A000364 -id:A000364 - cp's OEIS frontend

A241171 Triangle read by rows: Joffe's central differences of zero, T(n,k), 1 <= k <= n.

1, 1, 6, 1, 30, 90, 1, 126, 1260, 2520, 1, 510, 13230, 75600, 113400, 1, 2046, 126720, 1580040, 6237000, 7484400, 1, 8190, 1171170, 28828800, 227026800, 681080400, 681080400, 1, 32766, 10663380, 494053560, 6972966000, 39502663200, 95351256000, 81729648000, 1, 131070, 96461910, 8203431600, 196556560200, 1882311631200, 8266953895200, 16672848192000, 12504636144000

Offset: 1

N. J. A. Sloane, Apr 22 2014

T(n,k) gives the number of ordered set partitions of the set {1,2,...,2*n} into k even sized blocks. An example is given below. Cf. A019538 and A156289. - Peter Bala, Aug 20 2014

			Triangle begins:
1,
1, 6,
1, 30, 90,
1, 126, 1260, 2520,
1, 510, 13230, 75600, 113400,
1, 2046, 126720, 1580040, 6237000, 7484400,
1, 8190, 1171170, 28828800, 227026800, 681080400, 681080400,
1, 32766, 10663380, 494053560, 6972966000, 39502663200, 95351256000, 81729648000,
...
From _Peter Bala_, Aug 20 2014: (Start)
Row 2: [1,6]
k  Ordered set partitions of {1,2,3,4} into k blocks    Number
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1   {1,2,3,4}                                             1
2   {1,2}{3,4}, {3,4}{1,2}, {1,3}{2,4}, {2,4}{1,3},       6
    {1,4}{2,3}, {2,3}{1,4}
(End)

H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 283.
S. A. Joffe, Calculation of the first thirty-two Eulerian numbers from central differences of zero, Quart. J. Pure Appl. Math. 47 (1914), 103-126.
S. A. Joffe, Calculation of eighteen more, fifty in all, Eulerian numbers from central differences of zero, Quart. J. Pure Appl. Math. 48 (1917-1920), 193-271.

Muniru A Asiru, Rows n = 1..50 of triangle, flattened

Case m=2 of the polynomials defined in A278073.
Cf. A000680 (diagonal), A094088 (row sums), A000364 (alternating row sums), A281478 (central terms), A327022 (refinement).
Diagonals give A002446, A213455, A241172, A002456.
Cf. A019538, A036969, A156289.

Flat(List([1..10],n->List([1..n],k->1/(2^(k-1))*Sum([1..k],j->(-1)^(k-j)*Binomial(2*k,k-j)*j^(2*n))))); # Muniru A Asiru, Feb 27 2019

T := proc(n,k) option remember;
if k > n then 0
elif k=0 then k^n
elif k=1 then 1
else k*(2*k-1)*T(n-1,k-1)+k^2*T(n-1,k); fi;
end: # Minor edit to make it also work in the (0,0)-offset case. Peter Luschny, Sep 03 2022
for n from 1 to 12 do lprint([seq(T(n,k), k=1..n)]); od:

T[n_, k_] /; 1 <= k <= n := T[n, k] = k(2k-1) T[n-1, k-1] + k^2 T[n-1, k]; T[, 1] = 1; T[, ] = 0; Table[T[n, k], {n, 1, 9}, {k, 1, n}] (* _Jean-François Alcover, Jul 03 2019 *)

@cached_function
def A241171(n, k):
    if n == 0 and k == 0: return 1
    if k < 0 or k > n: return 0
    return (2*k^2 - k)*A241171(n - 1, k - 1) + k^2*A241171(n - 1, k)
for n in (1..6): print([A241171(n, k) for k in (1..n)]) # Peter Luschny, Sep 06 2017

T(n,k) = 0 if k <= 0 or k > n, = 1 if k=1, otherwise T(n,k) = k*(2*k-1)*T(n-1,k-1) + k^2*T(n-1,k).
Related to Euler numbers A000364 by A000364(n) = (-1)^n*Sum_{k=1..n} (-1)^k*T(n,k). For example, A000364(3) = 61 = 90 - 30 + 1.
From Peter Bala, Aug 20 2014: (Start)
T(n,k) = 1/(2^(k-1))*Sum_{j = 1..k} (-1)^(k-j)*binomial(2*k,k-j)*j^(2*n).
T(n,k) = k!*A156289(n,k) = k!*(2*k-1)!!*A036969.
E.g.f.: A(t,z) := 1/( 1 - t*(cosh(z) - 1) ) = 1 + t*z^2/2! + (t + 6*t^2)*z^4/4! + (t + 30*t^2 + 90*t^3)*z^6/6! + ... satisfies the partial differential equation d^2/dz^2(A) = D(A), where D = t^2*(2*t + 1)*d^2/dt^2 + t*(5*t + 1)*d/dt + t.
Hence the row polynomials R(n,t) satisfy the differential equation R(n+1,t) = t^2*(2*t + 1)*R''(n,t) + t*(5*t + 1)*R'(n,t) + t*R(n,t) with R(0,t) = 1, where ' indicates differentiation w.r.t. t. This is equivalent to the above recurrence equation.
Recurrence for row polynomials: R(n,t) = t*( Sum_{k = 1..n} binomial(2*n,2*k)*R(n-k,t) ) with R(0,t) := 1.
Row sums equal A094088(n) for n >= 1.
A100872(n) = (1/2)*R(n,2). (End)

A002439 Glaisher's T numbers.

