A122045 -id:A122045 - cp's OEIS frontend

A173774 The arithmetic mean of (21k + 8)binomial(2*k,k)^3 (k=0..n-1).

8, 120, 3680, 144760, 6427008, 306745824, 15364514880, 796663553400, 42395640372800, 2302336805317120, 127078484504270208, 7108177964254013920, 402042028998035350400, 22954860061516225396800

Offset: 1

Zhi-Wei Sun, Feb 24 2010

nonn

On Feb 10 2010, Zhi-Wei Sun introduced the sequence and conjectured that each term a(n) is an integer divisible by 4*binomial(2*n,n). On Feb 11 2011, Kasper Andersen confirmed this conjecture by noting that the sequence b(n) = a(n)/(4*binomial(2*n,n)), for n > 0, coincides with A112029. It was proved that for every prime p and positive integer a we have a(p^a) == 8 + 16*p^3*B_(p-3) (mod p^4), where B_0, B_1, B_2, ... are Bernoulli numbers. Given a prime p, it has been conjectured that Sum_{k=0..(p-1)/2} (21*k + 8)*binomial(2*k,k)^3 == 8*p + (-1)^((p-1)/2)*32*p^3*E_(p-3) (mod p^4) if p > 3 (where E_0, E_1, E_2, ... are Euler numbers), and that Sum_{k=0..floor(2p^a/3)} (21*k + 8)*binomial(2*k,k)^3 == 8*p^a (mod p^(a + 5 + (-1)^p)) if a is a positive integer with p^a == 1 (mod 3). He also observed that b(n) = a(n)/(4*binomial(2*n,n)) is odd if and only if n is a power of two.

			For n=2 we have a(2)=120 since (8*binomial(0,0)^3 + (21+8)*binomial(2,1)^3)/2 = 120.

G. C. Greubel, Table of n, a(n) for n = 1..500
Kasper Andersen, Re: A somewhat surprising conjecture
Zhi-Wei Sun, A somewhat surprising conjecture
Zhi-Wei Sun, Re: A somewhat surprising conjecture
Zhi-Wei Sun, Open conjectures on congruences, preprint, arXiv:0911.5665 [math.NT], 2009-2011.
Zhi-Wei Sun, Super congruences and Euler numbers, preprint, arXiv:1001.4453 [math.NT], 2010-2011.

Cf. A000984, A112029, A122045.

[(&+[(21*j+8)*(j+1)^3*Catalan(j)^2: j in [0..n-1]])/n: n in [1..30]]; // G. C. Greubel, Jul 06 2021

a[n_]:= Sum[(21*k+8)*Binomial[2*k,k]^3, {k,0,n-1}]/n; Table[a[n], {n, 25}]

[(1/n)*sum((21*j+8)*binomial(2*j,j)^3 for j in (0..n-1)) for n in (1..30)] # G. C. Greubel, Jul 06 2021

a(n) = (1/n)*Sum_{k=0..n-1} (21*k + 8)*binomial(2*k,k)^3.
(n+1)*a(n+1) = n*a(n) + 8*(21*n + 8)*binomial(2*n-1, n)^3, n > 0, with a(1) = 8.
a(n) ~ 2^(6*n) / (3 * (Pi*n)^(3/2)). - Vaclav Kotesovec, Jan 24 2019
a(n) = (1/n)*Sum_{j=0..n-1} (21*j + 8)*(j+1)^3*Catalan(j)^3. - G. C. Greubel, Jul 06 2021

A073448 Decimal expansion of sec(1).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Aug 01 2002

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

			1.85081571768092561791175324139...

Mohammad K. Azarian, Forty-Five Nested Equilateral Triangles and cosecant of 1 degree, Problem 813, College Mathematics Journal, Vol. 36, No. 5, November 2005, pp. 413-414.
Mohammad K. Azarian, Solution of Forty-Five Nested Equilateral Triangles and cosecant of 1 degree, Problem 813, College Mathematics Journal, Vol. 37, No. 5, November 2006, pp. 394-395.
Index entries for transcendental numbers

Cf. A049470 (cos(1)=1/A073448), A049469 (sin(1)), A049471 (tan(1)), A073447 (csc(1)), A073449 (cot(1)), A122045.

RealDigits[Sec[1],10,120][[1]] (* Harvey P. Dale, Mar 13 2013 *)

PARI
```
1/cos(1)
```

Equals Sum_{k>=0} (-1)^k * E(2*k) / (2*k)!, where E(k) is the k-th Euler number (A122045). - Amiram Eldar, May 15 2021

A126156 Expansion of e.g.f. sqrt(sec(sqrt(2)*x)), showing coefficients of only the even powers of x.

Original entry on oeis.org

1, 1, 7, 139, 5473, 357721, 34988647, 4784061619, 871335013633, 203906055033841, 59618325600871687, 21297483077038703899, 9127322584507530151393, 4621897483978366951337161, 2730069675607609356178641127, 1860452328661957054823447670979, 1448802510679254790311316267306753

Offset: 0

Paul D. Hanna, Dec 20 2006

nonn

Previous name was: Column 0 and row sums of symmetric triangle A126155.

