A002105 -id:A002105 - cp's OEIS frontend

A261042 Generating function g(0) where g(k) = 1 - x2(k+1)(k+2)/(x2(k+1)(k+2) - 1/g(k+1)).

1, 4, 64, 2176, 126976, 11321344, 1431568384, 243680935936, 53725527801856, 14893509177769984, 5070334006399074304, 2079588119566033616896, 1011390382859091900891136, 575501120339508919401447424, 378784713733072451034702413824, 285539131625477547496925147693056

Offset: 0

Peter Luschny, Aug 08 2015

More generally let G(y) defined by the Taylor expansion of the continued fraction
g(y,k) = 1 - (y*x*(k+1)*(k+2)) / ((y*x*(k+1)*(k+2)) - 1/g(y,k+1)). Then
G(1/2) -> A002105, G(1) -> A000182, G(2) -> A261042, G(4) -> A253165 and G(1/8)(n) *2^(n-1+padic(n,2)) -> A002425.

Cf. A000182, A002105, A002425, A126156 (example section), A253165.

eulerCF := proc(f, len) local g, k; g := 1;
for k from len-2 by -1 to 0 do g := 1 - f(k)/(f(k)-1/g) od;
PolynomialTools:-CoefficientList(convert(series(g, x, len), polynom), x) end:
A261042_list := len -> eulerCF(k -> x*2*(k+1)*(k+2), len): A261042_list(16);
# Alternative:
ser := series(cos(x/sqrt(2))^(-2), x, 32):
seq(2^(2*n)*(2*n)!*coeff(ser, x, 2*n), n = 0..15); # Peter Luschny, Sep 03 2022

fracGen[f_, len_] := Module[{g, k}, g[len] = 1; For[k = len-1, k >= 0, k--, g[k] = 1-f[k]/(f[k]-1/g[k+1])]; CoefficientList[g[0] + O[x]^(len+1), x] ]; A261042list[len_] := fracGen[x*2*(#+1)*(#+2)&, len-1]; A261042list[16] (* Jean-François Alcover, Aug 08 2015, after Peter Luschny *)

def A261042_list(len):
    f = lambda k: x*2*(k+1)*(k+2)
    g = 1
    for k in range(len-2,-1,-1):
        g = (1-f(k)/(f(k)-1/g)).simplify_rational()
    return taylor(g, x, 0, len-1).list()
A261042_list(16)

a(n) = 2^(2*n)*(2*n)!*[x^(2*n)] cos(x/sqrt(2))^(-2). - Peter Luschny, Sep 03 2022

A325222 E.g.f.: D(x,k) = dn( i * Integral C(x,k) dx, k) such that C(x,k) = cn( i * Integral C(x,k) dx, k), where D(x,k) = Sum_{n>=0} Sum_{j=0..n} T(n,j) * x^(2n)k^(2j)/(2n)!, as a triangle of coefficients T(n,j) read by rows.

Original entry on oeis.org

1, 0, 1, 0, 8, 1, 0, 136, 88, 1, 0, 3968, 6240, 816, 1, 0, 176896, 513536, 195216, 7376, 1, 0, 11184128, 51880064, 39572864, 5352544, 66424, 1, 0, 951878656, 6453433344, 8258202240, 2458228480, 139127640, 597864, 1, 0, 104932671488, 978593947648, 1889844670464, 994697838080, 137220256000, 3535586112, 5380832, 1, 0, 14544442556416, 178568645312512, 485265505927168, 398800479698944, 102950036177920, 7233820923904, 88992306208, 48427552, 1

Offset: 0

Paul D. Hanna, Apr 13 2019

Equals a row reversal of triangle A322231.

Compare to dn(x,k) = 1 - k^2 * Integral sn(x,k)*cn(x,k) dx, where sn(x,k), cn(x,k), and dn(x,k) are Jacobi elliptic functions.