Original entry on oeis.org

1, 23, 1681, 257543, 67637281, 27138236663, 15442193173681, 11828536957233383, 11735529528739490881, 14639678925928297567703, 22427641105413135505628881, 41393949926819051111431239623, 90592214447886493688036507587681, 231969423543894989257690172433129143

Offset: 0

N. J. A. Sloane

Kashaev's invariant for the (3,2)-torus knot. See Hikami 2003. For other Kashaev invariants see A208679, A208680, and A208681. - Peter Bala, Mar 01 2012
From Peter Bala, Dec 18 2021: (Start)
Glaisher's T numbers occur in the evaluation of the L-function L(X_12,s) := Sum_{k >= 1} X_12(k)/k^s for positive even values of s, where X_12(n) = A110161(n) is a nonprincipal Dirichlet character mod 12: the result is L(X_12,2*n+2) = a(n)/(6*sqrt(3)*36^n*(2*n+1)!) * Pi^(2*n+2).
We make the following conjectures:
1) Taking the sequence modulo an integer k gives an eventually periodic sequence with period dividing phi(k). For example, the sequence taken modulo 50 begins [1, 23, 31, 43, 31, 13, 31, 33, 31, 3, 31, 23, 31, 43, 31, 13, 31, 33, 31, 3, 31, 23, ...] and appears to have a pre-period of length 1 and a period of length 10 = (1/2)*phi(50).
2) Let i >= 0 and define a_i(n) = a(n+i). Then for each i the Gauss congruences a_i(n*p^k) == a_i(n*p^(k-1)) ( mod p^k ) hold for all prime p and positive integers n and k.
If true, then for each i the expansion of exp( Sum_{n >= 1} a_i(n)*x^n/n ) has integer coefficients.
3)(i)   a(m*n) == a(m)^n (mod 2^k) for k = 2*v_2(m) + 7, where v_p(i) denotes the p-adic valuation of i.
  (ii)  a(m*n) == a(m)^n (mod 3^k) for k = 2*v_3(m) + 2.
4)(i)   a(2*m*n) == a(n)^(2*m) (mod 2^k) for k = v_2(m) + 7
  (ii)  a((2*m+1)*n) == a(n)^(2*m+1) (mod 2^k) for k = v_2(m) + 7.
5)(i)   a(3*m*n) == a(n)^(3*m) (mod 3^k) for k = v_3(m) + 2
  (ii)  a((3*m+1)*n) == a(n)^(3*m+1) (mod 3^k) for k = v_3(m) + 2
  (iii) a((3*m+2)*n) == a(n)^(3*m+2) (mod 3^2).
6) For prime p >= 5, a((p-1)/2*n*m) == a((p-1)/2*n)^m (mod p^k) for k = v_p(m-1) + 1. (End)

			G.f. = 1 + 23*x + 1681*x^2 +257543*x^3 + 67637281*x^4 + 27138236663*x^5 + ...

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 76.
J. W. L. Glaisher, Messenger of Math., 28 (1898), 36-79, see esp. p. 76.
J. W. L. Glaisher, On the Bernoullian function, Q. J. Pure Appl. Math., 29 (1898), 1-168.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..100
G. E. Andrews, J. Jimenez-Urroz and K. Ono, q-series identities and values of certain L-functions, Duke Math J., Volume 108, No.3 (2001), 395-419.
Peter Bala, Some S-fractions related to the expansions of sin(ax)/cos(bx) and cos(ax)/cos(bx)
J. Bryson, K. Ono, S. Pitman and R. C. Rhoades, Unimodal Sequences and Quantum and Mock Modular Forms, P. 2.
Frank Garvan, Congruences and relations for r-Fishburn numbers, arXiv:1406.5611 [math.NT], 2014.
J. W. L. Glaisher, On a set of coefficients analogous to the Eulerian numbers, Proc. London Math. Soc., 31 (1899), 216-235.
K. Hikami, Volume Conjecture and Asymptotic Expansion of q-Series, Experimental Mathematics Vol. 12, Issue 3 (2003).
Michael E. Hoffman, Derivative Polynomials, Euler Polynomials, and Associated Integer Sequences, The Electronic Journal of Combinatorics, Volume 6.1 (1999): Research paper R21, 13 p.
Hsien-Kuei Hwang and Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.
J. C. P. Miller, Letter to N. J. A. Sloane, Mar 26 1971
Hjalmar Rosengren, Elliptic pfaffians and solvable lattice models, arXiv preprint arXiv:1605.02915 [math-ph], 2016.
A. Vieru, Agoh's conjecture: its proof, its generalizations, its analogues, arXiv preprint arXiv:1107.2938 [math.NT], 2011.
Don Zagier, Vassiliev invariants and a strange identity related to the Dedekind eta-function, Topology, vol.40, pp.945-960 (2001).
Index entries for sequences related to Glaisher's numbers

Cf. A000364, A000464, A002105, A079144, A158690.
Bisections: A156175, A156176.
Twice this sequence gives A000191. A208679, A208680, A208681.

m:=32; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Sin(2*x)/(2*Cos(3*x)) )); [Factorial(2*n-1)*b[2*n-1]: n in [1..Floor((m-2)/2)]]; // G. C. Greubel, Jul 04 2019

A002439 := proc(n) option remember; if n = 0 then 1; else (-4)^n-add((-9)^k*binomial(2*n+1, 2*k)*procname(n-k), k=1..n+1) ; end if; end proc:

a[n_] := a[n] = (-4)^n - Sum[(-9)^k*Binomial[2n + 1, 2k]*a[n-k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 11}] (* Jean-François Alcover, Dec 05 2011, after Maple *)
With[{nn=30},Take[CoefficientList[Series[Sin[2x]/(2Cos[3x]),{x,0,nn}], x]Range[0,nn-1]!,{2,-1,2}]] (* Harvey P. Dale, Feb 05 2012 *)
a[n_] := -(-4)^n 3^(1 + 2 n) EulerE[1 + 2 n, 1/6]  (* Bill Gosper, Oct 12 2015 *)

{a(n) = my(m=n+1); if( m<2, m>0, (-4)^(m-1) - sum(k=1, m, (-9)^k * binomial(2*m-1, 2*k) * a(n-k)))}; /* Michael Somos, Dec 11 1999 */

m = 32; T = taylor(sin(2*x)/(2*cos(3*x)), x, 0, m); [factorial(2*n+1)*T.coefficient(x, 2*n+1) for n in (0..(m-2)/2)] # G. C. Greubel, Jul 04 2019

