A014307 -id:A014307 - cp's OEIS frontend

A259852 Numerators of the terms of Lehmer's series S_2(2), where S_k(x) = Sum_{n>=1} n^kx^n/binomial(2n, n).

1, 8, 18, 128, 200, 192, 784, 8192, 10368, 25600, 30976, 147456, 173056, 401408, 10240, 8388608, 9469952, 7077888, 23658496, 20971520, 38535168, 253755392, 277348352, 268435456, 2621440000, 5670699008, 6115295232, 3758096384, 28219277312, 60397977600

Offset: 1

Jean-François Alcover, Jul 07 2015

			1/1, 8/3, 18/5, 128/35, 200/63, 192/77, 784/429, ... = A259852/A259853.

F. J. Dyson, N. E. Frankel, and M. L. Glasser, Lehmer's Interesting Series, arXiv:1009.4274 [math-ph], 2010-2011.
F. J. Dyson, N. E. Frankel, and M. L. Glasser, Lehmer's interesting series, Amer. Math. Monthly, 120 (2013), 116-130.
D. H. Lehmer, Interesting series involving the central binomial coefficient, Amer. Math. Monthly, 92(7) (1985), 449-457.

Cf. A014307, A180875, A259853 (denominators).

Table[2^n*n^2/Binomial[2*n, n] // Numerator, {n, 1, 40}]

vector(40, n, numerator(n^2*2^n/binomial(2*n,n))) \\ Michel Marcus, Jul 07 2015

a(n) = numerator(n^2*2^n/C(2*n,n)).

S_2(2) = Sum_{n>=1} 2^n*n^2/binomial(2*n, n) = 3F2([2,2,2]; [1,3/2]; 1/2) = 11 + 7*Pi/2. [Corrected by Petros Hadjicostas, May 14 2020]

A259853 Denominators of the terms of Lehmer's series S_2(2), where S_k(x) = Sum_{n>=1} n^kx^n/binomial(2n, n).

Original entry on oeis.org

1, 3, 5, 35, 63, 77, 429, 6435, 12155, 46189, 88179, 676039, 1300075, 5014575, 215441, 300540195, 583401555, 756261275, 4418157975, 6892326441, 22427411435, 263012370465, 514589420475, 895766768975, 15801325804719, 61989816618513, 121683714103007

Offset: 1

Jean-François Alcover, Jul 07 2015

The first 14 terms are identical to A052468.

			1/1, 8/3, 18/5, 128/35, 200/63, 192/77, 784/429, ... = A259852/A259853.

F. J. Dyson, N. E. Frankel, M. L. Glasser, Lehmer's Interesting Series, arXiv:1009.4274 [math-ph], 2010-2011.
F. J. Dyson, N. E. Frankel, and M. L. Glasser, Lehmer's interesting series, Amer. Math. Monthly, 120 (2013), 116-130.
D. H. Lehmer, Interesting series involving the central binomial coefficient, Amer. Math. Monthly, 92(7) (1985), 449-457.

Cf. A014307, A052468, A180875, A259852 (numerators).

Table[2^n*n^2/Binomial[2*n, n] // Denominator, {n, 1, 40}]

vector(40, n, denominator(n^2*2^n/binomial(2*n,n))) \\ Michel Marcus, Jul 07 2015

a(n) = denominator(n^2*2^n/C(2*n,n)).

A184962 Triangle T(n,k), read by rows, given by (0, 1, 2, 2, 4, 3, 6, 4, 8, 5, 10, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 3, 1, 0, 13, 15, 6, 1, 0, 75, 95, 45, 10, 1, 0, 541, 735, 390, 105, 15, 1, 0, 4683, 6727, 3885, 1190, 210, 21, 1, 0, 47293, 71127, 43918, 14805, 3010, 378, 28, 1, 0, 545835

Offset: 0

Philippe Deléham, Dec 22 2011

The Bell transform of the Fubini numbers. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 29 2016

			Triangle begins :
1
0, 1
0, 1, 1
0, 3, 3, 1
0, 13, 15, 6, 1
0, 75, 95, 45, 10, 1

Row sums are A014307(n).

Cf. A000670, A079641, A195204.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> (polylog(-n,1/2)+0^n)/2, 10); # Peter Luschny, Jan 29 2016

(* The function BellMatrix is defined in A264428. *)
bm = BellMatrix[(PolyLog[-#, 1/2] + Boole[n == 0])/2 &, 10]; Table[bm[[n, k]], {n, 1, Length[bm]}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 31 2016, after Peter Luschny *)

Sum_{k, 0<=k<=n} T(n,k)*x^k = A000007(n), A014307(n), A000629(n) for x = 0, 1, 2 respectively.

A280939 Expansion of e.g.f.: 2*sinh(x/2) / sqrt(2 - exp(x)).

Original entry on oeis.org

1, 1, 4, 19, 121, 946, 8779, 94249, 1148746, 15667741, 236396029, 3909054304, 70297156021, 1365847397461, 28512838809004, 636437585232559, 15125744356058821, 381337518656892106, 10164860714961807079, 285635253778131491389, 8438962752941736017146, 261512261403795336646801, 8481542634943973943517129, 287325556922319462615912544, 10148442521179099638781764121

Offset: 1

Paul D. Hanna, Jan 11 2017

nonn

			E.g.f.: A(x) = x + x^2/2! + 4*x^3/3! + 19*x^4/4! + 121*x^5/5! + 946*x^6/6! + 8779*x^7/7! + 94249*x^8/8! + 1148746*x^9/9! + 15667741*x^10/10! + 236396029*x^11/11! + 3909054304*x^12/12! + ...