			E.g.f.: D(x,k) = 1 + k^2*x^2/2! + (8*k^2 + 1*k^4)*x^4/4! + (136*k^2 + 88*k^4 + 1*k^6)*x^6/6! + (3968*k^2 + 6240*k^4 + 816*k^6 + 1*k^8)*x^8/8! + (176896*k^2 + 513536*k^4 + 195216*k^6 + 7376*k^8 + 1*k^10)*x^10/10! + (11184128*k^2 + 51880064*k^4 + 39572864*k^6 + 5352544*k^8 + 66424*k^10 + 1*k^12)*x^12/12! + (951878656*k^2 + 6453433344*k^4 + 8258202240*k^6 + 2458228480*k^8 + 139127640*k^10 + 597864*k^12 + 1*k^14)*x^14/14! + ...
such that D(x,k) = dn( i * Integral C(x,k) dx, k) where C(x,k) = cn( i * Integral C(x,k) dx, k).
This triangle of coefficients T(n,j) of x^(2*n)*k^(2*j)/(2*n)! in e.g.f. D(x,k) begins:
1;
0, 1;
0, 8, 1;
0, 136, 88, 1;
0, 3968, 6240, 816, 1;
0, 176896, 513536, 195216, 7376, 1;
0, 11184128, 51880064, 39572864, 5352544, 66424, 1;
0, 951878656, 6453433344, 8258202240, 2458228480, 139127640, 597864, 1;
0, 104932671488, 978593947648, 1889844670464, 994697838080, 137220256000, 3535586112, 5380832, 1;
0, 14544442556416, 178568645312512, 485265505927168, 398800479698944, 102950036177920, 7233820923904, 88992306208, 48427552, 1; ...
RELATED SERIES.
The related series S(x,k), where D(x,k)^2 - k^2*S(x,k)^2 = 1, starts
S(x,k) = x + (2 + 1*k^2)*x^3/3! + (16 + 28*k^2 + 1*k^4)*x^5/5! + (272 + 1032*k^2 + 270*k^4 + 1*k^6)*x^7/7! + (7936 + 52736*k^2 + 36096*k^4 + 2456*k^6 + 1*k^8)*x^9/9! + (353792 + 3646208*k^2 + 4766048*k^4 + 1035088*k^6 + 22138*k^8 + 1*k^10)*x^11/11! + (22368256 + 330545664*k^2 + 704357760*k^4 + 319830400*k^6 + 27426960*k^8 + 199284*k^10 + 1*k^12)*x^13/13! + (1903757312 + 38188155904*k^2 + 120536980224*k^4 + 93989648000*k^6 + 18598875760*k^8 + 702812568*k^10 + 1793606*k^12 + 1*k^14)*x^15/15! + ...
The related series C(x,k), where C(x,k)^2 - S(x,k)^2 = 1, starts
C(x,k) = 1 + x^2/2! + (5 + 4*k^2)*x^4/4! + (61 + 148*k^2 + 16*k^4)*x^6/6! + (1385 + 6744*k^2 + 2832*k^4 + 64*k^6)*x^8/8! + (50521 + 410456*k^2 + 383856*k^4 + 47936*k^6 + 256*k^8)*x^10/10! + (2702765 + 32947964*k^2 + 54480944*k^4 + 17142784*k^6 + 780544*k^8 + 1024*k^10)*x^12/12! + (199360981 + 3402510924*k^2 + 8760740640*k^4 + 5199585280*k^6 + 686711040*k^8 + 12555264*k^10 + 4096*k^12)*x^14/14! + ...
which also satisfies C(x,k) = cn( i * Integral C(x,k) dx, k).

Cf. A325220 (S), A325221(C).

Cf. A322231 (row reversal).

N=10;
{S=x; C=1; D=1; for(i=1, 2*N, S = intformal(C^2*D +O(x^(2*N+1))); C = 1 + intformal(S*C*D); D = 1 + k^2*intformal(S*C^2)); }
{T(n,j) = (2*n)!*polcoeff(polcoeff(D, 2*n, x), 2*j, k)}
for(n=0, N, for(j=0, n, print1( T(n,j), ", ")) ; print(""))

