A009747 -id:A009747 - cp's OEIS frontend

A109449 Triangle read by rows, T(n,k) = binomial(n,k)*A000111(n-k), 0 <= k <= n.

1, 1, 1, 1, 2, 1, 2, 3, 3, 1, 5, 8, 6, 4, 1, 16, 25, 20, 10, 5, 1, 61, 96, 75, 40, 15, 6, 1, 272, 427, 336, 175, 70, 21, 7, 1, 1385, 2176, 1708, 896, 350, 112, 28, 8, 1, 7936, 12465, 9792, 5124, 2016, 630, 168, 36, 9, 1, 50521, 79360, 62325, 32640, 12810, 4032, 1050, 240, 45, 10, 1

Offset: 0

Philippe Deléham, Aug 27 2005

The boustrophedon transform {t} of a sequence {s} is given by t_n = Sum_{k=0..n} T(n,k)*s(k). Triangle may be called the boustrophedon triangle.
The 'signed version' of the triangle is the exponential Riordan array [sech(x) + tanh(x), x]. - Peter Luschny, Jan 24 2009
Up to signs, the matrix is self-inverse: T^(-1)(n,k) = (-1)^(n+k)*T(n,k). - R. J. Mathar, Mar 15 2013

			Triangle starts:
      1;
      1,     1;
      1,     2,     1;
      2,     3,     3,     1;
      5,     8,     6,     4,     1;
     16,    25,    20,    10,     5,    1;
     61,    96,    75,    40,    15,    6,    1;
    272,   427,   336,   175,    70,   21,    7,   1;
   1385,  2176,  1708,   896,   350,  112,   28,   8,  1;
   7936, 12465,  9792,  5124,  2016,  630,  168,  36,  9,  1;
  50521, 79360, 62325, 32640, 12810, 4032, 1050, 240, 45, 10, 1; ...

Reinhard Zumkeller, Rows n = 0..125 of table, flattened
Peter Luschny, The Swiss-Knife polynomials.
J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A 44-54 1996 (Abstract, pdf, ps).
Wikipedia, Boustrophedon transform
Index entries for sequences related to boustrophedon transform

Cf. A000111, A000667, A000795, A002084, A003719, A007318, A009747.
See also : A000182, A000964, A009739, A062161, A062272.
Cf. A153641, A162660.
Cf. A000667 (row sums), A247453.

a109449 n k = a109449_row n !! k
a109449_row n = zipWith (*)
                (a007318_row n) (reverse $ take (n + 1) a000111_list)
a109449_tabl = map a109449_row [0..]
-- Reinhard Zumkeller, Nov 02 2013

f:= func< n,x | Evaluate(BernoulliPolynomial(n+1), x) >;
A109449:= func< n,k | k eq n select 1 else 2^(2*n-2*k+1)*Binomial(n,k)*Abs(f(n-k,3/4) - f(n-k,1/4) + f(n-k,1) - f(n-k,1/2))/(n-k+1) >;
[A109449(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Jul 10 2025

From Peter Luschny, Jul 10 2009, edited Jun 06 2022: (Start)
A109449 := (n,k) -> binomial(n, k)*A000111(n-k):
seq(print(seq(A109449(n, k), k=0..n)), n=0..9);
B109449 := (n,k) -> 2^(n-k)*binomial(n, k)*abs(euler(n-k, 1/2)+euler(n-k, 1)) -`if`(n-k=0, 1, 0): seq(print(seq(B109449(n, k), k=0..n)), n=0..9);
R109449 := proc(n, k) option remember; if k = 0 then A000111(n) else R109449(n-1, k-1)*n/k fi end: seq(print(seq(R109449(n, k), k=0..n)), n=0..9);
E109449 := proc(n) add(binomial(n, k)*euler(k)*((x+1)^(n-k)+ x^(n-k)), k=0..n) -x^n end: seq(print(seq(abs(coeff(E109449(n), x, k)), k=0..n)), n=0..9);
sigma := n -> ifelse(n=0, 1, [1,1,0,-1,-1,-1,0,1][n mod 8 + 1]/2^iquo(n-1,2)-1):
L109449 := proc(n) add(add((-1)^v*binomial(k, v)*(x+v+1)^n*sigma(k), v=0..k), k=0..n) end: seq(print(seq(abs(coeff(L109449(n), x, k)), k=0..n)), n=0..9);
X109449 := n -> n!*coeff(series(exp(x*t)*(sech(t)+tanh(t)), t, 24), t, n): seq(print(seq(abs(coeff(X109449(n), x, k)), k=0..n)), n=0..9);
(End)

lim = 10; s = CoefficientList[Series[(1 + Sin[x])/Cos[x], {x, 0, lim}], x] Table[k!, {k, 0, lim}]; Table[Binomial[n, k] s[[n - k + 1]], {n, 0, lim}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 24 2015, after Jean-François Alcover at A000111 *)
T[n_, k_] := (n!/k!) SeriesCoefficient[(1 + Sin[x])/Cos[x], {x, 0, n - k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 27 2019 *)

