A117442 -id:A117442 - cp's OEIS frontend

A117440 A cyclically signed version of Pascal's triangle.

1, 1, 1, -1, 2, 1, -1, -3, 3, 1, 1, -4, -6, 4, 1, 1, 5, -10, -10, 5, 1, -1, 6, 15, -20, -15, 6, 1, -1, -7, 21, 35, -35, -21, 7, 1, 1, -8, -28, 56, 70, -56, -28, 8, 1, 1, 9, -36, -84, 126, 126, -84, -36, 9, 1, -1, 10, 45, -120, -210, 252, 210, -120, -45, 10, 1

Offset: 0

Paul Barry, Mar 16 2006

			Triangle begins:
   1;
   1,  1;
  -1,  2,   1;
  -1, -3,   3,   1;
   1, -4,  -6,   4,   1;
   1,  5, -10, -10,   5,   1;
  -1,  6,  15, -20, -15,   6, 1;
  -1, -7,  21,  35, -35, -21, 7, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A007318, A009545 (row sums), A117441 (diagonal sums), A117442 (inverse).

Table[Binomial[n, k]*(Cos[Pi*(n-k)/2] +Sin[Pi*(n-k)/2]), {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 01 2021 *)

flatten([[binomial(n,k)*( cos(pi*(n-k)/2) + sin(pi*(n-k)/2) ) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 01 2021

Column k has e.g.f.: (x^k/k!)*(cos(x) + sin(x)).
T(n, k) = binomial(n,k)*(cos(Pi*(n-k)/2) + sin(Pi*(n-k)/2)).
Sum_{k=0..n} T(n, k) = A009545(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A117441(n) (upward diagonal sums).
G.f.: (1 + x - x*y)/(1 - 2*x*y + x^2*(1 + y^2)). - Stefano Spezia, Mar 10 2024

A161722 Generalized Bernoulli numbers B_n(X,0), X a Dirichlet character modulus 8.

Original entry on oeis.org

0, 2, -44, 2166, -196888, 28730410, -6148123332, 1813990148894, -705775346640176, 350112935442888018, -215681051222514096220, 161537815119247080938182, -144555133640020128085896264, 152323571317104251881943249786

Offset: 0

Peter Bala, Jun 18 2009

Let X be a periodic arithmetical function with period m. The generalized Bernoulli polynomials B_n(X,x) attached to X are defined by means of the generating function
(1)... t*exp(t*x)/(exp(m*t)-1) * Sum_{r = 0..m-1} X(r)*exp(r*t) = Sum_{n >= 0} B_n(X,x)*t^n/n!.
The values B_n(X,0) are generalizations of the Bernoulli numbers (case X = 1). For the theory and properties of these polynomials and numbers see [Cohen, Section 9.4]. In the present case, X is chosen to be the Dirichlet character modulus 8 given by
(2)... X(8*n+1) = X(8*n+7) = 1; X(8*n+3) = X(8*n+5) = -1; X(2*n) = 0.
The odd-indexed generalized Bernoulli numbers B_(2*n+1)(X,0) vanish. The current sequence lists the even-indexed values B_(2*n)(X,0).
The coefficients of the generalized Bernoulli polynomials B_n(X,x) are listed in A151751.

H. Cohen, Number Theory - Volume II: Analytic and Modern Tools, Graduate Texts in Mathematics. Springer-Verlag.

Cf. A000464, A153641, A151751, A117442.

G := x*sinh(x)/cosh(2*x): ser := series(G, x, 30):
seq((2*n)!*coeff(ser, x, 2*n), n = 0..14); # Peter Luschny, Nov 26 2020
# After an observation of F. Chapoton in A117442:
A161722 := proc(n) 4^n*add(binomial(2*n, k)*euler(k)*((x+1)/2)^(2*n-k), k=0..2*n);
coeff(%, x, 1) end: seq(A161722(n), n=0..13); # Peter Luschny, Nov 26 2020

terms = 13;
(CoefficientList[x(Sinh[x]/Cosh[2x]) + O[x]^(2terms+3), x] Range[0, 2terms+2]!)[[ ;; ;; 2]] (* Jean-François Alcover, Nov 16 2020 *)