E.g.f. D = D(x,k) = Sum_{n>=0} Sum_{j=0..n} T(n,j) * x^(2*n)*k^(2*j)/(2*n)!, along with related series S = S(x,k) and C = C(x,k), satisfies:
(1a) S = Integral C^2*D dx.
(1b) C = 1 + Integral S*C*D dx.
(1c) D = 1 + k^2 * Integral S*C^2 dx.
(2a) C^2 - S^2 = 1.
(2b) D^2 - k^2*S^2 = 1.
(3a) C + S = exp( Integral C*D dx ).
(3b) D + k*S = exp( k * Integral C^2 dx ).
(4a) S = sinh( Integral C*D dx ).
(4b) S = sinh( k * Integral C^2 dx ) / k.
(4c) C = cosh( Integral C*D dx ).
(4d) D = cosh( k * Integral C^2 dx ).
(5a) d/dx S = C^2*D.
(5b) d/dx C = S*C*D.
(5c) d/dx D = k^2 * S*C^2.
Given sn(x,k), cn(x,k), and dn(x,k) are Jacobi elliptic functions, with i^2 = -1, k' = sqrt(1-k^2), then
(6a) S = -i * sn( i * Integral C dx, k),
(6b) C = cn( i * Integral C dx, k),
(6c) D = dn( i * Integral C dx, k).
(7a) S = sc( Integral C dx, k') = sn(Integral C dx, k')/cn(Integral C dx, k'),
(7b) C = nc( Integral C dx, k') = 1/cn(Integral C dx, k'),
(7c) D = dc( Integral C dx, k') = dn(Integral C dx, k')/cn(Integral C dx, k').
Row sums equal ( (2*n)!/(n!*2^n) )^2 = A001818(n), the squares of the odd double factorials.
Column T(n,n+1) = 2^n*A002105(n+1), for n>=0, where A002105 gives the reduced tangent numbers.

A087736 Triangle T(n,k) read by rows given by [0, 1, 3, 6, 10, 15, 21, ...] DELTA [1, 3, 6, 10, 15, 21, 28,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 4, 0, 4, 23, 34, 0, 34, 249, 606, 496, 0, 496, 4354, 14181, 20434, 11056, 0, 11056, 112238, 450097, 894838, 885032, 349504, 0, 349504, 4008024, 18911670, 47136095, 65613780, 48468804, 14873104

Offset: 0

Philippe Deléham, Sep 30 2003, Jul 17 2007

			Triangle begins:
1;
0, 1;
0, 1, 4;
0, 4, 23, 34;
0, 34, 249, 606, 496;
0, 496, 4354, 14181, 20434, 11056;
0, 11056, 112238, 450097, 894838, 885032, 349504;
0, 349504, 4008024, 18911670, 47136095, 65613780, 48468804, 14873104 ;...

Diagonals give A002105: [1, 1, 4, 34, 496, ...] Row sums give A000364 : [1, 1, 5, 61, 1385, ...] Euler numbers.

Cf. A000217 A000384 A002105 A084938.

Sum_{k, 0<=k<=n}T(n,k)*x^(n-k)=A000012(n), A011782(n), A001147(n), A002105(n+1), A000364(n), A126151(n), A126156(n) for n = -3,-2,-1,0,1,2,3 respectively .

Corrected and edited. - Philippe Deléham, Nov 24 2008

A177389 Expansion of o.g.f.: Sum_{n>=0} Product_{k=1..n} tanh(k*arctanh(x)).

Original entry on oeis.org

1, 1, 2, 6, 22, 98, 514, 3110, 21334, 163650, 1388162, 12902086, 130391830, 1423632546, 16699055490, 209432697830, 2796597560150, 39613075175554, 593253347702530, 9366042608039814, 155466234198142998

Offset: 0

Paul D. Hanna, May 15 2010

nonn

			O.g.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 22*x^4 + 98*x^5 + 514*x^6 + ...
Let G(x) = Sum_{n>=0} A002105(n+1)*x^n/n!, so that
G(x) = 1 + x + 4*x^2/2! + 34*x^3/3! + 496*x^4/4! + 11056*x^5/5! + ...
then A(x) = G(arctanh(x)).
G.f.: 1 + x + x*(2x/(1+x^2)) + x*(2x/(1+x^2))*((3x+x^3)/(1+3x^2)) + x*(2x/(1+x^2))*((3x+x^3)/(1+3x^2))*((4x+4x^3)/(1+6x^2+x^4)) + ... - _Paul D. Hanna_, May 22 2010

Vaclav Kotesovec, Table of n, a(n) for n = 0..280

Cf. A002105.