This is the square root of the Euler numbers (A122045) with respect to the Cauchy type product as described by J. Singh (see link and the second Maple program) normalized by 2^n. A241885 shows the corresponding sqrt of the Bernoulli numbers. - Peter Luschny, May 07 2014

			E.g.f.: A(x) = 1 + x^2/2! + 7*x^4/4! + 139*x^6/6! + 5473*x^8/8! + 357721*x^10/10! + ...
where the logarithm begins:
log(A(x)) = x^2/2! + 4*x^4/4! + 64*x^6/6! + 2176*x^8/8! + 126976*x^10/10! + 11321344*x^12/12! + ...
compare the logarithm to
A(x)^4 = 1 + 4*x^2/2! + 64*x^4/4! + 2176*x^6/6! + 126976*x^8/8! + 11321344*x^10/10! + ...

H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 366.

G. C. Greubel, Table of n, a(n) for n = 0..200
Peter Bala, A triangle for calculating A126156
Alain Connes, Caterina Consani and Henri Moscovici, Zeta zeros and prolate wave operators, arXiv:2310.18423 [math.NT], Oct 2023, p.31.
Denis S. Grebenkov, Vittoria Sposini, Ralf Metzler, Gleb Oshanin, and Flavio Seno, Exact distributions of the maximum and range of random diffusivity processes, New J. Phys. (2021) Vol. 23, 023014.
Jitender Singh, On an arithmetic convolution, arXiv:1402.0065 [math.NT], 2014.

Diagonals: A126157, A126158.

Cf. A126155, A365672.

A126156 := proc(n)
        sqrt(sec(sqrt(2)*z)) ;
        coeftayl(%,z=0,2*n) ;
        %*(2*n)! ;
end;
seq(A126156(n),n=0..10) ; # Sergei N. Gladkovskii, Oct 31 2011
g := proc(f, n) option remember; local g0, m; g0 := sqrt(f(0));
if n=0 then g0 else if n=1 then 0 else add(binomial(n, m)*g(f,m)* g(f,n-m), m=1..n-1) fi; (f(n)-%)/(2*g0) fi end:
a := n -> (-2)^n*g(euler, 2*n);
seq(a(n), n=0..14); # Peter Luschny, May 07 2014
# Alternative: an algorithm as described by Peter Bala, see also A365672:
T := proc(n, k) option remember; if k = 0 then 1 else if k = n then
T(n, k-1) else (n - k + 1) * (2 * (n - k) + 1) * T(n, k - 1) + T(n - 1, k)
fi fi end:
a := n -> T(n, n): seq(a(n), n = 0..14);  # Peter Luschny, Sep 29 2023

a[n_] := SeriesCoefficient[ Sqrt[ Sec[ Sqrt[2]*x]], {x, 0, 2 n}]*(2*n)!; Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Nov 29 2013, after Sergei N. Gladkovskii *)

a(n):=if n=0 then 1 else 1/(4*n)*sum(binomial(2*n,2*k)*((2^(2*k)-1)*2^(3*k)*(-1)^((k-1))*bern(2*k)*a(n-k)),k,1,n); /* Vladimir Kruchinin, Feb 25 2015 */

a[n]:=if n=0 then 1 else sum(a[n-k]*binomial(2*n,2*k)*(k/(2*n)-1)*(-2)^k,k,1,n);
makelist(a[n],n,0,30); /* Tani Akinari, Sep 11 2023 */

/* E.g.f. A(x) = exp( Integral^2 A(x)^4 dx^2 ): */
{a(n)=local(A=1+x*O(x)); for(i=1, n, A=exp(intformal(intformal(A^4 + x*O(x^(2*n))))) ); (2*n)!*polcoeff(A, 2*n, x)}
for(n=0,20,print1(a(n),", "))

{a(n) = local(A=1+x); for(i=1,n, A = exp( intformal( A^2 * intformal( 1/A^2 + x*O(x^n)) ) ) ); n!*polcoeff(A,n)}
for(n=0,20,print1(a(2*n),", "))

{a(n)=-(n<1)-sum(j=0,n,sum(k=0,j/2,(2*n+1)!*(2*k-j)^(2*n)/(n!*(2*j+1)*(n-j)!*k!*(j-k)!*(-2)^(n+j-1))))}; /* Tani Akinari, Sep 28 2023 */

def A126156(n): return A126155(n, 0)
print([A126156(n) for n in range(17)])  # Peter Luschny, Dec 14 2023

