A249940 -id:A249940

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A249939

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

+10
10

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

a(n) = 4*A242858(2*n) for n>0.

a(n) = A249940(n)/3.

a(n) == 4 (mod 96) for n>0.

LINKS

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

FORMULA

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.

EXAMPLE

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 +...

PROG

(PARI) /* 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), ", "))

(PARI) /* 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), ", "))

(PARI) /* 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), ", "))

CROSSREFS

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

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

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 19 2014

STATUS

approved

A247082

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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.

LINKS

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

FORMULA

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

EXAMPLE

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 +...

MATHEMATICA

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 *)

PROG

(PARI) /* 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), ", "))

(PARI) /* 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), ", "))

(PARI) /* 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.)))

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 28 2014

STATUS

approved

A249941

E.g.f.: Sum_{n>=0} exp(n^3*x) / 2^(n+1).

+10
4

1, 13, 4683, 7087261, 28091567595, 230283190977853, 3385534663256845323, 81124824998504073881821, 2958279121074145472650648875, 155897763918621623249276226253693, 11403568794011880483742464196184901963, 1120959742203056268267494209293006882589981

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Number of ordered partitions of 3*n.

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..141

FORMULA

a(n) = Sum_{k=0..3*n} k! * Stirling2(3*n, k) for n>=0.

a(n) = Sum_{k=1..[(3*n+1)/2]} (2*k-1)! * Stirling2(3*n+1, 2*k) for n>0 with a(0)=1.

a(n) = A000670(3*n), where A000670 is the Fubini numbers.

a(n) ~ (3*n)! / (2 * (log(2))^(3*n+1)). - Vaclav Kotesovec, May 04 2015

a(n) = Sum_{k>=0} k^(3*n) / 2^(k + 1). - Ilya Gutkovskiy, Dec 19 2019

EXAMPLE

E.g.f.: A(x) = 1 + 13*x + 4683*x^2/2! + 7087261*x^3/3! + 28091567595*x^4/4! +...

where the e.g.f. equals the infinite series:

A(x) = 1/2 + exp(x)/2^2 + exp(8*x)/2^3 + exp(27*x)/2^4 + exp(64*x)/2^5 + exp(125*x)/2^6 + exp(216*x)/2^7 + exp(343*x)/2^8 +...

MATHEMATICA

Table[Sum[k! * StirlingS2[3*n, k], {k, 0, 3*n}], {n, 0, 20}] (* Vaclav Kotesovec, May 04 2015 *)

Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)*(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; a[n_] := Fubini[3n, 1]; a[0] = 1; Table[a[n], {n, 0, 11}] (* Jean-François Alcover, Mar 30 2016 *)

PROG

(PARI) /* E.g.f.: Sum_{n>=0} exp(n^3*x)/2^(n+1) */

\p100 \\ set precision

{a(n) = round( n!*polcoeff(sum(m=0, 600, exp(m^3*x +x*O(x^n))/2^(m+1)*1.), n) )}

for(n=0, 20, print1(a(n), ", "))

(PARI) /* Formula for a(n): */

{a(n) = sum(k=0, 3*n, k! * stirling(3*n, k, 2))}

for(n=0, 20, print1(a(n), ", "))

(PARI) /* Formula for a(n): */

{a(n) = if(n==0, 1, sum(k=1, (3*n+1)\2, (2*k-1)! * stirling(3*n+1, 2*k, 2)))}

for(n=0, 20, print1(a(n), ", "))

CROSSREFS

Cf. A000670, A068942, A249940.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 19 2014

STATUS

approved

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