A109449(n,k)=binomial(n,k)*if(n>k,2*abs(polylog(k-n,I)),1) \\ M. F. Hasler, Oct 05 2017

R = PolynomialRing(ZZ, 'x')
@CachedFunction
def skp(n, x) :
    if n == 0 : return 1
    return add(skp(k, 0)*binomial(n, k)*(x^(n-k)-(n+1)%2) for k in range(n)[::2])
def A109449_row(n):
    x = R.gen()
    return [abs(c) for c in list(skp(n,x)-skp(n,x-1)+x^n)]
for n in (0..10) : print(A109449_row(n)) # Peter Luschny, Jul 22 2012

Sum_{k>=0} T(n, k) = A000667(n).
Sum_{k>=0} T(2n, 2k) = A000795(n).
Sum_{k>=0} T(2n, 2k+1) = A009747(n).
Sum_{k>=0} T(2n+1, 2k) = A003719(n).
Sum_{k>=0} T(2n+1, 2k+1) = A002084(n).
Sum_{k>=0} T(n, 2k) = A062272(n).
Sum_{k>=0} T(n, 2k+1) = A062161(n).
Sum_{k>=0} (-1)^(k)*T(n, k) = A062162(n). - Johannes W. Meijer, Apr 20 2011
E.g.f.: exp(x*y)*(sec(x)+tan(x)). - Vladeta Jovovic, May 20 2007
T(n,k) = 2^(n-k)*C(n,k)*|E(n-k,1/2) + E(n-k,1)| - [n=k] where C(n,k) is the binomial coefficient, E(m,x) are the Euler polynomials and [] the Iverson bracket. - Peter Luschny, Jan 24 2009
From Reikku Kulon, Feb 26 2009: (Start)
A109449(n, 0) = A000111(n), approx. round(2^(n + 2) * n! / Pi^(n + 1)).
A109449(n, n - 1) = n.
A109449(n, n) = 1.
For n > 0, k > 0: A109449(n, k) = A109449(n - 1, k - 1) * n / k. (End)
From Peter Luschny, Jul 10 2009: (Start)
Let p_n(x) = Sum_{k=0..n} Sum_{v=0..k} (-1)^v C(k,v)*F(k)*(x+v+1)^n, where F(0)=1 and for k>0 F(k)=-1 + s_k 2^floor((k-1)/2), s_k is 0 if k mod 8 in {2,6}, 1 if k mod 8 in {0,1,7} and otherwise -1. T(n,k) are the absolute values of the coefficients of these polynomials.
Another way to express the polynomials p_n(x) is
p_n(x) = -x^n + Sum_{k=0..n} binomial(n,k)*Euler(k)((x+1)^(n-k) + x^(n-k)). (End)
From Peter Bala, Jan 26 2011: (Start)
An explicit formula for the n-th row polynomial is
x^n + i*Sum_{k=1..n}((1+i)/2)^(k-1)*Sum_{j=0..k} (-1)^j*binomial(k,j)*(x+i*j)^n, where i = sqrt(-1). This is the triangle of connection constants between the polynomial sequences {Z(n,x+1)} and {Z(n,x)}, where Z(n,x) denotes the zigzag polynomials described in A147309.
Denote the present array by M. The first column of the array (I-x*M)^-1 is a sequence of rational functions in x whose numerator polynomials are the row polynomials of A145876 - the generalized Eulerian numbers associated with the zigzag numbers. (End)
Let skp{n}(x) denote the Swiss-Knife polynomials A153641. Then
T(n,k) = [x^(n-k)] |skp{n}(x) - skp{n}(x-1) + x^n|. - Peter Luschny, Jul 22 2012
T(n,k) = A007318(n,k) * A000111(n - k), k = 0..n. - Reinhard Zumkeller, Nov 02 2013
T(n,k) = abs(A247453(n,k)). - Reinhard Zumkeller, Sep 17 2014

Edited, formula corrected, typo T(9,4)=2016 (before 2816) fixed by Peter Luschny, Jul 10 2009

A000667 Boustrophedon transform of all-1's sequence.