a(n) = Sum_{k=0..n} A087736(n,k)*3^(n-k). - Philippe Deléham, Jul 17 2007
E.g.f.: Sum_{n>=0} a(n)*x^(2*n)/(2*n)! = sqrt(sec(sqrt(2)*x)). - David Callan, Jan 03 2011
E.g.f. satisfies: A(x) = exp( Integral Integral A(x)^4 dx dx ), where A(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)! and the constant of integration is zero. - Paul D. Hanna, May 30 2015
E.g.f. satisfies: A(x) = exp( Integral A(x)^2 * Integral 1/A(x)^2 dx dx ), where A(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)! and the constant of integration is zero. - Paul D. Hanna, Jun 02 2015
G.f.: 1/(1-x/(1-6*x/(1-15*x/(1-28*x/(1-45*x/(1-66*x/(1-91*x/(1-... or 1/U(0) where U(k) = 1-x*(k+1)*(2*k+1)/U(k+1); (continued fraction). [See Wall.] - Sergei N. Gladkovskii, Oct 31 2011
G.f.: 1/U(0) where U(k) = 1 - (4*k+1)*(4*k+2)*x/(2 - (4*k+3)*(4*k+4)*x/ U(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Oct 24 2012
G.f.: 1/G(0) where G(k) = 1 -x*(k+1)*(2*k+1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 11 2013
G.f.: Q(0), where Q(k) = 1 - x*(2*k+1)*(k+1)/( x*(2*k+1)*(k+1) - 1/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Oct 09 2013
a(n) ~ 2^(5*n+2) * n^(2*n) / (exp(2*n) * Pi^(2*n+1/2)). - Vaclav Kotesovec, Jul 13 2014
a(n) = (1/(4*n))*Sum_{k=1..n} binomial(2*n,2*k)*((2^(2*k)-1)*2^(3*k)*(-1)^((k-1))*Bernoulli(2*k)*a(n-k)), a(0)=1. - Vladimir Kruchinin, Feb 25 2015
a(n) = Sum_{k=1..n} a(n-k)*binomial(2*n,2*k)*(k/(2*n)-1)*(-2)^k, a(0)=1. - Tani Akinari, Sep 11 2023
For n > 0, a(n) = -Sum_{j=0..n} Sum_{k=0..floor(j/2)} (2*n+1)!*(2*k-j)^(2*n)/(n!*(2*j+1)*(n-j)!*k!*(j-k)!*(-2)^(n+j-1)). - Tani Akinari, Sep 28 2023

New name based on a comment of David Callan, Peter Luschny, May 07 2014

A248727 A046802(x,y) --> A046802(x,y+1), transform of e.g.f. for the graded number of positroids of the totally nonnegative Grassmannians G+(k,n); enumerates faces of the stellahedra.

Original entry on oeis.org

1, 2, 1, 5, 5, 1, 16, 24, 10, 1, 65, 130, 84, 19, 1, 326, 815, 720, 265, 36, 1, 1957, 5871, 6605, 3425, 803, 69, 1, 13700, 47950, 65646, 44240, 15106, 2394, 134, 1, 109601, 438404, 707840, 589106, 267134, 63896, 7094, 263, 1

Offset: 0

Tom Copeland, Oct 12 2014

This is a transform of A046802 treating it as an array of h-vectors, so y is replaced by (y+1) in the e.g.f. for A046802.
An e.g.f. for the reversed row polynomials with signs is given by exp(a.(0;t)x) = [e^{(1+t)x} [1+t(1-e^(-x))]]^(-1) = 1 - (1+2t)x + (1+5t+5t^2)x^2/2! + ... . The reciprocal is an e.g.f. for the reversed face polynomials of the simplices A074909, i.e., exp(b.(0;t)x) = e^{(1+t)x} [1+t(1-e^(-x))] = 1 + (1+2t)x +(1+3t+3t^2) x^2/2! + ... , so the relations of A133314 apply between the two sets of polynomials. In particular, umbrally [a.(0;t)+b.(0;t)]^n vanishes except for n=0 for which it's unity, implying the two sets of Appell polynomials formed from the two bases, a_n(z;t) = (a.(0;t)+z)^n and b_n(z;t) = (b.(0;t) + z)^n, are an umbral compositional inverse pair, i.e., b_n(a.(x;t);t)= x^n = a_n(b.(x;t);t). Raising operators for these Appell polynomials are related to the polynomials of A028246, whose reverse polynomials are given by A123125 * A007318. Compare: A248727 = A007318 * A123125 * A007318 and A046802 = A007318 * A123125. See A074909 for definitions and related links. - Tom Copeland, Jan 21 2015
The o.g.f. for the umbral inverses is Og(x) = x / (1 - x b.(0;t)) = x / [(1-tx)(1-(1+t)x)] = x + (1+2t) x^2 + (1+3t+3t^2) x^3 + ... . Its compositional inverse is an o.g.f for signed A033282, the reverse f-polynomials for the simplicial duals of the Stasheff polytopes, or associahedra of type A, Oginv(x) =[1+(1+2t)x-sqrt[1+2(1+2t)x+x^2]] / (2t(1+t)x) = x - (1+2t) x^2 + (1+5t+5t^2) x^3 + ... . Contrast this with the o.g.f.s related to the corresponding h-polynomials in A046802. - Tom Copeland, Jan 24 2015
Face vectors, or coefficients of the face polynomials, of the stellahedra, or stellohedra. See p. 59 of Buchstaber and Panov. - Tom Copeland, Nov 08 2016
See A008279 for a relation between the e.g.f.s enumerating the faces of permutahedra and stellahedra. - Tom Copeland, Nov 14 2016

			The triangle T(n, k) starts:
n\k    0     1     2     3     4    5   6  7 ...
1:     1
2:     2     1
3:     5     5     1
4:    16    24    10     1
5:    65   130    84    19     1
6:   326   815   720   265    36    1
7:  1957  5871  6605  3425   803   69   1
8: 13700 47950 65646 44240 15106 2394 134  1
... reformatted, _Wolfdieter Lang_, Mar 27 2015

P. Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
L. Berry, S. Forcey, M. Ronco, and P. Showers, Polytopes and Hopf algebras of painted trees: Fan graphs and Stellohedra, arXiv:1608.08546 [math.CO], 2018.
L. Berry, S. Forcey, M. Ronco, and P. Showers, Species substitution, graph suspension, and graded Hopf algebras of painted tree polytopes, arXiv:1608.08546 [math.CO], 2019.
V. Buchstaber and T. Panov, Toric Topology, arXiv:1210.2368v3 [math.AT], 2014.
R. Da Rosa, D. Jensen, and D. Ranganathan, Toric graph associahedra and compactifications of M_(0,n), arXiv:1411.0537 [math.AG], 2015.
S. Forcey, M. Ronco, and P. Showers, Polytopes and algebras of grafted trees: Stellohedra, arXiv:1608.08546v2 [math.CO], 2016.
Stefan Forcey, The Hedra Zoo
I. Limonchenko, Moment-angle manifolds, 2-truncated cubes and Massey operations, arXiv:1510.07778 [math.AT], 2017.
M. Lin, Graph Cohomology, 2016, (Fig. 2.5 is a stellahedron).
T. Manneville and V. Pilaud, Compatibility fans for graphical nested complexes, arXiv:1501.07152v3 [math.CO], 2015-2016.
MathOverflow, Analogue of conic sections for the permutohedra, associahedra, and noncrossing partitions, an MO question posed by T. Copeland, 2017. (See Buchstaber references therein.)
V. Pilaud, The Associahedron and its Friends, presentation for Séminaire Lotharingien de Combinatoire, April 4-6, 2016. [From _Tom Copeland_, Jun 26 2018]

Cf. A046802, A007047, A122045, A000522, A036918, A001339, A052944.
Cf. A074909, A028246, A133314, A007318, A123125, A033282.
Cf. A008279, A008292, A090582, A097808, A130850.
Cf. A019538, A119879.

(* t = A046802 *) t[, 1] = 1; t[n, n_] = 1; t[n_, 2] = 2^(n - 1) - 1; t[n_, k_] = Sum[((i - k + 1)^i*(k - i)^(n - i - 1) - (i - k + 2)^i*(k - i - 1)^(n - i - 1))*Binomial[n - 1, i], {i, 0, k - 1}]; T[n_, j_] := Sum[Binomial[k, j]*t[n + 1, k + 1], {k, j, n}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 23 2015, after Tom Copeland *)

Let M(n,k)= sum{i=0,..,k-1, C(n,i)[(i-k)^i*(k-i+1)^(n-i)- (i-k+1)^i*(k-i)^(n-i)]} with M(n,0)=1. Then M(n,k)= A046802(n,k), and T(n,j)= sum(k=j,..,n, C(k,j)*M(n,k)) for j>0 with T(n,0)= 1 + sum(k=1,..,n, M(n,k)) for n>0 and T(0,0)=1.
E.g.f: y * exp[x*(y+1)]/[y+1-exp(x*y)].
Row sums are A007047. Row polynomials evaluated at -1 are unity. Row polynomials evaluated at -2 are A122045.
First column is A000522. Second column appears to be A036918/2 = (A001339-1)/2 = n*A000522(n)/2.
Second diagonal is A052944. (Changed from conjecture to fact on Nov 08 2016.)
The raising operator for the reverse row polynomials with row signs is R = x - (1+t) - t e^(-D) / [1 + t(1-e^(-D))] evaluated at x = 0, with D = d/dx. Also R = x - d/dD log[exp(a.(0;t)D], or R = - d/dz log[e^(-xz) exp(a.(0;t)z)] = - d/dz log[exp(a.(-x;t)z)] with the e.g.f. defined in the comments and z replaced by D. Note that t e ^(-D) / [1+t(1-e^(-D))] = t - (t+t^2) D + (t+3t^2+2t^3) D^2/2! - ... is an e.g.f. for the signed reverse row polynomials of A028246. - Tom Copeland, Jan 23 2015
Equals A007318*(padded A090582)*A007318*A097808 = A007318*(padded (A008292*A007318))*A007318*A097808 = A007318*A130850 = A007318*(mirror of A028246). Padded means in the same way that A097805 is padded A007318. - Tom Copeland, Nov 14 2016
Umbrally, the row polynomials are p_n(x) = (1 + q.(x))^n, where (q.(x))^k = q_k(x) are the row polynomials of A130850. - Tom Copeland, Nov 16 2016
From the previous umbral statement, OP(x,d/dy) y^n =  (y + q.(x))^n, where OP(x,y) = exp[y * q.(x)] = x/((1+x)*exp(-x*y) - 1), the e.g.f. of A130850, so OP(x,d/dy) y^n evaluated at y = 1 is  p_n(x), the n-th row polynomial of this entry, with offset 0. - Tom Copeland, Jun 25 2018
Consolidating some formulas in this entry and A046082, in umbral notation for concision, with all offsets 0: Let A_n(x;y) = (y + E.(x))^n, an Appell sequence in y where E.(x)^k = E_k(x) are the Eulerian polynomials of A123125. Then the row polynomials of A046802 (the h-polynomials of the stellahedra) are given by h_n(x) = A_n(x;1); the row polynomials of this entry (A248727, the face polynomials of the stellahedra), by f_n(x) = A_n(1 + x;1); the Swiss-knife polynomials of A119879, by Sw_n(x) = A_n(-1;1 + x); and the row polynomials of the Worpitsky triangle (A130850), by w_n(x) = A(1 + x;0). Other specializations of A_n(x;y) give A090582 (the f-polynomials of the permutohedra, cf. also A019538) and A028246 (another version of the Worpitsky triangle). - Tom Copeland, Jan 24 2020