{a(n)=local(X=x+x*O(x^n),Egf);Egf=sum(m=0,n,prod(k=1,m,tanh(k*atanh(X))));polcoeff(Egf,n)}

{a(n)=polcoeff(sum(m=0,n,prod(k=1,m,((1+x)^k-(1-x)^k)/((1+x)^k+(1-x)^k+x*O(x^n)))),n)} \\ Paul D. Hanna, May 22 2010

O.g.f.: A(x) = Sum_{n>=0} A002105(n+1)*arctanh(x)^n/n!, where A002105 is the reduced tangent numbers.
G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^k - (1-x)^k)/((1+x)^k + (1-x)^k). - Paul D. Hanna, May 22 2010
a(n) ~ 2^(3*n+9/2) * n^(n+1) / (exp(n) * Pi^(2*n+2)). - Vaclav Kotesovec, Nov 06 2014

A218260 E.g.f.: Sum_{n>=0} Product_{k=1..n} tanh((2k-1)x).

Original entry on oeis.org

1, 1, 6, 88, 2280, 92416, 5393376, 428428288, 44450655360, 5836916064256, 946245183223296, 185613384522661888, 43330332249288714240, 11871318610487327850496, 3772031142226151742038016, 1375871976238663365598117888

Offset: 0

Paul D. Hanna, Oct 24 2012

nonn

			E.g.f.: A(x) = 1 + x + 6*x^2/2! + 88*x^3/3! + 2280*x^4/4! + 92416*x^5/5! +...
where
A(x) = 1 + tanh(x) + tanh(x)*tanh(3*x) + tanh(x)*tanh(3*x)*tanh(5*x) + tanh(x)*tanh(3*x)*tanh(5*x)*tanh(7*x) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..150

Cf. A002105, A221080.

{a(n)=local(X=x+x*O(x^n),Egf);Egf=sum(m=0,n,prod(k=1,m,tanh((2*k-1)*X)));n!*polcoeff(Egf,n)}
for(n=0,30,print1(a(n),", "))

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

a(n) ~ 2^(4*n+7/2) * n^(2*n+1) / (exp(2*n) * Pi^(2*n+1)). - Vaclav Kotesovec, Nov 02 2014

A305533 Expansion of 1/(1 - x/(1 - 1x/(1 - 3x/(1 - 6x/(1 - 10x/(1 - ... - (k(k + 1)/2)x/(1 - ...))))))), a continued fraction.

Original entry on oeis.org

1, 1, 2, 7, 47, 592, 12287, 374857, 15639302, 851542747, 58536120467, 4953497262712, 505784457870707, 61300510121162077, 8698776162350603222, 1428545280744850604767, 268795232754158224790687, 57445320930331531152213232, 13837791987711934467999437927

Offset: 0

Ilya Gutkovskiy, Jun 04 2018

nonn

Invert transform of reduced tangent numbers (A002105).

N. J. A. Sloane, Transforms
Index entries for sequences related to Bernoulli numbers

Cf. A000217, A002105, A305532.

T := proc(n, k) option remember; if k = 0 then 1 else if k = n then T(n, k-1)
else (((k - n - 1)*(k - n)) / 2) * T(n, k - 1) + T(n - 1, k) fi fi end:
a := n -> T(n, n): seq(a(n), n = 0..18);  # Peter Luschny, Oct 01 2023

nmax = 18; CoefficientList[Series[1/(1 - x/(1 + ContinuedFractionK[-k (k + 1)/2 x, 1, {k, 1, nmax}])), {x, 0, nmax}], x]
nmax = 18; CoefficientList[Series[1/(1 - Sum[PolyGamma[2 k - 1, 1/2]/(2^(k - 2) Pi^(2 k)) x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[2^k (2^(2 k) - 1) Abs[BernoulliB[2 k]]/k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

a(n) ~ 2^(3*n + 2) * n^(2*n - 1/2) / (exp(2*n) * Pi^(2*n - 1/2)). - Vaclav Kotesovec, Jun 08 2019

A309522 Generalized Blasius numbers, square array read by ascending antidiagonals, A(n, k) for n, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 4, 6, 14, 1, 1, 11, 34, 24, 42, 1, 1, 36, 375, 496, 120, 132, 1, 1, 127, 6306, 27897, 11056, 720, 429, 1, 1, 463, 129256, 3156336, 3817137, 349504, 5040, 1430, 1, 1, 1717, 2877883, 514334274, 3501788976, 865874115, 14873104, 40320, 4862

Offset: 0

Petros Hadjicostas and Peter Luschny, Aug 06 2019

The generalized Blasius o.d.e. of order n whose infinite series solution involves row n of this square array appears in Salié (1955). Rows n = 2 and n = 3 of this array appear in Kuba and Panholzer (2014, 2016), who give combinatorial interpretations to the numbers in those two rows.

