / (25 - 24*cosh(x)).
ALTERNATE GENERATING FUNCTION.
E.g.f.: A(x) = 1 + 7*x + 1015*x^2/2! + 367927*x^3/3! + 248956855*x^4/4! +...
where
12*A(x) = 9 + exp(x)*(3/4) + exp(4*x)*(3/4)^2 + exp(9*x)*(3/4)^3 + exp(16*x)*(3/4)^4 + exp(25*x)*(3/4)^5 + exp(36*x)*(3/4)^6 +...

Seiichi Manyama, Table of n, a(n) for n = 0..186

Cf. A249938, A249939, A247082, A250915, A050352.

nmax=20; Table[(CoefficientList[Series[(18-17*Cosh[x]) / (25-24*Cosh[x]), {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[n]],{n,1,2*nmax+2,2}] (* Vaclav Kotesovec, Nov 29 2014 *)

/* E.g.f.: (18 - 17*cosh(x)) / (25 - 24*cosh(x)): */
{a(n) = local(X=x+O(x^(2*n+1))); (2*n)!*polcoeff( (18 - 17*cosh(X)) / (25 - 24*cosh(X)) , 2*n)}
for(n=0, 20, print1(a(n), ", "))

/* Formula for a(n): */
{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = if(n==0, 1, sum(k=0, 2*n, 3^(k-1) * k! * Stirling2(2*n, k) ))}
for(n=0, 20, print1(a(n), ", "))

/* As the Sum of an Infinite Series: */
\p60 \\ set precision
Vec(serlaplace(2/3 + 1/12*sum(n=0,2000,exp(n^2*x)*(3/4)^n*1.)))

E.g.f.: 2/3 + (1/12)*Sum_{n>=0} exp(n^2*x) * (3/4)^n  =  Sum_{n>=0} a(n)*x^n/n!.
a(n) = Sum_{k=0..2*n} 3^(k-1) * k! * Stirling2(2*n, k) for n>0 with a(0)=1.
a(n) ~ (2*n)! / (12 * (log(4/3))^(2*n+1)). - Vaclav Kotesovec, Nov 29 2014

A250915 E.g.f.: (32 - 31cosh(x)) / (41 - 40cosh(x)).

Original entry on oeis.org

1, 9, 2169, 1306809, 1469709369, 2656472295609, 7042235448544569, 25740278881968596409, 124066865052334175027769, 762445058190042799428289209, 5818666543923901596429593478969, 53987940899344324456042542132654009, 598504142090716188282023260396781018169

Offset: 0

Paul D. Hanna, Nov 28 2014

nonn

The number of 5-level labeled linear rooted trees with 2*n leaves.
A bisection of A050353.
a(n) == 9 (mod 2160) for n>0.

			E.g.f.: E(x) = 1 + 9*x^2/2! + 2169*x^4/4! + 1306809*x^6/6! + 1469709369*x^8/8! +...
where E(x) = (32 - 31*cosh(x)) / (41 - 40*cosh(x)).
ALTERNATE GENERATING FUNCTION.
E.g.f.: A(x) = 1 + 9*x + 2169*x^2/2! + 1306809*x^3/3! + 1469709369*x^4/4! +...
where
20*A(x) = 16 + exp(x)*(4/5) + exp(4*x)*(4/5)^2 + exp(9*x)*(4/5)^3 + exp(16*x)*(4/5)^4 + exp(25*x)*(4/5)^5 + exp(36*x)*(4/5)^6 +...

Seiichi Manyama, Table of n, a(n) for n = 0..180

Cf. A249938, A249939, A247082, A250914, A050353.

nmax=20; Table[(CoefficientList[Series[(32-31*Cosh[x]) / (41-40*Cosh[x]), {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[n]],{n,1,2*nmax+2,2}] (* Vaclav Kotesovec, Nov 29 2014 *)

/* E.g.f.: (32 - 31*cosh(x)) / (41 - 40*cosh(x)): */
{a(n) = local(X=x+O(x^(2*n+1))); (2*n)!*polcoeff( (32 - 31*cosh(X)) / (41 - 40*cosh(X)) , 2*n)}
for(n=0, 20, print1(a(n), ", "))

/* Formula for a(n): */
{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = if(n==0, 1, sum(k=0, 2*n, 4^(k-1) * k! * Stirling2(2*n, k) ))}
for(n=0, 20, print1(a(n), ", "))

/* As the Sum of an Infinite Series: */
\p100 \\ set precision
Vec(serlaplace(3/4 + 1/20*sum(n=0,3000,exp(n^2*x)*(4/5)^n*1.)))

E.g.f.: 3/4 + (1/20)*Sum_{n>=0} exp(n^2*x) * (4/5)^n  =  Sum_{n>=0} a(n)*x^n/n!.
a(n) = Sum_{k=0..2*n} 4^(k-1) * k! * Stirling2(2*n, k) for n>0 with a(0)=1.
a(n) ~ (2*n)! / (20 * (log(5/4))^(2*n+1)). - Vaclav Kotesovec, Nov 29 2014

A068942 a(n) = Bo(n^2), n=0,1..., where Bo(n) are the ordered Bell numbers, A000670.