Title expanded by Tom Copeland, Nov 08 2016

A120337 Euler-irregular primes p dividing E(2k) for some 2k < p-1.

Original entry on oeis.org

19, 31, 43, 47, 61, 67, 71, 79, 101, 137, 139, 149, 193, 223, 241, 251, 263, 277, 307, 311, 349, 353, 359, 373, 379, 419, 433, 461, 463, 491, 509, 541, 563, 571, 577, 587, 619, 677, 691, 709, 739, 751, 761, 769, 773, 811, 821, 877, 887, 907, 929, 941, 967, 971, 983

Offset: 1

Stefan Krämer, Jun 22 2006

nonn

Conjecture (Ernvall and Metsänkylä, 1978): The asymptotic density of this sequence within the primes is 1 - 1/sqrt(e) = 0.393469... (A290506), the same as the corresponding conjectured density of the irregular primes (A000928). - Amiram Eldar, Dec 06 2022

			a(1) = 19 because 19 divides E(10) = -19*2659 and 10 + 1 < 19.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000
Reijo Ernvall, On the distribution mod 8 of the E-irregular primes, Annales Academiae Scientiarum Fennicae, Series A. I. Mathematica, Vol. 1, 1975, pp. 195-198.
Reijo Ernvall and Tauno Metsänkylä, Cyclotomic invariants and E-irregular primes, Mathematics of Computation, Vol. 32, No. 142 (1978), pp. 617-629; Corrigenda, ibid., Vol. 33, No. 145 (1979), p. 433.
Su Hu and Min-Soo Kim, A note on the irregular primes with respect to Euler polynomials, arXiv:1510.01558 [math.NT], 2015.
Su Hu, Min-Soo Kim, Pieter Moree and Min Sha, Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture, arXiv:1809.08431 [math.NT], 2018.
Romeo Mestrovic, A search for primes p such that Euler number E_{p-3} is divisible by p, arXiv preprint arXiv:1212.3602 [math.NT], 2012. - From _N. J. A. Sloane_, Jan 25 2013
Prime Pages, Euler Irregular
Samuel S. Wagstaff, Prime divisors of the Bernoulli and Euler numbers, Number theory for the millennium, III, 2002, pp. 357-374, 2002. MR 1956285.

Cf. A000928, A092218, A120115, A122045, A290506.

A120337_list := proc(bound)
local ae, F, p, m, maxp; F := NULL;
for m from 2 by 2 to bound do
  p := nextprime(m+1);
  ae := abs(euler(m));
  maxp := min(ae, bound);
  while p <= maxp do
      if ae mod p = 0
      then F := F,p fi;
      p := nextprime(p);
   od;
od;
sort([F]) end: # Peter Luschny, Apr 25 2011

fQ[p_] := Block[{k = 1}, While[ 2k +1 < p && Mod[ EulerE[ 2k], p] != 0, k++]; p > 2k +1]; Select[ Prime@ Range@ 168, fQ@# &] (* Robert G. Wilson v, Dec 10 2014 *)

The (trivial) divisors of E(2n) are given by the theorem of Sylvester (1861): Let p prime with p=1 (mod 4), p-1|2n, p^k|2n then p^{k+1} | E(2n).

Terms 251 through 983 from Peter Luschny, Apr 25 2011

A119880 Expansion of e.g.f. exp(2x)*sech(x).

Original entry on oeis.org

1, 2, 3, 2, -3, 2, 63, 2, -1383, 2, 50523, 2, -2702763, 2, 199360983, 2, -19391512143, 2, 2404879675443, 2, -370371188237523, 2, 69348874393137903, 2, -15514534163557086903, 2, 4087072509293123892363, 2, -1252259641403629865468283, 2, 441543893249023104553682823

Offset: 0

Paul Barry, May 26 2006

Transform of 2^n under the matrix A119879.

Also the Swiss-Knife polynomials A153641 evaluated at x=2. - Peter Luschny, Nov 23 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, Eulerian-Dowling Polynomials as Moments, Using Riordan Arrays, arXiv:1702.04007 [math.CO], 2017.

Cf. A119879, A119880, A122045.

