A003701 -id:A003701 - cp's OEIS frontend

A172396 G.f. satisfies: A(x) = G(x/A(x)) where o.g.f. G(x) = A(xG(x)) = Sum_{n>=0} A003701(n)x^n.

1, 1, 1, 0, 3, 0, 38, 0, 947, 0, 37394, 0, 2120190, 0, 162980012, 0, 16330173251, 0, 2070201641498, 0, 324240251016266, 0, 61525045423103316, 0, 13913915097436287598, 0, 3698477457114061621492, 0

Offset: 0

Paul D. Hanna, Feb 07 2010

nonn

The e.g.f. of A003701 is exp(x)/cos(x) = Sum_{n>=0} A003701(n)*x^n/n!.

Compare to A157308 and A157310.

			G.f.: A(x) = 1 + x + x^2 + 3*x^4 + 38*x^6 + 947*x^8 + 37394*x^10 +...
where G(x) = A(x*G(x)) is the o.g.f. of A003701:
G(x) = 1 + x + 2*x^2 + 4*x^3 + 12*x^4 + 36*x^5 + 152*x^6 + 624*x^7 +...
while the e.g.f. of A003701 is given by:
exp(x)/cos(x) = 1 + x + 2*x^2/2! + 4*x^3/3! + 12*x^4/4! + 36*x^5/5! +...

Cf. A003701, A158119, A157308, A157310.

{a(n)=local(X=x+x*O(x^n),G=sum(m=0,n,m!*polcoeff(exp(X)/cos(X),m)*x^m)+x*O(x^n)); polcoeff(x/serreverse(x*G),n)}

a(n) = |A157308(n)| = |A157310(n)| for n>=0.
a(2n) = A158119(n) for n>=0; a(2n-1) = 0 for n>=2, with a(1)=1.
G.f. A = A(x) satisfies: A(x) = 1/(1-x/A - (x/A)^2/(1-x/A - 2^2*(x/A)^2/(1-x/A - 3^2*(x/A)^2/(1-x/A - 4^2*(x/A)^2/(1-x/A - 5^2*(x/A)^2/(1-x/A -...)))))), a recursive continued fraction. [From Paul D. Hanna, Jan 05 2012]

A055209 a(n) = Product_{i=0..n} i!^2.

Original entry on oeis.org

1, 1, 4, 144, 82944, 1194393600, 619173642240000, 15728001190723584000000, 25569049282962188245401600000000, 3366980847587422591723894776791040000000000, 44337041641882947649156022595410930014617600000000000000

Offset: 0

N. J. A. Sloane, Jul 18 2000

a(n) is the discriminant of the polynomial x(x+1)(x+2)...(x+n). - Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 13 2003
This is the Hankel transform (see A001906 for definition) of the sequence: 1, 0, 1, 0, 5, 0, 61, 0, 1385, 0, 50521, ... (see A000364: Euler numbers). - Philippe Deléham, Apr 06 2005
Also, for n>0, the quotient of (-1)^(n-1)S(u)^(n^2)/S(un) and the determinant of the (n-1) X (n-1) square matrix [P'(u), P''(u), ..., P^(n-1)(u); P''(u), P'''(u), ..., P^(n)(u); P'''(u), P^(4)(u), ..., P^(n+1)(u); ...; P^(n-1)(u), P^(n)(u), ..., P^(2n-3)(u)] where S and P are the Weierstrass Sigma and The Weierstrass P-function, respectively and f^(n) is the n-th derivative of f. See the King and Schwarz & Weierstrass references. - Balarka Sen, Jul 31 2013
a(n) is the number of idempotent monotonic labeled magmas. That is, prod(i,j) >= max(i,j) and prod(i,i) = i. - Chad Brewbaker, Nov 03 2013
Ramanujan's infinite nested radical sqrt(1+2*sqrt(1+3*sqrt(1+...))) = 3 can be written sqrt(1+sqrt(4+sqrt(144+...))) = sqrt(a(1)+sqrt(a(2)+sqrt(a(3)+...))). Vijayaraghavan used that to prove convergence of Ramanujan's formula. - Petros Hadjicostas and Jonathan Sondow, Mar 22 2014
a(n) is the determinant of the (n+1)-th order Hankel matrix whose (i,j)-entry is equal to A000142(i+j), i,j = 0,1,...,n. - Michael Shmoish, Sep 02 2020

R. Bruce King, Beyond The Quartic Equation, Birkhauser Boston, Berlin, 1996, p. 72.
Srinivasa Ramanujan, J. Indian Math. Soc., III (1911), 90 and IV (1912), 226.
T. Vijayaraghavan, in Collected Papers of Srinivasa Ramanujan, G.H. Hardy, P.V. Seshu Aiyar and B.M. Wilson, eds., Cambridge Univ. Press, 1927, p. 348; reprinted by Chelsea, 1962.