Eq. (22) in Kuba and Panholzer (2014, p. 23) and Eq. (5) in Kuba and Panholzer (2016, p. 233) are general o.d.e.s for generating infinite sequences of numbers with some combinatorial properties. Even though they bear some similarity to Salié's general o.d.e., it is not clear whether either one can be used to give combinatorial interpretation to the numbers in rows n >= 4 of the current square array.

			Table A(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
[0] 1, 1,  2,    5,      14,         42,           132, ...  A000108
[1] 1, 1,  2,    6,      24,        120,           720, ...  A000142
[2] 1, 1,  4,   34,     496,      11056,        349504, ...  A002105
[3] 1, 1, 11,  375,   27897,    3817137,     865874115, ...  A018893
[4] 1, 1, 36, 6306, 3156336, 3501788976, 7425169747776, ...
     A260878|

Heinrich Blasius, Grenzschichten in Flüssigkeiten mit kleiner Reibung, Z. Math. u. Physik 56 (1908), 1-37; see p. 8. [This article was based on his PhD thesis. He corrected c_6 = A(n=3, k=6) but his "correction" of c_7 = A(n=3, k=7) was not right!]
Heinrich Blasius, Grenzschichten in Flüssigkeiten mit kleiner Reibung, Z. Math. u. Physik 56 (1908), 1-37 [English translation by J. Vanier on behalf of the National Advisory Committee for Aeronautics (NACA), 1950]; see p. 8. [This is a translation of Blasius' article. The value of c_6 = A(n=3, k=6) was corrected in the article and the translation, but the "correction" for c_7 = A(n=3, k=7) in both documents is wrong.]
Markus Kuba and Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, arXiv:1411.4587 [math.CO], 2014.
Markus Kuba and Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, Discrete Mathematics 339(1) (2016), 227-254.
Hans Salié, Über die Koeffizienten der Blasiusschen Reihen, Math. Nachr. 14 (1955), 241-248 (1956). [In the article the array is denoted by c^{(n)}_v for n, v >= 1. We have A(n, k) = c^{(n)}_{k+1} for n >= 1 and k >= 0. The Catalan numbers (row n = 0 for A(n, k)) do not appear in Salié's article.]

Rows include A000108, A000142, A002105 (shifted), A018893.
Columns include A260878.
Cf. A256522 (Blasius constant), A260876.

A := proc(n, k) option remember; if k = 0 then 1 else
add(binomial(n*k-1, n*v)*A(n, v)*A(n, k-1-v), v=0..k-1) fi end:
seq(seq(A(n-k, k), k=0..n), n=0..9);

A[n_, k_] := A[n, k] = If[k == 0, 1, Sum[Binomial[n*k - 1, n*v]*A[n, v]* A[n, k - 1 - v], {v, 0, k - 1}]];
Table[A[n - k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 26 2019, from Maple *)

A(n, k) = Sum_{v=0..k-1} binomial(n*k-1, n*v)*A(n, v)*A(n, k-1-v) for k > 0 and A(n, 0) = 1.
A(n, 2) = A260876(n, 2) = binomial(2*n - 1, n) + 1 for n >= 0.
A(n, 3) = A260876(n, 2) + A260876(n, 3) - 1 = (binomial(3*n - 1, 2*n) + 1) * (binomial(2*n - 1, n) + 1) + binomial(3*n - 1, n) for n >= 1.

A113897 Triangle read by rows: number of simsun n-permutations with k descents.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 11, 4, 1, 26, 34, 1, 57, 180, 34, 1, 120, 768, 496, 1, 247, 2904, 4288, 496, 1, 502, 10194, 28768, 11056, 1, 1013, 34096, 166042, 141584, 11056, 1, 2036, 110392, 868744, 1372088, 349504, 1, 4083, 349500, 4247720, 11204160, 6213288, 349504

Offset: 1

Chak-On Chow (cchow(AT)alum.mit.edu), Jan 28 2006

Is this A094503 after removal of the top row? - R. J. Mathar, Aug 13 2008

Yes. See formula of Peter Bala, Jun 26 2012 in A094503. - Stefano Spezia, Aug 09 2023

			Triangle begins
   1;
   1,   1;
   1,   4;
   1,  11,   4;
   1,  26,  34;
   1,  57, 180,  34;
   ...