(1)... a(n) = (-1)^(n+1)*2*n*A000464(n-1).
The sequence of generalized Bernoulli numbers
(2)... [B_n(X,0)]n>=2 = [2,0,-44,0,2166,0,...]
has the e.g.f.
(3)... t*(exp(t)-exp(3*t)-exp(5*t)+exp(7*t))/(exp(8*t)-1),
which simplifies to
(4)... t*sinh(t)/cosh(2*t) = 2*t^2/2! - 44*t^4/4! + ....
Hence
(5)... B_(2*n)(X,0) = (-1)^(n+1)*2*n*A000464(n-1) and B_(2*n+1)(X,0) = 0.
a(n) = (-1/2)*16^n*n*euler(2*n-1, 1/4) for n >= 1 after a formula of Peter Bala in A000464. - Peter Luschny, Nov 26 2020

Cross-reference corrected by Peter Bala, Jun 22 2009

Offset set to 0 and a(0) = 0 prepended by Peter Luschny, Nov 26 2020

A117443 Expansion of e.g.f.: exp(x)/(cos(x) + sin(x)).

Original entry on oeis.org

1, 0, 2, -4, 28, -160, 1272, -11184, 114448, -1309440, 16680992, -233587264, 3569157568, -59075960320, 1053056675712, -20111857791744, 409715696197888, -8868323731660800, 203247024658514432, -4916860703228314624, 125206830774036241408, -3347784042587048058880

Offset: 0

Paul Barry, Mar 16 2006

Row sums of number triangle A117442. Binomial transform of alternating sign Springer numbers (-1)^n*A001586(n).

G. C. Greubel, Table of n, a(n) for n = 0..425

Cf. A117442.

CoefficientList[Series[E^x/(Cos[x]+Sin[x]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 04 2014 *)
A117442[n_, k_]:= (-1)^(n-k)*Binomial[n, k]*Abs[Numerator[EulerE[n-k, 1/4]]]; Table[Sum[A117442[n, k], {k, 0, n}], {n, 0, 30}] (* G. C. Greubel, Jun 02 2021 *)

my(x='x+O('x^30)); Vec(serlaplace(exp(x)/(cos(x) + sin(x)))) \\ Michel Marcus, Jun 02 2021

@CachedFunction
def f(n): return (-1/4)^n*sum( binomial(n, j)*2^j*euler_number(j) for j in (0..n) ) # f(n) = Euler(n, 1/4)
def A117443(n): return sum( (-1)^(n+k)*binomial(n,k)*abs(numerator(f(n-k))) for k in (0..n) )
[A117443(n) for n in (0..30)] # G. C. Greubel, Jun 02 2021

E.g.f.: 1/Q(0); Q(k)=1-(x^2)/((4*k+1)*(2*k+1)+2*x*(4*k+1)*(2*k+1)/(4*k+3-2*x-x*(4*k+3)/(x-(4*k+4)/Q(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Nov 28 2011
G.f.: 1/Q(0) where Q(k) = 1 + 4*k*x - 2*x^2*(2*k + 1)^2/( 1 + (4*k+2)*x - 2*x^2*(2*k + 2)^2/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 10 2013
a(n) ~ (-1)^n * n! * 2^(2*n+3/2) / (Pi^(n+1) * exp(Pi/4)). - Vaclav Kotesovec, Aug 04 2014
a(n) = Sum_{k=0..n} (-1)^(n+k) * binomial(n, k) * abs(numerator( Euler(n-k, 1/4) )), where Euler(n, x) is the Euler number polynomial. - G. C. Greubel, Jun 02 2021

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A117440 A cyclically signed version of Pascal's triangle.

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A161722 Generalized Bernoulli numbers B_n(X,0), X a Dirichlet character modulus 8.

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A117443 Expansion of e.g.f.: exp(x)/(cos(x) + sin(x)).

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