G. C. Greubel, Table of n, a(n) for n = 0..32
Paul Barry, A Note on Three Families of Orthogonal Polynomials defined by Circular Functions, and Their Moment Sequences, Journal of Integer Sequences, Vol. 15 (2012), #12.7.2.
Richard Ehrenborg, The Hankel determinant of exponential polynomials, Amer. Math. Monthly, Vol. 107, No. 6 (2000), pp. 557-560.
William Q. Erickson and Jan Kretschmann, The structure and normalized volume of Monge polytopes, arXiv:2311.07522 [math.CO], 2023. See p. 7.
John W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Christian Radoux, Déterminants de Hankel et théorème de Sylvester, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.
H. A. Schwarz and K. Weierstrass, Formeln und Lehrsätze zum Gebrauche der elliptischen Functionen, Springer, Berlin, 1893, p. 19.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl., Vol. 332, No. 1 (2007), pp. 292-314; see pp. 305-306.
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
Index entries for sequences related to factorial numbers

Cf. A055209 is the Hankel transform (see A001906 for definition) of A000023, A000142, A000166, A000522, A003701, A010842, A010843, A051295, A052186, A053486, A053487.

Cf. A112302, A258619.

[1] cat [(&*[(Factorial(k))^2: k in [1..n]]): n in [1..10]]; // G. C. Greubel, Oct 14 2018

seq(mul(mul(j^2,j=1..k), k=0..n), n=0..10); # Zerinvary Lajos, Sep 21 2007

Table[Product[(i!)^2,{i,n}],{n,0,11}] (* Harvey P. Dale, Jul 06 2011 *)
Table[BarnesG[n + 2]^2, {n, 0, 11}] (* Jan Mangaldan, May 07 2014 *)

a(n)=prod(i=1,n,i!)^2 \\ Charles R Greathouse IV, Jan 12 2012

def A055209(n) :
   return prod(factorial(i)^(2) for i in (0..n))
[A055209(n) for n in (0..11)] # Jani Melik, Jun 06 2015

a(n) = A000178(n)^2. - Philippe Deléham, Mar 06 2004
a(n) = Product_{i=0..n} i^(2*n - 2*i + 2). - Charles R Greathouse IV, Jan 12 2012
Asymptotic: a(n) ~ exp(2*zeta'(-1)-3/2*(1+n^2)-3*n)*(2*Pi)^(n+1)*(n+1)^ (n^2+2*n+5/6). - Peter Luschny, Jun 23 2012
lim_{n->infinity} a(n)^(2^(-(n+1))) = 1. - Vaclav Kotesovec, Jun 06 2015
Sum_{n>=0} 1/a(n) = A258619. - Amiram Eldar, Nov 17 2020

A009739 E.g.f. tan(x)*exp(x).

Original entry on oeis.org

0, 1, 2, 5, 12, 41, 142, 685, 3192, 19921, 116282, 887765, 6219972, 56126201, 458790022, 4776869245, 44625674352, 526589630881, 5534347077362, 72989204937125, 852334810990332, 12424192360405961, 159592488559874302, 2547879762929443405, 35703580441464231912

Offset: 0

R. H. Hardin

Robert Israel, Table of n, a(n) for n = 0..484

Cf. A003701.

G(x):=exp(x)*tan(x): f[0]:=G(x): for n from 1 to 54 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..22 ); # Zerinvary Lajos, Apr 05 2009
# Alternative:
S:= series(exp(x)*tan(x),x, 51):
seq(coeff(S,x,j)*j!,j=0..50); # Robert Israel, Sep 22 2019

x='x+O('x^66); concat([0],Vec(serlaplace(tan(x)*exp(x)))) \\ Joerg Arndt, Apr 26 2013

a(2n) = A009747(n), a(2n+1) = A003719(n).
E.g.f.: exp(x)*tan(x). - Zerinvary Lajos, Apr 05 2009
G.f.: 1/(x-1)/Q(0), where Q(k)= 1 - 1/x - (k+1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 26 2013
G.f.: x/(1-x)/Q(0), where Q(k)= 1 - x - x^2*(k+1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 26 2013
G.f.: G(0)*x/(1-x)^2, where G(k) = 1 - x^2*(k+1)*(k+2)/(x^2*(k+1)*(k+2) - (1-x)^2/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 23 2014
a(n) ~ 2^(3/2 + n)*(exp(Pi) - (-1)^n)*exp(-Pi/2 - n)*Pi^(-1/2 - n)*n^(1/2 + n). - Robert Israel, Sep 22 2019