Original entry on oeis.org

1, 1, 75, 7087261, 5315654681981355, 106697365438475775825583498141, 144199280951655469628360978109406917583513090155, 27656793065414932606012896651489726461435178241015434306518713649426461

Offset: 0

Karol A. Penson, Mar 09 2002

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..20

Cf. A000670, A068939, A249938, A249941.

a[n_] := PolyLog[-n^2, 1/2]/2; a[0] = 1; Table[a[n], {n, 0, 7}] (* Jean-François Alcover, Mar 30 2016 *)
Table[Sum[k!*StirlingS2[n^2, k], {k, 0, n^2}], {n, 0, 10}] (* Vaclav Kotesovec, Jun 08 2021 *)

a(n) = sum(k=0, n^2, k!*stirling(n^2, k, 2)); \\ Seiichi Manyama, Jan 17 2022

a(n) = Sum_{k>=1} (k^(n^2))/2^(k+1); this is the analog of the Dobinski formula.
Integral representation as n-th moment of a positive function on a positive half-axis, in Maple notation: a(n)=int(x^n*(sum(exp(-ln(x)^2/(4*ln(k))) / (2^k*sqrt(ln(k))), k=2..infinity)/(4*sqrt(Pi)*x)+Dirac(x-1)/4), x=0..infinity).
a(n) ~ (n^2)! / (2 * log(2)^(n^2 + 1)). - Vaclav Kotesovec, Jun 08 2021

A340837 a(n) = (1/2) * Sum_{k>=0} (k*(k - 1))^n / 2^k.

Original entry on oeis.org

1, 2, 52, 3272, 382672, 71819552, 19755648832, 7489898916992, 3743721038908672, 2385494267756237312, 1887436919680269939712, 1815491288416066631616512, 2086364959404184854563049472, 2823211429546048668686123343872, 4443155724532239407325655263035392

Offset: 0

Ilya Gutkovskiy, Jan 23 2021

nonn

Cf. A000670, A020556, A052841, A080163, A098696, A122101, A249938.

Table[(1/2) Sum[(k (k - 1))^n/2^k, {k, 0, Infinity}], {n, 0, 14}]
Table[(1/2) Sum[(-1)^k Binomial[n, k] HurwitzLerchPhi[1/2, k - 2 n, 0], {k, 0, n}], {n, 0, 14}]

a(n) = Sum_{k=0..n} (-1)^k * binomial(n,k) * A000670(2*n-k).
a(n) = 2 * A080163(n) for n > 0. - Hugo Pfoertner, Jan 23 2021
a(n) = A122101(2*n,n). - Alois P. Heinz, Jun 23 2023

A308865 a(n) = Sum_{k>=0} k^(2*n+1)/2^(k+1).

Original entry on oeis.org

1, 13, 541, 47293, 7087261, 1622632573, 526858348381, 230283190977853, 130370767029135901, 92801587319328411133, 81124824998504073881821, 85438451336745709294580413, 106697365438475775825583498141, 155897763918621623249276226253693, 263478385263023690020893329044576861

Offset: 0

Ilya Gutkovskiy, Jun 29 2019

nonn

Cf. A000670, A008277, A099977, A249938.

Table[Sum[k^(2 n + 1)/2^(k + 1), {k, 0, Infinity}], {n, 0, 14}]
Table[Sum[k! StirlingS2[2 n + 1, k], {k, 0, 2 n + 1}], {n, 0, 14}]

a(n) = Sum_{k=0..2*n+1} k!*Stirling2(2*n+1,k).
a(n) = A000670(2*n+1).
a(n) ~ sqrt(Pi) * 2^(2*n + 1) * n^(2*n + 3/2) / (exp(2*n) * (log(2))^(2*n + 2)). - Vaclav Kotesovec, Sep 25 2019

A340838 a(n) = (1/2) * Sum_{k>=0} (k*(k + n))^n / 2^k.

Original entry on oeis.org

1, 4, 139, 11928, 1909787, 491329088, 185373016419, 96425597012608, 66139668570414571, 57840395870803141632, 62813828698519808489915, 82933938539372018962724864, 130828514220436815006398809563, 243020960809424084526916839817216, 525038425527430196237626528753654867

Offset: 0

Ilya Gutkovskiy, Jan 23 2021

nonn

Cf. A000670, A098696, A249938, A292916, A340822.

Table[(1/2) Sum[(k (k + n))^n/2^k, {k, 0, Infinity}], {n, 0, 14}]
Join[{1}, Table[(1/2) Sum[Binomial[n, k] HurwitzLerchPhi[1/2, k - 2 n, 0] n^k, {k, 0, n}], {n, 1, 14}]]

a(n) = Sum_{k=0..n} binomial(n,k) * A000670(2*n-k) * n^k.

cp's OEIS Frontend

A249939 E.g.f.: 1/(5 - 4*cosh(x)).

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A247082 E.g.f.: (8 - 7*cosh(x)) / (13 - 12*cosh(x)).

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A250914 E.g.f.: (18 - 17*cosh(x)) / (25 - 24*cosh(x)).

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A250915 E.g.f.: (32 - 31*cosh(x)) / (41 - 40*cosh(x)).

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A068942 a(n) = Bo(n^2), n=0,1..., where Bo(n) are the ordered Bell numbers, A000670.

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A340837 a(n) = (1/2) * Sum_{k>=0} (k*(k - 1))^n / 2^k.

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A308865 a(n) = Sum_{k>=0} k^(2*n+1)/2^(k+1).

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A247082 E.g.f.: (8 - 7cosh(x)) / (13 - 12cosh(x)).

A250914 E.g.f.: (18 - 17cosh(x)) / (25 - 24cosh(x)).

A250915 E.g.f.: (32 - 31cosh(x)) / (41 - 40cosh(x)).