Q_{2n+1}(sqrt(3))/sqrt(3), where the polynomials Q_n() are defined in A104035. - N. J. A. Sloane, Nov 06 2009
E.g.f.: sin(2*x)/(2*cos(3*x)) = Sum a(n)*x^(2*n+1)/(2*n+1)!.
With offset 1 instead of 0: a(1)=1, a(n)=(-4)^(n-1) - Sum_{k=1..n} (-9)^k*C(2*n-1, 2*k)*a(n-k).
a(n) = -(-4)^n*3^(2n+1)*E_{2n+1}(1/6), where E is an Euler polynomial. - Bill Gosper, Aug 08 2001, corrected Oct 12 2015.
From Peter Bala, Mar 24 2009: (Start)
Basic hypergeometric generating function: exp(-t)*Sum {n = 0..inf} Product {k = 1..n} (1-exp(-24*k*t)) = 1 + 23*t + 1681*t^2/2! + .... For other sequences with generating functions of a similar type see A000364, A000464, A002105, A079144, A158690.
a(n) = (1/2)*(-1)^(n+1)*L(-2*n-1), where L(s) is a Dirichlet L-function for a Dirichlet character with modulus 12: L(s) = 1 - 1/5^s - 1/7^s + 1/11^s + - - + .... See the Andrew's link. (End)
From Peter Bala, Jan 21 2011: (Start)
Let I = sqrt(-1) and w = exp(2*Pi*I/6). Then
a(n) = I/sqrt(3) *sum {k = 0..2*n+2} w^(n-k) *sum {j = 1..2*n+2} (-1)^(k-j) *binomial(2*n+2,k-j) *(2*j-1)^(2*n+1).
This formula can be used to obtain congruences for a(n). For example, for odd prime p we find a(p-1) = 1 (mod p) and a((p-1)/2) = (-1)^((p-1)/2) (mod p).
Cf. A002437 and A182825. (End)
a(n) = (-1)^n/(4*n+4)*12^(2*n+1)*sum {k = 1..12} X(k)*B(2*n+2,k/12), where B(n,x) is a Bernoulli polynomial and X(n) is a periodic function modulo 12 given by X(n) = 0 except for X(12*n+1) = X(12*n+11) = 1 and X(12*n+5) = X(12*n+7) = -1. - Peter Bala, Mar 01 2012
a(n) ~ n^(2*n+3/2) * 2^(4*n+3) * 3^(2*n+3/2) / (exp(2*n) * Pi^(2*n+3/2)). - Vaclav Kotesovec, Mar 01 2014
From Peter Bala, May 11 2017: (Start)
Let X = 24*x. G.f. A(x) = 1/(1 + x - X/(1 - 2*X/(1 + x - 5*X/(1 - 7*X/(1 + x - 12*X/(1 - ...)))))) = 1 + 23*x + 1681*x^2 + ..., where the sequence [1, 2, 5, 7, 12, ...] of unsigned coefficients in the partial numerators of the continued fraction are generalized pentagonal numbers A001318.
A(x) = 1/(1 + 25*x - 2*X/(1 - X/(1 + 25*x - 7*X/(1 - 5*X/(1 + 25*x - 15*X/(1 - 12*X/(1 + 25*x - 26*X/(1 - 22*X/(1 + 25*x - ...))))))))), where the sequence [2, 1, 7, 5, 15, 12, 26, 22, ...] of unsigned coefficients in the partial numerators is obtained by swapping pairs of adjacent generalized pentagonal numbers.
G.f. as a J-fraction: A(x) = 1/(1 - 23*x - 2*X^2/(1 - 167*x - 5*7*X^2/(1 - 455*x - 12*15*X^2/(1 - 887*x - ...)))).
Let B(x) = 1/(1 - x)*A(x/(1 - x)), that is, B(x) is the binomial transform of A(x). Then B(x/24) is the o.g.f. for A079144. (End)
a(n) == 23^n ( mod (2^7)*(3^2) ). - Peter Bala, Dec 25 2021

More terms from Michael Somos

Offset changed from 1 to 0 by N. J. A. Sloane, Dec 11 1999

A009843 E.g.f. x/cos(x) (odd powers only).

Original entry on oeis.org

1, 3, 25, 427, 12465, 555731, 35135945, 2990414715, 329655706465, 45692713833379, 7777794952988025, 1595024111042171723, 387863354088927172625, 110350957750914345093747, 36315529600705266098580265, 13687860690719716241164167451, 5858139922124796551409938058945

Offset: 0

R. H. Hardin

Expanding x/cosh(x) gives alternated signed values at odd positions.

Related to the formulas sum(k>0,sin(kx)/k^(2n+1))=(-1)^(n+1)/2*x^(2n+1)/(2n+1)!*sum(i=0,2n,(2Pi/x)^i*B(i)*C(2n+1,i)) and if x=Pi/2 sum(k>0,(-1)^(k+1)/k^(2n+1))=(-1)^n*E(2n)*Pi^(2n+1)/2^(2n+2)/(2n)!. - Benoit Cloitre, May 01 2002

			x/cos(x) = x + 1/2*x^3 + 5/24*x^5 + 61/720*x^7 + 277/8064*x^9 + ...

R. P. Brent, Generalising Tuenter's binomial sums, arXiv:1407.3533 [math.CO], 2014.
R. B. Brent, Generalizing Tuenter's Binomial Sums, J. Int. Seq. 18 (2015) # 15.3.2.
Sylvie Corteel, Alexander Lazar, and Anna Vanden Wyngaerd, Decorated square paths at q = -1, arXiv:2408.10640 [math.CO], 2024. See p. 3.
Bishal Deb and Alan D. Sokal, Classical continued fractions for some multivariate polynomials generalizing the Genocchi and median Genocchi numbers, arXiv:2212.07232 [math.CO], 2022. See p. 12.
Peter Luschny, The lost Bernoulli numbers.
Wikipedia, Bernoulli Polynomials

Bisection of A009391, A009392, A065619, A083008.

Cf. A099028. Cf. A002111, A069852, A069994.

seq((2*i+1)!*coeff(series(x/cos(x),x,32),x,2*i+1),i=0..13);
A009843 := n -> (-1)^n*(2*n+1)*euler(2*n): # Peter Luschny

c = CoefficientList[Series[1/MittagLefflerE[2,z^2],{z,0,40}],z]; Table[(-1)^n* Factorial[2*n+1]*c[[2*n+1]], {n,0,16}] (* Peter Luschny, Jul 03 2016 *)

a(n)=(-1)^(n+1)*sum(i=0,2*n+1,binomial(2*n+1,i)*bernfrac(i)*4^i)

a(n)=subst(bernpol(2*n+1),'x,1/4)*4^(2*n+1)*(-1)^(n+1) \\ Charles R Greathouse IV, Dec 10 2014