Extended and signs tested by Olivier Gérard, Mar 15 1997

A081658 Triangle read by rows: T(n, k) = (-2)^kbinomial(n, k)Euler(k, 1/2).

Original entry on oeis.org

1, 1, 0, 1, 0, -1, 1, 0, -3, 0, 1, 0, -6, 0, 5, 1, 0, -10, 0, 25, 0, 1, 0, -15, 0, 75, 0, -61, 1, 0, -21, 0, 175, 0, -427, 0, 1, 0, -28, 0, 350, 0, -1708, 0, 1385, 1, 0, -36, 0, 630, 0, -5124, 0, 12465, 0, 1, 0, -45, 0, 1050, 0, -12810, 0, 62325, 0, -50521, 1, 0, -55, 0, 1650, 0, -28182, 0, 228525, 0, -555731, 0, 1, 0, -66, 0, 2475, 0

Offset: 0

Paul Barry, Mar 26 2003

These are the coefficients of the Swiss-Knife polynomials A153641. - Peter Luschny, Jul 21 2012
Nonzero diagonals of the triangle are of the form A000364(k)*binomial(n+2k,2k)*(-1)^k.
A363393 is the dual triangle ('dual' in the sense of Euler-tangent versus Euler-secant numbers). - Peter Luschny, Jun 05 2023

			The triangle begins
[0] 1;
[1] 1, 0;
[2] 1, 0,  -1;
[3] 1, 0,  -3, 0;
[4] 1, 0,  -6, 0,   5;
[5] 1, 0, -10, 0,  25, 0;
[6] 1, 0, -15, 0,  75, 0,  -61;
[7] 1, 0, -21, 0, 175, 0, -427, 0;
...
From _Peter Luschny_, Sep 17 2021: (Start)
The triangle shows the coefficients of the following polynomials:
[1] 1;
[2] 1 -    x^2;
[3] 1 -  3*x^2;
[4] 1 -  6*x^2 +   5*x^4;
[5] 1 - 10*x^2 +  25*x^4;
[6] 1 - 15*x^2 +  75*x^4 -  61*x^6;
[7] 1 - 21*x^2 + 175*x^4 - 427*x^6;
...
These polynomials are the permanents of the n X n matrices with all entries above the main antidiagonal set to 'x' and all entries below the main antidiagonal set to '-x'. The main antidiagonals consist only of ones. Substituting x <- 1 generates the Euler tangent numbers A155585. (Compare with A046739.)
(End)

Row reversed: A119879.

Cf. A000364, A046739, A155585, A363393.

ogf := n -> euler(n) / (1 - x)^(n + 1):
ser := n -> series(ogf(n), x, 16):
T := (n, k) -> coeff(ser(k), x, n - k):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, Jun 05 2023
T := (n, k) -> (-2)^k*binomial(n, k)*euler(k, 1/2):
seq(seq(T(n, k), k = 0..n), n = 0..9);  # Peter Luschny, Apr 03 2024

sk[n_, x_] := Sum[Binomial[n, k]*EulerE[k]*x^(n - k), {k, 0, n}];
Table[CoefficientList[sk[n, x], x] // Reverse, {n, 0, 12}] // Flatten (* Jean-François Alcover, Jun 04 2019 *)
Flatten@Table[Binomial[n, k] EulerE[k], {n, 0, 12}, {k, 0, n}] (* Oliver Seipel, Jan 14 2025 *)

from functools import cache
@cache
def T(n: int, k: int) -> int:
    if k == 0: return 1
    if k % 2 == 1:  return 0
    if k == n: return -sum(T(n, j) for j in range(0, n - 1, 2))
    return (T(n - 1, k) * n) // (n - k)
for n in range(10):
    print([T(n, k) for k in range(n + 1)])  # Peter Luschny, Jun 05 2023