EulerPoly:= func< n,x | (&+[ (&+[ (-1)^j*Binomial(k,j)*(x+j)^n : j in [0..k]])/2^k: k in [0..n]]) >;
A119880:= func< n | (-2)^n*EulerPoly(n, -1/2) >;
[A119880(n): n in [0..40]]; // G. C. Greubel, Jun 07 2023

A119880_list := proc(n) local S,A,m,k;
A := array(0..n-1,0..n-1); S := NULL;
for m from 0 to n-1 do
   A[m,0] := (-2)^m*euler(m,0);
   for k from m-1 by -1 to 0 do
       A[k,m-k] := A[k+1,m-k-1] + A[k,m-k-1] od;
    S := S,A[0,m] od;
S end:
A119880_list(31); # Peter Luschny, Jun 15 2012
P := proc(n,x) option remember; if n = 0 then 1 else
  (n*x-(1/2)*(1-x))*P(n-1,x)+x*(1-x)*diff(P(n-1,x),x);
  expand(%) fi end:
A119880 := n -> (-1)^n*subs(x=-1, P(n,x)):
seq(A119880(n), n=0..30);  # Peter Luschny, Mar 07 2014

Table[2 (1 + Zeta[-n] (2^n - 1) + 2^(2n+1) Zeta[-n, 3/4]), {n, 0, 30}] (* Peter Luschny, Jul 16 2013 *)
Range[0, 30]! CoefficientList[Series[Exp[2 x] Sech[x], {x, 0, 30}], x] (* Vincenzo Librandi, Mar 08 2014 *)

def skp(n, x):
    A = lambda k: 0 if (k+1)%4 == 0 else (-1)^((k+1)//4)*2^(-(k//2))
    return add(A(k)*add((-1)^v*binomial(k,v)*(v+x+1)^n for v in (0..k)) for k in (0..n))
A119880 = lambda n: skp(n,2)
[A119880(n) for n in (0..30)]  # Peter Luschny, Nov 23 2012

a(n) = Sum_{k=0..n} A119879(n,k) * 2^k.
From Sergei N. Gladkovskii, Oct 14 2012 to Dec 16 2013: (Start)
Continued fractions:
G.f.: 1/U(0) where U(k) = 1 - x - x*(k+1)/(1 + x*(k+1)/U(k+1)).
G.f.: 1/Q(0), where Q(k) = 1 - 3*x + x*(k+1)/(1-x*(k+1)/Q(k+1)).
G.f.: x/(1-x)/Q(0) + 1/(1-x), where Q(k) = 1 - x + x^2*(k+1)*(k+2)/Q(k+1).
G.f.: T(0)/(1-2*x), where T(k) = 1 - x^2*(k+1)^2/(x^2*(k+1)^2 + (1-2*x)^2/T(k+1)).
E.g.f.: 2/Q(0), where Q(k) = 1 + 3^k/( 1 - x/( x - 3^k*(k+1)/Q(k+1))). (End)
a(n) = 2*(1+zeta(-n)*(2^n-1)+2^(2*n+1)*zeta(-n,3/4)). - Peter Luschny, Jul 16 2013
a(n) = (-2)^n*Euler(n, -1/2). - Peter Luschny, Jul 21 2020

A163747 Expansion of e.g.f. 2exp(x)(1-exp(x))/(1+exp(2*x)).

Original entry on oeis.org

0, -1, -1, 2, 5, -16, -61, 272, 1385, -7936, -50521, 353792, 2702765, -22368256, -199360981, 1903757312, 19391512145, -209865342976, -2404879675441, 29088885112832, 370371188237525, -4951498053124096, -69348874393137901

Offset: 0

Roger L. Bagula, Aug 03 2009

sign

The real part of the exponential expansion of 2*((1+i)/(1+i*exp(z))-1) = (-1-i)*z + (-1/2+i/2)*z^2 + (1/3+i/3)*z^3 + (5/24-5i/24)*z^4 + (-2/15-2i/15)*z^5 + ...  where i is the imaginary unit.
From Paul Curtz, Mar 12 2013: (Start)
a(n) is an autosequence of the first kind; a(n) and successive differences are:
0,   -1,   -1,      2,      5,    -16,    -61;
-1,   0,    3,      3,    -21,    -45,    333;
1,    3,    0,    -24,    -24,    378,    780;
2,   -3,  -24,      0,    402,    402, -11214;
-5, -21,   24,    402,      0, -11616, -11616;
-16, 45,  378,   -402, -11616,      0, 514608;
61, 333, -780, -11214,  11616, 514608,      0;
The main diagonal is A000004. The inverse binomial transform is the signed sequence.
The first two upper diagonals are A002832 (median Euler numbers) signed.
Sum of the antidiagonals: 0, -2, 0, 10, 0, ... = 2*A122045(n+1) (End)

Robert Israel, Table of n, a(n) for n = 0..485
Toufik Mansour, Howard Skogman, and Rebecca Smith, Passing through a stack k times with reversals, arXiv:1808.04199 [math.CO], 2018.
A. Randrianarivony and J. Zeng, Une famille de polynomes qui interpole plusieurs suites classiques de nombres, Adv. Appl. Math. 17 (1996), 1-26. See Section 6 (negative of the zeroth column of matrix a_{n,k} on p. 18).

Variant: A163982.
Cf. A000004, A000111, A002832, A122045.
Minus the zeroth column of A323833.