# The objective of this implementation is efficiency.
# n -> [a(0), a(1), ..., a(n)] for n > 0.
def A009843_list(n):
    S = [0 for i in range(n+1)]
    S[0] = 1
    for k in range(1, n+1):
        S[k] = k*S[k-1]
    for k in range(1, n+1):
        for j in range(k, n+1):
            S[j] = (j-k)*S[j-1]+(j-k+1)*S[j]
        S[k] = (2*k+1)*S[k]
    return S
print(A009843_list(10)) # Peter Luschny, Aug 09 2011

a(n) = (2n+1)*A000364(n) = sum(i=0, 2n, B(i)*C(2n+1, i)*4^i)=(2n+1)*E(2n) where B(i) are the Bernoulli numbers, C(2n, i) the binomial numbers and E(2n) the Euler numbers. - Benoit Cloitre, May 01 2002
Recurrence: a(n) = -(-1)^n*Sum[i=0..n-1, (-1)^i*a(i)*C(2n+1, 2i+1) ]. - Ralf Stephan, Feb 24 2005
a(n) = 4^n |E_{2n}(1/2)+E_{2n}(1)| (2n+1) for n > 0; E_{n}(x) Euler polynomial. - Peter Luschny, Nov 25 2010
a(n) = (2*n+1)! * [x^(2*n+1)] x/cos(x).
From Sergei N. Gladkovskii, Nov 15 2011, Oct 19 2012, Nov 10 2012, Jan 14 2013, Apr 10 2013, Oct 13 2013, Dec 01 2013: (Start) Continued fractions:
E.g.f.: x / cos(x) = x+x^3/Q(0); Q(k) = 8k+2-x^2/(1+(2k+1)*(2k+2)/Q(k+1)).
E.g.f.: x + x^3/U(0) where U(k) = (2*k+1)*(2*k+2) - x^2 + x^2*(2*k+1)*(2*k+2)/U(k+1).
G.f.: 1/G(0) where G(k) = 1 - x*(8*k^2+8*k+3)-16*x^2*(k+1)^4/G(k+1).
E.g.f.: 2*x/(Q(0) + 1) where Q(k)= 1 - x/(2*k+1)/(2*k+2)/(1 - 1/(1 + 1/Q(k+1))).
Let A(x) = S_{n>=0}a(n)*x^n/(2*n+1)! then A(x) = 1 + Q(0)*x/(2-x) where Q(k) = 1 - x*(2*k+1)*(2*k+2)/(x*(2*k+1)*(2*k+2) + ((2*k+1)*(2*k+2) - x)*((2*k+3)*(2*k+4) - x)/Q(k+1)).
G.f.: T(0)/(1-3*x) where T(k) = 1 - 16*x^2*(k+1)^4/(16*x^2*(k+1)^4 - (1 - x*(8*k^2 +8*k+3)) *(1 - x*(8*k^2+24*k+19))/T(k+1)).
G.f.: 1/T(0) where T(k) = 1 + x - x*(2*k+2)^2/(1 - x*(2*k+2)^2/T(k+1)). (End)
a(n) = (-1)^n*2^(4*n+1)*(2*n+1)*(zeta(-2*n,1/4)-zeta(-2*n,3/4)). - Peter Luschny, Jul 22 2013
From Peter Bala, Mar 02 2015: (Start)
a(n) = (-1)^(n+1)*4^(2*n + 1)*B(2*n + 1,1/4), where B(n,x) denotes the n-th Bernoulli polynomial. Cf. A002111, A069852 and A069994.
Conjecturally, a(n) = the unsigned numerator of B(2*n+1,1/4).
G.f. for signed version of sequence: Sum_{n >= 0} { 1/(n + 1) * Sum_{k = 0..n} (-1)^k*binomial(n,k)/( (1 - (4*k + 1)*x)*(1 - (4*k + 3)*x) ) } = 1 - 3*x^2 + 25*x^4 - 427*x^6 + .... (End)
a(n) ~ (2*n+1)! * 2^(2*n+2)/Pi^(2*n+1). - Vaclav Kotesovec, Jul 04 2016
G.f.: 1/(1 + x - 4*x/(1 - 4*x/(1 + x - 16*x/(1 - 16*x/(1 + x - 36*x/(1 - 36*x/(1 + x - ...))))))). Cf. A005439. - Peter Bala, May 07 2017

Extended and signs tested by Olivier Gérard, Mar 15 1997

A349264 Generalized Euler numbers, a(n) = n![x^n](sec(4x)(sin(4x) + 1)).

Original entry on oeis.org

1, 4, 16, 128, 1280, 16384, 249856, 4456448, 90767360, 2080374784, 52975108096, 1483911200768, 45344872202240, 1501108249821184, 53515555843342336, 2044143848640217088, 83285910482761809920, 3605459138582973251584, 165262072909347030040576, 7995891855149741436305408

Offset: 0

Peter Luschny, Nov 20 2021

nonn

			Exponential generating functions of generalized Euler numbers in context:
egf1 = sec(1*x)*(sin(x) + 1).
   [A000111, A000364, A000182]
egf2 = sec(2*x)*(sin(x) + cos(x)).
   [A001586, A000281, A000464]
egf3 = sec(3*x)*(sin(2*x) + cos(x)).
   [A007289, A000436, A000191]
egf4 = sec(4*x)*(sin(4*x) + 1).
   [A349264, A000490, A000318]
egf5 = sec(5*x)*(sin(x) + sin(3*x) + cos(2*x) + cos(4*x)).
   [A349265, A000187, A000320]
egf6 = sec(6*x)*(sin(x) + sin(5*x) + cos(x) + cos(5*x)).
   [A001587, A000192, A000411]
egf7 = sec(7*x)*(-sin(2*x) + sin(4*x) + sin(6*x) + cos(x) + cos(3*x) - cos(5*x)).
   [A349266, A064068, A064072]
egf8 = sec(8*x)*2*(sin(4*x) + cos(4*x)).
   [A349267, A064069, A064073]
egf9 = sec(9*x)*(4*sin(3*x) + 2)*cos(3*x)^2.
   [A349268, A064070, A064074]

Matthew House, Table of n, a(n) for n = 0..389
William Y. C. Chen, Neil J. Y. Fan, Jeffrey Y. T. Jia, The generating function for the Dirichlet series Lm(s), Mathematics of Computation, Vol. 81, No. 278, pp. 1005-1023, April 2012.
Ruth Lawrence and Don Zagier, Modular forms and quantum invariants of 3-manifolds, Asian J. Math. 3 (1999), no. 1, 93-107.
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967) 689-694.
D. Shanks, Corrigendum: Generalized Euler and class numbers, Math. Comp. 22, (1968) 699.
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]

Row 4 of A349271.
Cf. A000111, A001586, A007289, A349265, A001587, A349266, A349267, A349268.
Secant  bisections: A000364, A000281, A000436, A000490, A000187, A000192, A064068, A064069, A064070.
Tangent bisections: A000182, A000464, A000191, A000318, A000320, A000411, A064072, A064073, A064074.
Cf. A225147, A349429.

sec(4*x)*(sin(4*x) + 1): series(%, x, 20): seq(n!*coeff(%, x, n), n = 0..19);

m = 19; CoefficientList[Series[Sec[4*x] * (Sin[4*x] + 1), {x, 0, m}], x] * Range[0, m]! (* Amiram Eldar, Nov 20 2021 *)

seq(n)={my(x='x + O('x^(n+1))); Vec(serlaplace((sin(4*x) + 1)/cos(4*x)))} \\ Andrew Howroyd, Nov 20 2021

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

A000464 Expansion of e.g.f. sin(x)/cos(2*x).