R = PolynomialRing(ZZ, 'x')
@CachedFunction
def p(n, x) :
    if n == 0 : return 1
    return add(p(k, 0)*binomial(n, k)*(x^(n-k)-(n+1)%2) for k in range(n)[::2])
def A081658_row(n) : return [R(p(n,x)).reverse()[i] for i in (0..n)]
for n in (0..8) : print(A081658_row(n)) # Peter Luschny, Jul 20 2012

Coefficients of the polynomials in k in the binomial transform of the expansion of 2/(exp(kx)+exp(-kx)).
From Peter Luschny, Jul 20 2012: (Start)
p{n}(0) = Signed Euler secant numbers A122045.
p{n}(1) = Signed Euler tangent numbers A155585.
p{n}(2) has e.g.f. 2*exp(x)/(exp(-2*x)+1) A119880.
2^n*p{n}(1/2) = Signed Springer numbers A188458.
3^n*p{n}(1/3) has e.g.f. 2*exp(4*x)/(exp(6*x)+1)
4^n*p{n}(1/4) has e.g.f. 2*exp(5*x)/(exp(8*x)+1).
Row sum: A155585 (cf. A009006). Absolute row sum: A003701.
The GCD of the rows without the first column: A155457. (End)
From Peter Luschny, Jun 05 2023: (Start)
T(n, k) = [x^(n - k)] Euler(k) / (1 - x)^(k + 1).
For a recursion see the Python program.
Conjecture: If n is prime then n divides T(n, k) for 1 <= k <= n-1. (End)

Typo in data corrected by Peter Luschny, Jul 20 2012

Error in data corrected and new name by Peter Luschny, Apr 03 2024

A302584 a(n) = n! * [x^n] exp(n*x)/cos(x).

Original entry on oeis.org

1, 1, 5, 36, 357, 4500, 68857, 1239504, 25661545, 600655824, 15684383021, 452001644864, 14249852124365, 487836995500608, 18022519535240417, 714658089577017600, 30275849571771536977, 1364687729891761740032, 65213822241378992547925, 3293203845745202062590976

Offset: 0

Ilya Gutkovskiy, Apr 10 2018

nonn

N. J. A. Sloane, Transforms

Cf. A000364, A003701, A062024, A115415, A115416, A302583, A302585, A302586, A302587.

Table[n! SeriesCoefficient[Exp[n x]/Cos[x], {x, 0, n}], {n, 0, 19}]
Table[(2 I)^n EulerE[n, (1 - I n)/2], {n, 0, 19}]

a(n) ~ n^n / cos(1). - Vaclav Kotesovec, Jun 08 2019

A301942 Expansion of e.g.f. arcsin(x)/cos(x) (odd powers only).

Original entry on oeis.org

1, 4, 44, 1016, 42384, 2908544, 306305856, 46659144832, 9760451385600, 2683733034474496, 936308392553036800, 403127865773461755904, 209562975305232836300800, 129255511221696545852424192, 93252273300325219683758915584, 77766048645578119241905858314240

Offset: 0

Ilya Gutkovskiy, Apr 09 2018

nonn

			arcsin(x)/cos(x) = x/1! + 4*x^3/3! + 44*x^5/5! + 1016*x^7/7! + 42384*x^9/9! + ...

Cf. A000182, A000364, A000795, A001818, A002084, A003701, A003702, A012782, A296741, A302444, A302542, A302543.

nmax = 16; Table[(CoefficientList[Series[ArcSin[x]/Cos[x], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

a(n) = (2*n+1)! * [x^(2*n+1)] arcsin(x)/cos(x).

A302444 Expansion of e.g.f. arcsinh(x)/cos(x) (odd powers only).

Original entry on oeis.org

1, 2, 24, 216, 15936, -77056, 90991744, -8523712768, 2731708067840, -684815907467264, 268028469798256640, -114888252320482000896, 62022733722259702579200, -38635369828053720937463808, 28349537098304682205749968896, -23874826868622028919177351004160

Offset: 0

Ilya Gutkovskiy, Apr 09 2018

sign

			arcsinh(x)/cos(x) = x/1! + 2*x^3/3! + 24*x^5/5! + 216*x^7/7! + 15936*x^9/9! - 77056*x^11/11! + ...