G. C. Greubel, Table of n, a(n) for n = 1..250

Cf. A014304, A014307.

m:=50; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(2*Sinh(x/2)/Sqrt(2 - Exp(x)))); [Factorial(n)*b[n]: n in [1..m-1]]; // G. C. Greubel, Oct 10 2018

seq(coeff(series(factorial(n)*(2*sinh(x/2)/sqrt(2-exp(x))),x,n+1), x, n), n = 1 .. 25); # Muniru A Asiru, Oct 11 2018

Rest[With[{nmax = 50}, CoefficientList[Series[2*Sinh[x/2]/Sqrt[2 - Exp[x]], {x, 0, nmax}], x]*Range[0, nmax]!]] (* G. C. Greubel, Oct 10 2018 *)

{a(n) = my(X=x+x*O(x^n)); n!*polcoeff( 2*sinh(X/2) / sqrt(2 - exp(X)),n)}
for(n=1,20,print1(a(n),", "))

a(n) ~ n^n / (sqrt(2) * log(2)^(n + 1/2) * exp(n)). - Vaclav Kotesovec, Jan 11 2017

A296545 Expansion of e.g.f. arcsinh(exp(x)-1).

Original entry on oeis.org

0, 1, 1, 0, -5, -15, 46, 735, 2185, -33390, -453479, -364155, 57806200, 681966285, -3289884779, -197798065920, -1815938249585, 33917006295885, 1155429901407646, 5691720408045315, -408736165211351795, -10271257189100959590, 23948813753053818421, 6626731340918542069425, 124356774945741129842320

Offset: 0

Ilya Gutkovskiy, Dec 15 2017

sign

			arcsinh(exp(x)-1) = x/1! + x^2/2! - 5*x^4/4! - 15*x^5/5! + 46*x^6/6! + 735*x^7/7! + ...

Robert Israel, Table of n, a(n) for n = 0..440

Cf. A014307, A024429, A121868.

S:= series(arcsinh(exp(x)-1),x,41):
seq(coeff(S,x,j)*j!,j=0..40); # Robert Israel, Dec 17 2017

nmax = 24; CoefficientList[Series[ArcSinh[Exp[x] - 1], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 24; CoefficientList[Series[-Log[1 - Exp[x] + Sqrt[1 + (1 - Exp[x])^2]], {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: -log(1 - exp(x) + sqrt(1 + (1 - exp(x))^2)).

A155520 Triangle read by rows: A(n,k) is the number of ordered trees with n edges having k drawings. A drawing of an ordered tree T with n edges is a sequence of trees (T_0, T_1, T_2, ..., T_n), such that T_n = T and T_{i-1} arises from T_i by deleting a leaf of T_i.

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 6, 1, 1, 5, 2, 6, 9, 1, 4, 4, 4, 2, 1, 2, 2

Offset: 1

Emeric Deutsch, Mar 19 2009

Row sums are the Catalan numbers (A000108).

Sum(k*A(n,k), k>0)=A014307(n).

			We represent ordered trees by their corresponding Dyck paths via the "glove" bijection.
The "tree" UDUUDD has 2 drawings:
* , UD, UUDD, UDUUDD and *, UD, UDUD, UDUUDD;
the "tree" UUDDUD has 2 drawings:
*, UD, UUDD, UUDDUD and *, UD, UUDD, UUDDUD.
Thus A(3,2)=2.
The "tree" UUUDDD has 1 drawing: *, UD, UUDD, UUUDDD;
the "tree" UUDUDD has 1 drawing: *, UD, UUDD, UUDUDD;
the "tree" UDUDUD has 1 drawing: *, UD, UDUD, UDUDUD.
Thus A(3,1)=3.
Triangle starts:
1;
2;
3, 2;
4, 2, 6, 1, 1;
5, 2, 6, 9, 1, 4, 4, 4, 2, 1, 2, 2;

M. Klazar, Twelve countings with rooted plane trees, European Journal of Combinatorics 18 (1997), 195-210; Addendum, 18 (1997), 739-740.

Keyword tabf added by Michel Marcus, Apr 09 2013

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A259852 Numerators of the terms of Lehmer's series S_2(2), where S_k(x) = Sum_{n>=1} n^k*x^n/binomial(2*n, n).

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A259853 Denominators of the terms of Lehmer's series S_2(2), where S_k(x) = Sum_{n>=1} n^k*x^n/binomial(2*n, n).

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A184962 Triangle T(n,k), read by rows, given by (0, 1, 2, 2, 4, 3, 6, 4, 8, 5, 10, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938.

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A280939 Expansion of e.g.f.: 2*sinh(x/2) / sqrt(2 - exp(x)).

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Magma

Maple

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A296545 Expansion of e.g.f. arcsinh(exp(x)-1).

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Maple

Mathematica

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A155520 Triangle read by rows: A(n,k) is the number of ordered trees with n edges having k drawings. A drawing of an ordered tree T with n edges is a sequence of trees (T_0, T_1, T_2, ..., T_n), such that T_n = T and T_{i-1} arises from T_i by deleting a leaf of T_i.

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A259852 Numerators of the terms of Lehmer's series S_2(2), where S_k(x) = Sum_{n>=1} n^kx^n/binomial(2n, n).

A259853 Denominators of the terms of Lehmer's series S_2(2), where S_k(x) = Sum_{n>=1} n^kx^n/binomial(2n, n).