Original entry on oeis.org

1, 11, 361, 24611, 2873041, 512343611, 129570724921, 44110959165011, 19450718635716001, 10784052561125704811, 7342627959965776406281, 6023130568334172003579011, 5858598896811701995459355761, 6667317162352419006959182803611, 8776621742176931117228228227924441

Offset: 0

N. J. A. Sloane

From Peter Bala, Dec 22 2021: (Start)
Conjectures:
1) Taking the sequence (a(n))n>=1 modulo an integer k gives a purely periodic sequence with period dividing phi(k). For example, the sequence taken modulo 21 begins [11, 4, 20, 10, 17, 1, 11, 4, 20, 10, 17, 1, 11, 4, 20, 10, 17, 1, ...] with an apparent period of length 6, which divides phi(21) = 12.
2) For i >= 0, define a_i(n) = a(n+i). Then for each i the Gauss congruences a_i(n*p^k) == a_i(n*p^(k-1)) ( mod p^k ) hold for all prime p and positive integers n and k. If true, then for each i the expansion of exp(Sum_{n >= 1} a_i(n)*x^n/n) has integer coefficients.
3) a(m*n) == a(m)^n (mod 2^k) for k = 2*v_2(m) + 4, where v_p(i) denotes the p-adic valuation of i.
4)(i)   a(2*m*n) == a(n)^(2*m) (mod 2^k) for k = v_2(m) + 4
  (ii)  a((2*m+1)*n) == a(n)^(2*m+1) (mod 2^k) for k = v_2(m) + 4. (End)

H. Cohen, Number Theory - Volume II: Analytic and Modern Tools, Graduate Texts in Mathematics. Springer-Verlag.
J. W. L. Glaisher, "On the coefficients in the expansions of cos x/ cos 2x and sin x/ cos 2x", Quart. J. Pure and Applied Math., 45 (1914), 187-222.
I. J. Schwatt, Intro. to Operations with Series, Chelsea, p. 278.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Matthew House, Table of n, a(n) for n = 0..215 (terms 0..50 from T. D. Noe)
G. E. Andrews, J. Jimenez-Urroz, and K. Ono, q-series identities and values of certain L-functions, Duke Math Jour., Volume 108, No.3 (2001), 395-419.
Peter Bala, Some S-fractions related to the expansions of sin(ax)/cos(bx) and cos(ax)/cos(bx)
D. Dumont, Further triangles of Seidel-Arnold type and continued fractions related to Euler and Springer numbers, Adv. Appl. Math., 16 (1995), 275-296.
Simon Plouffe, Table of n, a(n) for n=0..1023
D. Shanks, Generalized Euler and class numbers. Math. Comp. 21 (1967) 689-694.
D. Shanks, Corrigenda to: "Generalized Euler and class numbers", Math. Comp. 22 (1968), 699
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]
Alan D. Sokal, The Euler and Springer numbers as moment sequences, arXiv:1804.04498 [math.CO], 2018.
A. Vieru, Agoh's conjecture: its proof, its generalizations, its analogues, arXiv preprint arXiv:1107.2938 [math.NT], 2011.

Row 2 of A235606.
Cf. A064073. Bisection of A000822, A001586.
Cf. A000364, A000828, A002105, A002439, A079144, A158690.

a := n -> (-1)^n*2^(6*n+4)*(Zeta(0, -2*n-1, 5/8)-Zeta(0, -2*n-1, 7/8)):
seq(a(n), n=0..12); # Peter Luschny, Oct 15 2015

With[{nn=30},Take[CoefficientList[Series[Sin[x]/Cos[2x],{x,0,nn}],x] Range[0,nn-1]!,{2,-1,2}]] (* Harvey P. Dale, Mar 23 2012 *)
nmax = 15; km0 = 10; d[n_, km_] := Round[(2^(4n-1/2) (2n-1)! Sum[ JacobiSymbol[2, 2k+1]/(2k+1)^(2n), {k, 0, km}])/Pi^(2n)]; dd[km_] := dd[km] = Table[d[n, km], {n, 1, nmax}]; dd[km0]; dd[km = 2*km0]; While[dd[km] != dd[km/2, km = 2*km]]; A000464 = dd[km] (* Jean-François Alcover, Feb 08 2016 *)

a(n)=if(n<0, 0, n+=n+1; n!*polcoeff(sin(x+x*O(x^n))/cos(2*x+x*O(x^n)),n)) /* Michael Somos, Feb 09 2006 */