Original entry on oeis.org

1, 2, 4, 9, 24, 77, 294, 1309, 6664, 38177, 243034, 1701909, 13001604, 107601977, 959021574, 9157981309, 93282431344, 1009552482977, 11568619292914, 139931423833509, 1781662223749884, 23819069385695177, 333601191667149054, 4884673638115922509

Offset: 0

N. J. A. Sloane and Simon Plouffe

Fill in a triangle, like Pascal's triangle, beginning each row with a 1 and filling in rows alternately right to left and left to right.

Row sums of triangle A109449. - Reinhard Zumkeller, Nov 04 2013

			...............1..............
............1..->..2..........
.........4..<-.3...<-..1......
......1..->.5..->..8...->..9..

Alois P. Heinz, Table of n, a(n) for n = 0..485 (first 101 terms from T. D. Noe)
C. K. Cook, M. R. Bacon, and R. A. Hillman, Higher-order Boustrophedon transforms for certain well-known sequences, Fib. Q., 55(3) (2017), 201-208.
Peter Luschny, An old operation on sequences: the Seidel transform.
J. Millar, N. J. A. Sloane, and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory Ser. A, 76(1) (1996), 44-54 (Abstract, pdf, ps).
J. Millar, N. J. A. Sloane, and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory Ser. A, 76(1) (1996), 44-54.
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [USA access only through the HATHI TRUST Digital Library]
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [Access through ZOBODAT]
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, Transforms.
Wikipedia, Boustrophedon transform.
Index entries for sequences related to boustrophedon transform.

Absolute value of pairwise sums of A009337.
Column k=1 of A292975.
Cf. A000111, A000795, A009747, A002084, A003719, A109449, A227862.

a000667 n = if x == 1 then last xs else x
            where xs@(x:_) = a227862_row n
-- Reinhard Zumkeller, Nov 01 2013

With[{nn=30},CoefficientList[Series[Exp[x](Tan[x]+Sec[x]),{x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Nov 28 2011 *)
t[, 0] = 1; t[n, k_] := t[n, k] = t[n, k-1] + t[n-1, n-k];
a[n_] := t[n, n];
Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *)

x='x+O('x^33); Vec(serlaplace( exp(x)*(tan(x) + 1/cos(x)) ) ) \\ Joerg Arndt, Jul 30 2016

from itertools import islice, accumulate
def A000667_gen(): # generator of terms
    blist = tuple()
    while True:
        yield (blist := tuple(accumulate(reversed(blist),initial=1)))[-1]
A000667_list = list(islice(A000667_gen(),20)) # Chai Wah Wu, Jun 11 2022

# Algorithm of L. Seidel (1877)
def A000667_list(n) :
    R = []; A = {-1:0, 0:0}
    k = 0; e = 1
    for i in range(n) :
        Am = 1
        A[k + e] = 0
        e = -e
        for j in (0..i) :
            Am += A[k]
            A[k] = Am
            k += e
        # print [A[z] for z in (-i//2..i//2)]
        R.append(A[e*i//2])
    return R
A000667_list(10)  # Peter Luschny, Jun 02 2012

E.g.f.: exp(x) * (tan(x) + sec(x)).
Limit_{n->infinity} 2*n*a(n-1)/a(n) = Pi; lim_{n->infinity} a(n)*a(n-2)/a(n-1)^2 = 1 + 1/(n-1). - Gerald McGarvey, Aug 13 2004
a(n) = Sum_{k=0..n} binomial(n, k)*A000111(n-k). a(2*n) = A000795(n) + A009747(n), a(2*n+1) = A002084(n) + A003719(n). - Philippe Deléham, Aug 28 2005
a(n) = A227862(n, n * (n mod 2)). - Reinhard Zumkeller, Nov 01 2013
G.f.: E(0)*x/(1-x)/(1-2*x) + 1/(1-x), where E(k) = 1 - x^2*(k + 1)*(k + 2)/(x^2*(k + 1)*(k + 2) - 2*(x*(k + 2) - 1)*(x*(k + 3) - 1)/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 16 2014
a(n) ~ n! * exp(Pi/2) * 2^(n+2) / Pi^(n+1). - Vaclav Kotesovec, Jun 12 2015

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

A062161 Boustrophedon transform of n mod 2.