A163747 := proc(n) exp(t)*(1-exp(t))/(1+exp(2*t)) ; coeftayl(%,t=0,n) ; 2*%*n! ; end proc: # R. J. Mathar, Sep 11 2011
seq((euler(n) - 2^n*(2*euler(n,1)+euler(n,3/2)))/2 + 1, n=0..30); # Robert Israel, May 24 2016
egf := (2 - 2*I)/(exp(-x) + I); ser := series(egf, x, 24):
seq(n!*Re(coeff(ser, x, n)), n = 0..22); # Peter Luschny, Aug 09 2021

f[t_] = (1 + I)/(1 + I*Exp[t]) - 1;
Table[Re[2*n!*SeriesCoefficient[Series[f[t], {t, 0, 30}], n]], {n, 0, 30}]
max = 20; Clear[g]; g[max + 2] = 1; g[k_] := g[k] = 1 - x + (4*k+3)*(k+1)*x^2 /( 1 + (4*k+5)*(k+1)*x^2 / g[k+1]); gf = -x/g[0]; CoefficientList[Series[gf, {x, 0, max}], x] (* Vaclav Kotesovec, Jan 22 2015, after Sergei N. Gladkovskii *)
Table[(EulerE[n] - 2^n (2 EulerE[n, 1] + EulerE[n, 3/2]))/2 + 1, {n, 0, 20}] (* Benedict W. J. Irwin, May 24 2016 *)

G.f.: -x/W(0), where W(k) = 1 - x + (4*k+3)*(k+1)*x^2 / (1 + (4*k+5)*(k+1)*x^2 / W(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 22 2015
a(n) ~ n! * (cos(Pi*n/2) - sin(Pi*n/2)) * 2^(n+2) / Pi^(n+1). - Vaclav Kotesovec, Apr 23 2015
a(n) = (A122045(n) - 2^n(2*Euler(n,1) + Euler(n,3/2)))/2 + 1, where Euler(n,x) is the n-th Euler polynomial. - Benedict W. J. Irwin, May 24 2016
a(n) = 2*4^n*(HurwitzZeta(-n, 1/4) - HurwitzZeta(-n, 3/4)) + HurwitzZeta(-n, 1)*(4^(n+1) - 2^(n+1)). - Peter Luschny, Jul 21 2020
a(n) = 2^n*(Euler(n, 1/2) - Euler(n, 1)). - Peter Luschny, Mar 19 2021
a(n) = ((-2)^(n + 1)*(1 - 2^(n + 1))*Bernoulli(n + 1))/(n + 1) + Euler(n). - Peter Luschny, May 06 2021
a(n) = n!*Re([x^n]((2 - 2*i)/(i + exp(-x)))). - Peter Luschny, Aug 09 2021

A073746 Decimal expansion of sech(1).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Aug 07 2002

sech(x) = 2/(e^x + e^(-x)).

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 14 2019

			0.64805427366388539957497735322...

Samuel M. Selby (ed.), CRC Basic Mathematical Tables, CRC Press, 1970, p. 218.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Hyperbolic Secant.
Eric Weisstein's World of Mathematics, Hyperbolic Functions.
Index entries for transcendental numbers

Cf. A068118 (continued fraction), A073743 (cosh(1)=1/A073746), A073742 (sinh(1)), A073744 (tanh(1)), A073745 (csch(1)), A073747 (coth(1)), A122045.

RealDigits[Sech[1], 10, 100][[1]] (* Amiram Eldar, May 15 2021 *)

PARI
```
1/cosh(1)
```

Equals Sum_{k>=0} E(2*k) / (2*k)!, where E(k) is the k-th Euler number (A122045). - Amiram Eldar, May 15 2021

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

A220002 Numerators of the coefficients of an asymptotic expansion in even powers of the Catalan numbers.

Original entry on oeis.org

1, 5, 21, 715, -162877, 19840275, -7176079695, 1829885835675, -5009184735027165, 2216222559226679575, -2463196751104762933637, 1679951011110471133453965, -5519118103058048675551057049, 5373485053345792589762994345215, -12239617587594386225052760043303511

Offset: 0

Peter Luschny, Dec 27 2012

sign

Let N = 4*n+3 and A = sum_{k>=0} a(k)/(A123854(k)*N^(2*k)) then
C(n) ~ 8*4^n*A/(N*sqrt(N*Pi)), C(n) = (4^n/sqrt(Pi))*(Gamma(n+1/2)/ Gamma(n+2)) the Catalan numbers A000108.
The asymptotic expansion of the Catalan numbers considered here is based on the Taylor expansion of square root of the sine cardinal. This asymptotic series involves only even powers of N, making it more efficient than the asymptotic series based on Stirling's approximation to the central binomial which involves all powers (see for example: D. E. Knuth, 7.2.1.6 formula (16)). The series is discussed by Kessler and Schiff but is included as a special case in the asymptotic expansion given by J. L. Fields for quotients Gamma(x+a)/Gamma(x+b) and discussed by Y. L. Luke (p. 34-35), apparently overlooked by Kessler and Schiff.

			With N = 4*n+3 the first few terms of A are A = 1 + 5/(4*N^2) + 21/(32*N^4) + 715/(128*N^6) - 162877/(2048*N^8) + 19840275/(8192*N^10). With this A C(n) = round(8*4^n*A/(N*sqrt(N*Pi))) for n = 0..39 (if computed with sufficient numerical precision).