E.g.f.: Sum_{k>=0} a(k)x^(2k+1)/(2k+1)! = sin(x)/cos(2x).
a(n) = (-1)^n*L(X,-2n+1) where L(X,z) is the Dirichlet L-function L(X,z) = Sum_{k>=0} X(k)/k^z and where X(k) is the Dirichlet character Legendre(k,2) which begins 1,0,-1,0,-1,0,1,0,1,0,-1,0,-1,0,1,0,1,0,-1,0.... - Benoit Cloitre, Mar 22 2009 [This Dirichlet character is A091337. - Jianing Song, Oct 22 2023]
From Peter Bala, Mar 24 2009: (Start)
Basic hypergeometric generating function:
2*exp(-t)*Sum_{n = 0..inf} (Product_{k = 1..n} (1-exp(-16*k*t))/Product_{k = 1..n+1} (1+exp(-(16*k-8)*t))) = 1 + 11*t + 361*t^2/2! + 24611*t^3/3! + .... For other sequences with generating functions of a similar type see A000364, A002105, A002439, A079144 and A158690.
a(n) = (-1)^(n+1)*L(-2*n-1), where L(s) is the Dirichlet L-function L(s) = 1 - 1/3^s - 1/5^s + 1/7^s + - - + ... [Andrews et al., Theorem 5]. (End)
From Peter Bala, Jun 18 2009: (Start)
a(n) = (-1)^n*B_(2*n+2)(X)/(2*n+2), where B_n(X) denotes the X-Bernoulli number with X a Dirichlet character modulus 8 given by X(8*n+1) = X(8*n+7) = 1, X(8*n+3) = X(8*n+5) = -1 and X(2*n) = 0. See A161722 for the values of B_n(X).
For the theory and properties of the generalized Bernoulli numbers B_n(X) and the associated generalized Bernoulli polynomials B_n(X,x) see [Cohen, Section 9.4].
The present sequence also occurs in the evaluation of the finite sum of powers Sum_{i = 0..m-1} {(8*i+1)^n - (8*i+3)^n - (8*i+5)^n + (8*i+7)^n}, n = 1,2,... - see A151751 for details. (End)
G.f. 1/G(0) where G(k) = 1 + x - x*(4*k+3)*(4*k+4)/(1 - (4*k+4)*(4*k+5)*x/G(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Aug 11 2012
G.f.: 1/E(0) where E(k) = 1 - 11*x - 32*x*k*(k+1) - 16*x^2*(k+1)^2*(4*k+3)*(4*k+5)/E(k+1) (continued fraction, 1-step). - Sergei N. Gladkovskii, Sep 17 2012
a(n) ~ (2*n+1)! * 2^(4*n+7/2) / Pi^(2*n+2). - Vaclav Kotesovec, May 03 2014
a(n) = (-1)^n*2^(6*n+4)*(Zeta(-2*n-1,5/8)-Zeta(-2*n-1,7/8)). - Peter Luschny, Oct 15 2015
From Peter Bala, May 11 2017: (Start)
G.f. A(x) = 1 + 11*x + 361*x^2 + ... = 1/(1 + x - 12*x/(1 - 20*x/(1 + x - 56*x/(1 - 72*x/(1 + x - ... - 4*n*(4*n - 1)*x/(1 - 4*n*(4*n + 1)*x/((1 + x) - ...))))))).
A(x) = 1/(1 + 9*x - 20*x/(1 - 12*x/(1 + 9*x - 72*x/(1 - 56*x/(1 + 9*x - ... - 4*n*(4*n + 1)*x/(1 - 4*n*(4*n - 1)*x/(1 + 9*x - ...))))))).
It follows that the first binomial transform of A(x) and the ninth binomial transform of A(x) have continued fractions of Stieltjes-type (S-fractions). (End)
a(n) = (-1)^(n+1)*4^(2*n+1)*E(2*n+1,1/4), where E(n,x) is the n-th Euler polynomial. Cf. A002439. - Peter Bala, Aug 13 2017
From Peter Bala, Dec 04 2021: (Start)
F(x) = exp(x)*(exp(2*x) - 1)/(exp(4*x) + 1) = x - 11*x^3/3! + 361x^5/5! - 24611*x^7/7! + ... is the e.g.f. for the sequence [1, 0, -11, 0, 361, 0, -24611, 0, ...], a signed and aerated version of this sequence.
The binomial transform exp(x)*F(x) = x + 2*x^2/2! - 8*x^3/3! - 40*x^4/4! + + - - is an e.g.f. for a signed version of A000828 (omitting the initial term). (End)
From Peter Bala, Dec 22 2021: (Start)
a(1) = 1, a(n) = (-1)^(n-1) - Sum_{k = 1..n} (-4)^k*C(2*n-1,2*k)*a(n-k).
a(n) == 1 (mod 10); a(5*n+1) == 0 mod(11);
a(n) == - 23^(n+1) (mod 108); a(n) == (7^2)*59^n (mod 144);
a(n) == 11^n (mod 240); a(n) == (11^2)*131^n (mod 360). (End)

Better description, new reference, Aug 15 1995

A010686 Periodic sequence: repeat [1, 5].

Original entry on oeis.org

1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1

Offset: 0

N. J. A. Sloane

Also continued fraction expansion of (5+3*sqrt(5))/10. - Bruno Berselli, Sep 30 2011
From Gary Detlefs, May 19 2014: (Start)
This sequence can be generated by an infinite number of formulas all having the form a^(b*n) mod c subject to the following conditions. The number a is congruent to either 5,11,13,17,21, or 23 mod 24 and b is of the form 2k+1.
  1. If a = 5 mod 6 then c = 6.
  2. If a = 5 mod 8 then c = 8.
  3. If a = 5 mod 12 then c = 12.
  4. If a = 5 mod 24 then c = 24.
For example: a(n)= 13^(5*n) mod 8, a(n)= 29^(7*n) mod c where c is any number in {6,8,12,24}. (End)
Decimal expansion of 5/33. - Stefano Spezia, Feb 09 2025

			0.15151515151515151515151515151515151515151...

Burkard Polster, Juggling, maths and a beautiful mind. [From Parthasarathy Nambi, Nov 20 2009]
Vladimir Ivanovich Smirnov, A course of higher mathematics, vol. 1 , Pergamon Press, 1964, p. 2.
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A000364.

[seq (modp((4*n+1),8),n=0..80)]; # Zerinvary Lajos, Dec 01 2006

PadRight[{},120,{1,5}] (* Harvey P. Dale, Aug 19 2012 *)

A010686(n):=if evenp(n) then 1 else 5$
makelist(A010686(n),n,0,30); /* Martin Ettl, Nov 09 2012 */

a(n)=n%2*4+1 \\ Charles R Greathouse IV, Sep 24 2015

[pow(5,n,12) for n in range(51)] # Zerinvary Lajos, Nov 25 2009

From Paul Barry, Jun 03 2003: (Start)
G.f.: (1+5*x)/((1-x)*(1+x)).
E.g.f.: 3*exp(x)-2*exp(-x).
a(n) = 3-2(-1)^n.
a(n) = 5^((1-(-1)^n)/2) = 5^(ceiling(n/2)-floor(n/2)). (End)
a(n) = 5^n mod 24. - Paul Curtz, Jan 09 2008
a(n) = 5^n mod 12. - Zerinvary Lajos, Nov 25 2009
a(n) = A000364(n+1) mod 10. - Paul Curtz, Feb 09 2010
a(n) = 11^n mod 6. - Vincenzo Librandi, Jun 01 2016

Definition rewritten by Bruno Berselli, Sep 30 2011

A000795 Salié numbers: expansion of cosh x / cos x = Sum_{n >= 0} a(n)*x^(2n)/(2n)!.