Original entry on oeis.org

0, 1, 2, 4, 12, 36, 142, 624, 3192, 18256, 116282, 814144, 6219972, 51475776, 458790022, 4381112064, 44625674352, 482962852096, 5534347077362, 66942218896384, 852334810990332, 11394877025289216, 159592488559874302, 2336793875186479104, 35703580441464231912

Offset: 0

Frank Ellermann, Jun 10 2001

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
Peter Luschny, An old operation on sequences: the Seidel transform
J. Millar, N. J. A. Sloane, and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A 44-54 1996 (Abstract, pdf, ps).
Wikipedia, Boustrophedon transform.
Index entries for sequences related to boustrophedon transform.

Cf. A000035, A062272.

a062161 n = sum $ zipWith (*) (a109449_row n) $ cycle [0,1]
-- Reinhard Zumkeller, Nov 03 2013

With[{nn=30},CoefficientList[Series[(Sec[x]+Tan[x])Sinh[x],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Feb 16 2013 *)

from itertools import accumulate, islice
def A062161_gen(): # generator of terms
    blist, m = tuple(), 1
    while True:
        yield (blist := tuple(accumulate(reversed(blist),initial=(m := 1-m))))[-1]
A062161_list = list(islice(A062161_gen(),40)) # Chai Wah Wu, Jun 12 2022

# Generalized algorithm of L. Seidel (1877)
def A062161_list(n) :
    R = []; A = {-1:0, 0:0}
    k = 0; e = 1
    for i in range(n) :
        Am = 1 if e == -1 else 0
        A[k + e] = 0
        e = -e
        for j in (0..i) :
            Am += A[k]
            A[k] = Am
            k += e
        # print [A[z] for z in (-i//2..i//2)]
        R.append(A[e*i//2])
    return R
A062161_list(10) # Peter Luschny, Jun 02 2012

a(2n) = A009747(n), a(2n+1) = A002084(n).
E.g.f.: (sec(x)+tan(x))*sinh(x); a(n)=(A000667(n)-A062162(n))/2. - Paul Barry, Jan 21 2005
a(n) = Sum{k, k>=0} binomial(n, 2k+1)*A000111(n-2k-1). - Philippe Deléham, Aug 28 2005
a(n) = Sum_{k=0..n} A109449(n,k) * (k mod 2). - Reinhard Zumkeller, Nov 03 2013 [corrected by Jason Yuen, Jan 07 2025]

A087800 a(n) = 12*a(n-1) - a(n-2), with a(0) = 2 and a(1) = 12.

Original entry on oeis.org

2, 12, 142, 1692, 20162, 240252, 2862862, 34114092, 406506242, 4843960812, 57721023502, 687808321212, 8195978831042, 97663937651292, 1163771272984462, 13867591338162252, 165247324784962562, 1969100306081388492

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Oct 11 2003

a(n+1)/a(n) converges to (6+sqrt(35)) = 11.9160797... a(0)/a(1)=2/12; a(1)/a(2)=12/142; a(2)/a(3)=142/1692; a(3)/a(4)=1692/20162; ... etc. Lim_{n->infinity} a(n)/a(n+1) = 0.0839202... = 1/(6+sqrt(35)) = (6-sqrt(35)).

Except for the first term, positive values of x (or y) satisfying x^2 - 12xy + y^2 + 140 = 0. - Colin Barker, Feb 25 2014

			a(4) = 20162 = 12a(3) - a(2) = 12*1692 - 142 = (6+sqrt(35))^4 + (6-sqrt(35))^4 = 20161.9999504 + 0.00004959 = 20162.
G.f. = 2 + 12*x + 142*x^2 + 1692*x^3 + 20162*x^4 + 240252*x^5 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for linear recurrences with constant coefficients, signature (12,-1).

Cf. A009747, A086928, A001927, A023038.