Donald E. Knuth, The Art of Computer Programming, Volume 4, Fascicle 4: Generating All Trees—History of Combinatorial Generation, 2006.
Y. L. Luke, The Special Functions and their Approximations, Vol. 1. Academic Press, 1969.

Peter Luschny, Table of n, a(n) for n = 0..99
J. L. Fields, A note on the asymptotic expansion of a ratio of gamma functions, Proc. Edinburgh Math. Soc. 15 (1) (1966), 43-45.
D. Kessler and J. Schiff, The asymptotics of factorials, binomial coefficients and Catalan numbers. April 2006.

The logarithmic version is A220422. Appears in A193365 and A220466.

Cf. A220412.

A220002 := proc(n) local s; s := n -> `if`(n > 0, s(iquo(n,2))+n, 0);
(-1)^n*mul(4*i+2, i = 1..2*n)*2^s(iquo(n,2))*coeff(taylor(sqrt(sin(x)/x), x,2*n+2), x, 2*n) end: seq(A220002(n), n = 0..14);
# Second program illustrating J. L. Fields expansion of gamma quotients.
A220002 := proc(n) local recF, binSum, swing;
binSum := n -> add(i,i=convert(n,base,2));
swing := n -> n!/iquo(n, 2)!^2;
recF := proc(n, x) option remember; `if`(n=0, 1, -2*x*add(binomial(n-1,2*k+1)*bernoulli(2*k+2)/(2*k+2)*recF(n-2*k-2,x),k=0..n/2-1)) end: recF(2*n,-1/4)*2^(3*n-binSum(n))*swing(4*n+1) end:

max = 14; CoefficientList[ Series[ Sqrt[ Sinc[x]], {x, 0, 2*max+1}], x^2][[1 ;; max+1]]*Table[ (-1)^n*Product[ (2*k+1), {k, 1, 2*n}], {n, 0, max}] // Numerator (* Jean-François Alcover, Jun 26 2013 *)

length = 15; T = taylor(sqrt(sin(x)/x),x,0,2*length+2)
def A005187(n): return A005187(n//2) + n if n > 0 else 0
def A220002(n):
    P = mul(4*i+2 for i in (1..2*n)) << A005187(n//2)
    return (-1)^n*P*T.coefficient(x, 2*n)
[A220002(n) for n in range(length)]

# Second program illustrating the connection with the Euler numbers.
def A220002_list(n):
    S = lambda n: sum((4-euler_number(2*k))/(4*k*x^(2*k)) for k in (1..n))
    T = taylor(exp(S(2*n+1)),x,infinity,2*n-1).coefficients()
    return [t[0].numerator() for t in T][::-1]
A220002_list(15)

Let [x^n]T(f(x)) denote the coefficient of x^n in the Taylor expansion of f(x) then r(n) = (-1)^n*prod_{i=1..2n}(2i+1)*[x^(2*n)]T(sqrt(sin(x)/x)) is the rational coefficient of the asymptotic expansion (in N=4*n+3) and a(n) = numerator(r(n)) = r(n)*2^(3*n-bs(n)), where bs(n) is the binary sum of n (A000120).
Also a(n) = numerator([x^(2*n)]T(exp(S))) where S = sum_{k>=1}((4-E(2*k))/ (4*k)*x^(2*k)) and E(n) the Euler numbers A122045.
Also a(n) = sf(4*n+1)*2^(3*n-bs(n))*F_{2*n}(-1/4) where sf(n) is the swinging factorial A056040, bs(n) the binary sum of n and F_{n}(x) J. L. Fields' generalized Bernoulli polynomials A220412.
In terms of sequences this means
r(n) = (-1)^n*A103639(n)*A008991(n)/A008992(n),
a(n) = (-1)^n*A220371(n)*A008991(n)/A008992(n).
Note that a(n) = r(n)*A123854(n) and A123854(n) = 2^A004134(n) = 8^n/2^A000120(n).
Formula from Johannes W. Meijer:
a(n) = d(n+1)*A098597(2*n+1)*(A008991(n)/A008992(n)) with d(1) = 1 and
d(n+1) = -4*(2*n+1)*A161151(n)*d(n),
d(n+1) = (-1)^n*2^(-1)*(2*(n+1))!*A060818(n)*A048896(n).

cp's OEIS Frontend

A173774 The arithmetic mean of (21*k + 8)*binomial(2*k,k)^3 (k=0..n-1).

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A073448 Decimal expansion of sec(1).

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A126156 Expansion of e.g.f. sqrt(sec(sqrt(2)*x)), showing coefficients of only the even powers of x.

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A248727 A046802(x,y) --> A046802(x,y+1), transform of e.g.f. for the graded number of positroids of the totally nonnegative Grassmannians G+(k,n); enumerates faces of the stellahedra.

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A120337 Euler-irregular primes p dividing E(2k) for some 2k < p-1.

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A119880 Expansion of e.g.f. exp(2x)*sech(x).

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A173774 The arithmetic mean of (21k + 8)binomial(2*k,k)^3 (k=0..n-1).

A163747 Expansion of e.g.f. 2exp(x)(1-exp(x))/(1+exp(2*x)).

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