Original entry on oeis.org

1, 2, 12, 152, 3472, 126752, 6781632, 500231552, 48656756992, 6034272215552, 929327412759552, 174008703107274752, 38928735228629389312, 10255194381004799025152, 3142142941901073853366272, 1107912434323301224813002752, 445427836895850552387642130432

Offset: 0

N. J. A. Sloane

Named after the German mathematician Hans Salié (1902-1978). - Amiram Eldar, Jun 10 2021

			cosh x / cos x = Sum_{n>=0} a(n)*x^(2n)/(2n)! = 1 + x^2 + (1/2)*x^4 + (19/90)*x^6 + (31/360)*x^8 + (3961/113400)*x^10 + ...
G.f. = 1 + 2*x + 12*x^2 + 252*x^3 + 3472*x^4 + 126752*x^5 + 6781632*x^6 + ...

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 86, Problem 32.
Hans Salié, Arithmetische Eigenschaften der Koeffizienten einer speziellen Hurwitzschen Potenzreihe, Wiss. Z. Karl-Marx-Univ. Leipzig Math.-Natur. Reihe 12 (1963), pp. 617-618.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
Peter Bala, A triangle for calculating A000795, 2017.
L. Carlitz, The coefficients of cosh x/ cos x, Monatshefte für Mathematik, Vol. 69, No. 2 (1965), pp. 129-135.
Timothy Chow and Richard Stanley, Salié permutations and fair permutations, MathOverflow, 2012.
Marc Deléglise and Jean-Louis Nicolas, On the Largest Product of Primes with Bounded Sum, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.8.
J. M. Gandhi, The coefficients of cosh x/ cos x and a note on Carlitz's coefficients of sinh x / sin x, Math. Magazine, Vol. 31, No. 4 (1958), pp. 185-191..
J. M. Gandhi and V. S. Taneja, The coefficients of cosh x / cos x, Fib. Quart., Vol. 10, No. 4 (1972), pp. 349-353.
M. S. Krick, On the coefficients of cosh x / cos x, Math. Mag., Vol. 34, No. 1 (1960), pp. 37-40.
Peter Luschny, An old operation on sequences: the Seidel transform.

A005647(n) = a(n)/2^n.

Cf. A000364, A086646.

A000795 := proc(n)
        (2*n)!*coeftayl( cosh(x)/cos(x),x=0,2*n) ;
end proc: # R. J. Mathar, Oct 20 2011

max = 16; se = Series[ Cosh[x] / Cos[x], {x, 0, 2*max} ]; a[n_] := SeriesCoefficient[ se, 2*n ]*(2*n)!; Table[ a[n], {n, 0, max} ] (* Jean-François Alcover, Apr 02 2012 *)
With[{nn=40},Take[CoefficientList[Series[Cosh[x]/Cos[x],{x,0,nn}],x] Range[ 0,nn]!,{1,-1,2}]] (* Harvey P. Dale, May 11 2012 *)
a[ n_] := If[ n < 0, 0, With[ {m = 2 n}, m! SeriesCoefficient[ Cosh[ x] / Cos[ x], {x, 0, m}]]]; (* Michael Somos, Aug 15 2015 *)

# Generalized algorithm of L. Seidel (1877)
def A000795_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
        if e == -1 : R.append(A[-i//2])
    return R
A000795_list(10) # Peter Luschny, Jun 02 2012

a(n) = Sum_{k=0..n} binomial(2n, 2k)*A000364(n-k). - Philippe Deléham, Dec 16 2003
a(n) = Sum_{k>=0} (-1)^(n+k)*2^(2n-k)*A065547(n, k). - Philippe Deléham, Feb 26 2004
a(n) = Sum_{k>=0} A086646(n, k). - Philippe Deléham, Feb 26 2004
G.f.: 1 / (1 - (1^2+1)*x / (1 - 2^2*x / (1 - (3^2+1)*x / (1 - 4^2*x / (1 - (5^2+1)*x / (1 - 6^2*x / ...)))))). - Michael Somos, May 12 2012
G.f.: Q(0)/(1-2*x), where Q(k) = 1 - 8*x^2*(2*k^2+2*k+1)*(k+1)^2/( 8*x^2*(2*k^2+2*k+1)*(k+1)^2 - (1 - 8*x*k^2 - 4*x*k -2*x)*(1 - 8*x*k^2 - 20*x*k -14*x)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 22 2013
a(n) ~ (2*n)! * 2^(2*n+2) * cosh(Pi/2) / Pi^(2*n+1). - Vaclav Kotesovec, Mar 08 2014
a(n) = 1 - Sum_{k=1..n} (-1)^k * binomial(2*n,2*k) * a(n-k). - Ilya Gutkovskiy, Mar 09 2022

A002436 E.g.f.: Sum_{n >= 0} a(n)x^(2n)/(2n)! = sec(2x).

Original entry on oeis.org

1, 4, 80, 3904, 354560, 51733504, 11070525440, 3266330312704, 1270842139934720, 630424777638805504, 388362339077351014400, 290870261262635870715904, 260290690801376575335956480, 274278793184290987427604987904, 336150887870579862992197737512960

Offset: 0

N. J. A. Sloane

			G.f. = 1 + 4*x + 80*x^2 + 3904*x^3 + 354560*x^4 + 51733504*x^5 + 11070525440*x^6 + ...

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 75.
J. W. L. Glaisher, On the last two figures in certain coefficients analogous to the Eulerian numbers, Quart. J. Pure Appl. Math., 44 (1913), 105-112.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..216

Cf. A000364, A000816, A000831.