Chak-On Chow and Wai Chee Shiu, Counting Simsun Permutations by Descents, Ann. Comb. 15, 625-635 (2011). See p. 627.
Ming-Jian Ding and Bao-Xuan Zhu, Some results related to Hurwitz stability of combinatorial polynomials, Advances in Applied Mathematics, Volume 152, (2024), 102591. See p. 35.
R. P. Stanley, Flag f-vectors and the cd-index, Math. Zeitschrift 216 (1994), 483-499.
S. Sundaram, Plethysm, Partitions with an Even Number of Blocks and Euler Numbers, in "Formal Power Series and Algebraic Combinatorics 1994," DIMACS Series in Discrete Mathematics and Theoretical Computer Science 24, AMS (1996).

Cf. A000111, A000295, A002105.

Table[SeriesCoefficient[(2t-1)*(Sec[x*Sqrt[2t-1]/2]/(Sqrt[2t-1]- Tan[x*Sqrt[2t-1]/2]))^2,{x,0,n},{t,0,k}]n!,{n,11},{k,0,Floor[n/2]}]//Flatten (* Stefano Spezia, Aug 09 2023 *)

T(n, k) = (k+1)*T(n-1, k) + (n-2k+1)*T(n-1, k-1);
Row g.f.: T(n, t) = Sum_{k=0..floor(n/2)} T(n, k)*t^k,
T(n, t) = ((n-1)*t + 1)*T(n-1, t) + t*(1-2t)*T(n-1, t)'.
E.g.f.: Sum_{n>=1} T(n, t)*x^n/n! = (2t-1)*(sec(x*sqrt(2t-1)/2)/(sqrt(2t-1) - tan(x*sqrt(2t-1)/2)))^2.

Corrected and extended by Vladeta Jovovic, Jan 30 2006

A187802 E.g.f.: Sum_{n>=0} Product_{k=1..n} tanh(nkx).

Original entry on oeis.org

1, 1, 16, 970, 146176, 44183536, 23478931456, 20054284098640, 25800626187206656, 47592874959658936576, 121099500781576410628096, 411996060596290629454466560, 1826628916277875316651443879936, 10329535274999799577516027932553216, 73156530986984637348101331408897703936

Offset: 0

Paul D. Hanna, Jan 06 2013

nonn

Compare to the e.g.f. of A002105, the reduced tangent numbers:

Sum_{n>=0} Product_{k=1..n} tanh(k*x).

			E.g.f.: A(x) = 1 + x + 16*x^2/2! + 970*x^3/3! + 146176*x^4/4! +...
where
A(x) = 1 + tanh(x) + tanh(2*1*x)*tanh(2*2*x) + tanh(3*1*x)*tanh(3*2*x)*tanh(3*3*x) + tanh(4*1*x)*tanh(4*2*x)*tanh(4*3*x)*tanh(4*4*x) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..140

Cf. A002105.

With[{nn=20},CoefficientList[Series[Sum[Product[Tanh[n*k*x],{k,n}],{n,0,nn}],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 26 2024 *)

{a(n)=local(X=x+x*O(x^n), Egf); Egf=sum(m=0, n, prod(k=1, m, tanh(m*k*X))); n!*polcoeff(Egf, n)}
for(n=0, 20, print1(a(n), ", "))

a(n) ~ c * d^n * (n!)^3 / sqrt(n), where d = 2.67441747301630303932685879..., c = 0.4405132627693901422580367... . - Vaclav Kotesovec, Nov 02 2014

A203827 Table of coefficients of up-down polynomials P_n(m) = Sum_{i=0..floor(log_2(2n))} binomial(m,i).