I:=[2,12]; [n le 2 select I[n] else 12*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Nov 07 2018

a[0] = 2; a[1] = 12; a[n_] := 12a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 17}] (* Robert G. Wilson v, Jan 30 2004 *)
CoefficientList[Series[(2 - 12 x)/(1 - 12 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 28 2014 *)
a[ n_] := 2 ChebyshevT[ n, 6]; (* Michael Somos, May 28 2014 *)
LinearRecurrence[{12,-1},{2,12},20] (* Harvey P. Dale, Jan 29 2019 *)

Vec((2-12*x)/(1-12*x+x^2) + O(x^100)) \\ Colin Barker, Feb 25 2014

{a(n) = 2 * polchebyshev( n, 1, 6)}; /* Michael Somos, May 28 2014 */

[lucas_number2(n,12,1) for n in range(1,20)] # Zerinvary Lajos, Jun 25 2008

a(n) = (6+sqrt(35))^n + (6-sqrt(35))^n.
a(n) = 2*A023038(n).
G.f.: (2-12*x)/(1-12*x+x^2). - Philippe Deléham, Nov 17 2008
a(-n) = a(n). - Michael Somos, May 28 2014

A296462 Expansion of e.g.f. arcsin(x)*arctanh(x) (even powers only).

Original entry on oeis.org

0, 2, 12, 238, 9912, 708282, 77392260, 12002011110, 2507167177200, 678724656721650, 231129344455890300, 96694934804540934750, 48752132066414189721000, 29154453671147281799726250, 20403607225475633039372992500, 16520371586328834323725749873750, 15322889489994265975004588078700000

Offset: 0

Ilya Gutkovskiy, Dec 13 2017

nonn

			arcsin(x)*arctanh(x) = 2*x^2/2! + 12*x^4/4! + 238*x^6/6! + 9912*x^8/8! + 708282*x^10/10! + ...

Cf. A001818, A009744, A009747, A010050, A012723, A296463.

nmax = 16; Table[(CoefficientList[Series[ArcSin[x] ArcTanh[x], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
nmax = 16; Table[(CoefficientList[Series[I (Log[1 - x] - Log[1 + x]) Log[I x + Sqrt[1 - x^2]]/2, {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

E.g.f.: arcsinh(x)*arctan(x) (even powers only, absolute values).
E.g.f.: i*(log(1 - x) - log(1 + x))*log(i*x + sqrt(1 - x^2))/2, where i is the imaginary unit (even powers only).
a(n) ~ Pi * (2*n-1)! / 2. - Vaclav Kotesovec, Dec 13 2017

A024272 E.g.f. tan(x)*sinh(x)/2 (even powers only).

Original entry on oeis.org

0, 1, 6, 71, 1596, 58141, 3109986, 229395011, 22312837176, 2767173538681, 426167405495166, 79796244279937151, 17851790220732115956, 4702787739658825158421, 1440911869083478804851546, 508062238427253843822090491, 204262590490127231070131373936, 92884219961086104169154295141361

Offset: 0

R. H. Hardin

nonn

G. C. Greubel, Table of n, a(n) for n = 0..240

A009747.

With[{nn = 50}, Take[CoefficientList[Series[Tan[x]*Sinh[x]/2, {x, 0, nn}], x] Range[0, nn]!, {1, -1, 2}]] (* G. C. Greubel, Apr 12 2017 *)

x='x+O('x^66); v=Vec(serlaplace(tan(x)*sinh(x)/2)); concat([0],vector(#v\2,n,v[2*n-1])) \\ Joerg Arndt, Apr 26 2013

Extended and signs tested 03/97.

A296463 Expansion of e.g.f. arcsinh(x)*arctanh(x) (even powers only).

Original entry on oeis.org

0, 2, 4, 158, 3624, 427482, 29665260, 6948032310, 991515848400, 383952670412850, 93532380775766100, 53913667654307868750, 20087427376748637675000, 16096655588343149442026250, 8531309209053208518037597500, 9057367559484733295974741323750, 6486329752640392315697926589700000

Offset: 0

Ilya Gutkovskiy, Dec 13 2017

nonn

			arcsinh(x)*arctanh(x) = 2*x^2/2! + 4*x^4/4! + 158*x^6/6! + 3624*x^8/8! + 427482*x^10/10! + ..

Cf. A001818, A009744, A009747, A010050, A012752, A296462.

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

E.g.f.: arcsin(x)*arctan(x) (even powers only, absolute values).
E.g.f.: (log(1 + x) - log(1 - x))*log(x + sqrt(1 + x^2))/2 (even powers only).
a(n) ~ (2*n-1)! * log(1+sqrt(2)) * (1 - (-1)^n * sqrt(Pi) / (4 * log(1+sqrt(2)) * sqrt(n))). - Vaclav Kotesovec, Dec 13 2017

cp's OEIS Frontend

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A296462 Expansion of e.g.f. arcsin(x)*arctanh(x) (even powers only).

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