(-1)^n*a(n) give the alternating row sums of A060187(2*n), n >= 0. The alternating sums for odd numbered rows vanish. - Wolfdieter Lang, Jul 12 2017

m:=35; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (1+Tan(x))/(1-Tan(x)) )); [Factorial(n-1)*b[n]: n in [1..m by 2]]; // Vincenzo Librandi, May 30 2019

A := n -> (-4)^n*euler(2*n); # (Then A(n) = a(n+1) for n >= 0.) # Peter Luschny, Jan 27 2009

Rest@ Union[ Range[0, 24]! CoefficientList[ Series[ Sec[ 2x], {x, 0, 24}], x]] (* Robert G. Wilson v, Apr 16 2011 *)
a[ n_] :=  If[ n < 0, 0, 2 (-16)^n LerchPhi[ -1, -2 n, 1/2]]; (* Michael Somos, Oct 14 2014 *)
With[{nn=30},Take[CoefficientList[Series[Sec[2x],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, May 06 2018 *)

{a(n) = local(m); if( n<0, 0, m = 2*n; m! * polcoeff( 1 / cos( 2*x + x * O(x^m)), m))}; /* Michael Somos, Apr 16 2011 */

@CachedFunction
def sp(n,x) :
    if n == 0 : return 1
    return -add(2^(n-k)*sp(k,1/2)*binomial(n,k) for k in range(n)[::2])
A002436 = lambda n : abs(sp(2*(n-1),x))
[A002436(n) for n in (1..15)]   # Peter Luschny, Jul 30 2012

a(n) = A000831(2*n) = 4^n * A000364(n). a(n) = 2 * A000816(n) except n=0. - Michael Somos, Apr 26 2011
E.g.f.: sec(2*x) = 1 + 2*(x^2)/G(0); G(k) = (k+1)*(2*k+1) - 2*(x^2) + (x^2)*(2*k+1)*(2*k+2)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 01 2011
E.g.f.: sec(2*x) = 1/cos(2*x) = 1/(cos(x)^2 - sin(x)^2). - Arkadiusz Wesolowski, Jul 25 2012
From Sergei N. Gladkovskii, Oct 23 2012 (Start)
G.f.: 1/U(0) where U(k) = 1 - 2*(4*k+1)*(4*k+2)*x/(1 - 2*(4*k+3)*(4*k+4)*x/U(k+1)); (continued fraction, 2-step).
E.g.f.: 1/S(0) where S(k) = 1 - 2*x^2/((4*k+1)*(2*k+1) - x^2*(4*k+1)*(2*k+1)/(x^2 - (4*k+3)*(k+1)/S(k+1))); (continued fraction, 3rd kind, 3-step). (End)
G.f.: 1/U(0) where U(k) = 1 - (4*k+2)*(4*k+2)*x^2/(1 - (4*k+4)*(4*k+4)*x^2/U(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Nov 06 2012
G.f.: 1/G(0) where G(k) = 1 - 4*x*(k+1)^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 12 2013
a(n+1) = | 2*16^n*lerchphi(-1, -2*n, 1/2) |, n>=0. - Peter Luschny, Apr 27 2013
G.f.: Q(0), where Q(k) = 1 - x*(2*k+2)^2/( x*(2*k+2)^2 - 1/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 10 2013
E.g.f.: sec(2*x) = 1/cos(2*x) = 1 + 2*x^2/(1-2*x^2)*T(0), where T(k) = 1 - x^2*(2*k+1)*(2*k+2)/( x^2*(2*k+1)*(2*k+2) + ((k+1)*(2*k+1) - 2*x^2)*((k+2)*(2*k+3) - 2*x^2)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 25 2013
a(n) = (-1)^n*2^(6*n+1)*(Zeta(-2*n,1/4) - Zeta(-2*n, 3/4)), where Zeta(a, z) is the generalized Riemann zeta function. - Peter Luschny, Mar 11 2015

More terms from Michael Somos, Jun 21 2002

A092731 Decimal expansion of Pi^5.

Original entry on oeis.org

3, 0, 6, 0, 1, 9, 6, 8, 4, 7, 8, 5, 2, 8, 1, 4, 5, 3, 2, 6, 2, 7, 4, 1, 3, 1, 0, 0, 4, 3, 4, 3, 5, 6, 0, 6, 4, 8, 0, 3, 0, 0, 7, 0, 6, 6, 2, 8, 0, 7, 4, 9, 9, 0, 5, 5, 3, 4, 9, 2, 4, 4, 3, 6, 8, 6, 2, 3, 4, 9, 9, 2, 1, 3, 3, 6, 1, 4, 0, 2, 4, 4, 8, 5, 7, 8, 3, 5, 0, 0, 4, 7, 3, 5, 0, 5, 1, 1, 8, 9, 0, 4, 0, 3, 7

Offset: 3

Mohammad K. Azarian, Apr 12 2004

			306.0196847852814532

G. C. Greubel, Table of n, a(n) for n = 3..10000
Jean-Christophe Pain, The fifth power of Pi: new series representation involving the golden ratio and an application in physics, 2022.
Jean-Christophe Pain, New series representations for any positive power of Pi from a relation involving trigonometric functions, 2208.02624 [math.NT], 2022.
Kh. Hessami Pilehrood and Tatiana Hessami Pilehrood, Series acceleration formulas for beta values, Discr. Math. Theor. Comp. Sci. 12 (2) (2010) 223-236.
Index entries for transcendental numbers

Cf. A000796, A002161, A019692, A091925, A092735, A000364, A002437.

R:= RealField(100); (Pi(R))^5; // G. C. Greubel, Mar 09 2018

RealDigits[Pi^5, 10, 100][[1]] (* G. C. Greubel, Mar 09 2018 *)

PARI
```
Pi^5 \\ G. C. Greubel, Mar 09 2018
```

From Peter Bala, Oct 31 2019: (Start)
Pi^5 = (4!/(2*305)) * Sum_{n >= 0} (-1)^n*( 1/(n + 1/6)^5 + 1/(n + 5/6)^5 ), where 305 = ((3^5 + 1)/4)*A000364(2) = A002437(2).
Pi^5 = (4!/(2*3905)) * Sum_{n >= 0} (-1)^n*( 1/(n + 1/10)^5 - 1/(n + 3/10)^5 - 1/(n + 7/10)^5 + 1/(n + 9/10)^5 ), where 3905 = ((5^5 - 1)/4)*A000364(2).
Cf. A019692, A091925 and A092735. (End)

cp's OEIS Frontend

A241171 Triangle read by rows: Joffe's central differences of zero, T(n,k), 1 <= k <= n.

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A002439 Glaisher's T numbers.

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A009843 E.g.f. x/cos(x) (odd powers only).

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A349264 Generalized Euler numbers, a(n) = n!*[x^n](sec(4*x)*(sin(4*x) + 1)).

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

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A000464 Expansion of e.g.f. sin(x)/cos(2*x).

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A349264 Generalized Euler numbers, a(n) = n![x^n](sec(4x)(sin(4x) + 1)).

A002436 E.g.f.: Sum_{n >= 0} a(n)x^(2n)/(2n)! = sec(2x).