Original entry on oeis.org

1, -1, 1, -1, 0, 1, 1, -1, 1, -1, 0, 0, 1, 1, -1, 0, 2, 1, 0, -1, 2, -1, 1, -1, 1, -1, 0, 0, 0, 1, 1, -1, 0, 0, 3, 1, 0, -1, 0, 5, -1, 1, -1, 0, 3, 1, 0, 0, -1, 3, -1, 1, 0, -2, 5, -1, 0, 1, -2, 3, 1, -1, 1, -1, 1, -1, 0, 0, 0, 0, 1, 1, -1, 0, 0, 0, 4, 1, 0, -1, 0, 0, 9

Offset: 0

Vladimir Shevelev, Jan 06 2012

For a permutation s = (s_1,...,s_m), the number n = Sum_{j=1..m-1} b_j*2^(m-i-1), where b_j=1, if s_(j+1) > s_j, and b_j=0, if s_(j+1) < s_j, is called index of s. Up-down polynomial P_n(m) gives the number of permutations with index n.
If n = 2^(k_1-1) + 2^(k_2-1) + ... + 2^(k_r-1), k_1 > ... > k_r >= 1, then k_1,k_2,...,k_r are only positive roots of the polynomial P_n(m).
If F(m,x) = Sum_{n>=0} P_n(m)*x^n and t(x) = Sum_{n>=0} t_n*x^n (|x|<1), where t_n = (-1)^A010060(n), then F(m,x)/t(x) is a rational function.
The sequence {n_k} for which P_(n_k)(m) has a root m=-1 begins 2, 5, 8, 11, 23, ...
If n is in A089633, then P_n(m) has only real roots.
Remark from the author. Ivan Niven posed the problem of enumeration of permutations of n elements with a given up-down structure. He introduced (n-1)-dimensional parameter (Niven's signature) and did the enumeration in a determinant form, but did not find simple relations. Therefore, the title of his paper includes the word "problem". Instead of his many-dimensional parameter, the author introduced one-dimensional parameter (index). It allowed us to find many simple relations and properties for the enumeration numbers which is called up-down coefficients since they have many close properties with the binomial coefficients. In particular, they give a decomposition of Eulerian numbers. Many other relations will appear in the paper by the author and U. Spilker (to appear), where, in particular, we prove that the enumeration numbers are maximal when the index corresponds to the alternating (or Andre) permutations.

			Table begins
   1
  -1  1
  -1  0  1
   1 -1  1
  -1  0  0  1
   1 -1  0  2
   1  0 -1  2
  -1  1 -1  1
  -1  0  0  0  1
   1 -1  0  0  3
   1  0 -1  0  5
  -1  1 -1  0  3
   1  0  0 -1  3
  -1  1  0 -2  5
  -1  0  1 -2  3
   1 -1  1 -1  1
  -1  0  0  0  0  1
   1 -1  0  0  0  4
   1  0 -1  0  0  9

I. Niven, A combinatorial problem of finite sequences, Nieuw Arch. Wisk. 16(1968), 116-123.

V. Shevelev, The number of permutations with prescribed up-down structure as a function of two variables, INTEGERS 12 (2012), article #A1.

Cf. A010060, A000984, A000364, A002105, A008292.

Sum_{n=0..2^(m-1)} P_n(m) = m!;
Sum_{n=0..2^m-1} (-1)^n*P_n(m) = 0.
P_{2^m-1}(2*m) = binomial(2*m-1, m-1);
P_{2^m-1}(2*m+1) = binomial(2*m, m).
If m is an odd prime, then
(1) P_n(m) == t_n (mod m), where t_n = (-1)^A010060(n);
(2) P_((2^(m+1)-4)/6)(m) == (-1)^((m-1)/2) (mod m);
(3) P_((2^(2*m+1)-2)/6)(2*m) == 1 (mod 2*m).
For m >= 1, P_((2^(2^m+1)-2)/6)(2^m) == 1 (mod 2^m).
P_((4^m-1)/3)(2*m) = |E_(2*m)| (cf. A000364);
P_((2^(2*m-1)-1)/3) = |B_(2*m)|*4^m(4^m-1)/(2*m) (cf. A002105).
If n = 2^(k_1-1) + 2^(k_2-1) + ... + 2^(k_r-1), k_1 > k_2 > ... > k_r >= 1, then
(Recursion 1) P_n(m) = (-1)^r + Sum_{i=1..r} binomial(m,k_i)*P_(n-2^(k_i-1))(k_i) and
(Recursion 2) for h > k_1, P_(n+2^(h-1))(m) = binomial(m,h)*P_n(h) - P_n(m).

cp's OEIS Frontend

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A187802 E.g.f.: Sum_{n>=0} Product_{k=1..n} tanh(nkx).