Cf. A000182, A000364, A000795, A001818, A002084, A003701, A003702, A012821, A296742, A301942, A302542, A302543.

nmax = 16; Table[(CoefficientList[Series[ArcSinh[x]/Cos[x], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

a(n) = (2*n+1)! * [x^(2*n+1)] arcsinh(x)/cos(x).

A302542 Expansion of e.g.f. arctan(x)/cos(x) (odd powers only).

Original entry on oeis.org

1, 1, 29, -139, 31737, -1824151, 313750293, -51584719523, 13137192234225, -3947317975733039, 1522475446731094285, -702509124781480897211, 389722900767594460770025, -253710144786166583863030983, 192285396891961478711402819077, -167564604997707653568802119363795

Offset: 0

Ilya Gutkovskiy, Apr 09 2018

sign

			arctan(x)/cos(x) = x/1! + x^3/3! + 29*x^5/5! - 139*x^7/7! + 31737*x^9/9! - 1824151*x^11/11! + ...

Cf. A000182, A000364, A000795, A002084, A003701, A003702, A010050, A012801, A191003, A301942, A302444, A302543.

nmax = 16; Table[(CoefficientList[Series[ArcTan[x]/Cos[x], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

a(n) = (2*n+1)! * [x^(2*n+1)] arctan(x)/cos(x).

A302543 Expansion of e.g.f. arctanh(x)/cos(x) (odd powers only).

Original entry on oeis.org

1, 5, 69, 2001, 104073, 8723549, 1088372557, 190057979177, 44285819490065, 13267464006201781, 4964113699657822805, 2266816666007859759489, 1239999748307938170531225, 800189083150907165762837517, 601369618369661775955962338653, 520607107122686183781743903500505

Offset: 0

Ilya Gutkovskiy, Apr 09 2018

nonn

			arctanh(x)/cos(x) = x/1! + 5*x^3/3! + 69*x^5/5! + 2001*x^7/7! + 104073*x^9/9! + ...

Cf. A000182, A000364, A000795, A002084, A003701, A003702, A010050, A012840, A296743, A301942, A302444, A302542.

nmax = 16; Table[(CoefficientList[Series[ArcTanh[x]/Cos[x], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

a(n) = (2*n+1)! * [x^(2*n+1)] arctanh(x)/cos(x).

A348587 Expansion of e.g.f. exp(x) / (2 - cos(x)).

Original entry on oeis.org

1, 1, 0, -2, 2, 26, -30, -622, 982, 25846, -50910, -1639142, 3874862, 147434366, -406614390, -17851478062, 56266545142, 2799621404086, -9927225631470, -552054087163382, 2175042302117822, 133686372253841006, -579383205000618150, -39002628245713951102

Offset: 0

Ilya Gutkovskiy, Oct 24 2021

sign

Cf. A003701, A094088, A327034, A348580.

nmax = 23; CoefficientList[Series[Exp[x]/(2 - Cos[x]), {x, 0, nmax}], x] Range[0, nmax]!

my(x='x+O('x^40)); Vec(serlaplace(exp(x)/(2-cos(x)))) \\ Michel Marcus, Oct 24 2021

a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(n,2*k) * A094088(k).

cp's OEIS Frontend

A172396 G.f. satisfies: A(x) = G(x/A(x)) where o.g.f. G(x) = A(x*G(x)) = Sum_{n>=0} A003701(n)*x^n.

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A055209 a(n) = Product_{i=0..n} i!^2.

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Magma

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A009739 E.g.f. tan(x)*exp(x).

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A081658 Triangle read by rows: T(n, k) = (-2)^k*binomial(n, k)*Euler(k, 1/2).

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Maple

Mathematica

Python

Sage

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Extensions

A302584 a(n) = n! * [x^n] exp(n*x)/cos(x).

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Links

Crossrefs

Programs

Mathematica

Formula

A301942 Expansion of e.g.f. arcsin(x)/cos(x) (odd powers only).

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Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A302444 Expansion of e.g.f. arcsinh(x)/cos(x) (odd powers only).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A302542 Expansion of e.g.f. arctan(x)/cos(x) (odd powers only).

Views

A172396 G.f. satisfies: A(x) = G(x/A(x)) where o.g.f. G(x) = A(xG(x)) = Sum_{n>=0} A003701(n)x^n.

A081658 Triangle read by rows: T(n, k) = (-2)^kbinomial(n, k)Euler(k, 1/2).