A119879 -id:A119879 - cp's OEIS frontend

A130850 Triangle read by rows, 0 <= k <= n, T(n,k) = Sum_{j=0..n} A(n,j)*binomial(n-j,k) where A(n,j) are the Eulerian numbers A173018.

1, 1, 1, 2, 3, 1, 6, 12, 7, 1, 24, 60, 50, 15, 1, 120, 360, 390, 180, 31, 1, 720, 2520, 3360, 2100, 602, 63, 1, 5040, 20160, 31920, 25200, 10206, 1932, 127, 1, 40320, 181440, 332640, 317520, 166824, 46620, 6050, 255, 1, 362880, 1814400, 3780000, 4233600, 2739240, 1020600, 204630, 18660, 511, 1

Offset: 0

Philippe Deléham, Aug 20 2007

Old name was: Triangle T(n,k), 0<=k<=n, read by rows given by [1,1,2,2,3,3,4,4,5,5,...] DELTA [1,0,2,0,3,0,4,0,5,0,6,0,...] where DELTA is the operator defined in A084938.
Vandervelde (2018) refers to this as the Worpitzky number triangle - N. J. A. Sloane, Mar 27 2018 [Named after the German mathematician Julius Daniel Theodor Worpitzky (1835-1895). - Amiram Eldar, Jun 24 2021]
Triangle given by A123125*A007318 (as infinite lower triangular matrices), A123125 = Euler's triangle, A007318 = Pascal's triangle; A007318*A123125 gives A046802.
Taylor coefficients of Eulerian polynomials centered at 1. - Louis Zulli, Nov 28 2015
A signed refinement is A263634. - Tom Copeland, Nov 14 2016
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 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 this entry (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

			Triangle begins:
1
1      1
2      3       1
6      12      7       1
24     60      50      15      1
120    360     390     180     31      1
720    2520    3360    2100    602     63      1
5040   20160   31920   25200   10206   1932    127    1
40320  181440  332640  317520  166824  46620   6050   255   1
362880 1814400 3780000 4233600 2739240 1020600 204630 18660 511 1
...

G. C. Greubel, Table of n, a(n) for n = 0..1034 [a(438) and a(901) corrected by _Georg Fischer_, Nov 10 2021]
Nguyen-Huu-Bong, Some Combinatorial Properties of Summation Operators, J. Comb. Theory, Ser. A, Vo. 11, No. 3 (1971), pp. 213-221. See Table on page 214.
John K. Sikora, On Calculating the Coefficients of a Polynomial Generated Sequence Using the Worpitzky Number Triangles, arXiv:1806.00887 [math.NT], 2018.
Sam Vandervelde, The Worpitzky Numbers Revisited, Amer. Math. Monthly, Vol. 125, No. 3 (2018), pp. 198-206.

Essentially reverse of A028246.
Variants are A130850, A075263, A163626.
Cf. A000142, A001710, A005460, A005461, A005462, A005463, A005464, A263634.
Cf. A019538, A028246, A046802, A090582, A119879, A123125, A248727.

Table[(n-k)!*StirlingS2[n+1, n-k+1], {n, 0, 10}, {k, 0, n}] (* G. C. Greubel, Nov 15 2015 *)

t(n, k) = (n-k)!*stirling(n+1, n-k+1, 2);
tabl(nn) = for (n=0, 10, for (k=0, n, print1(t(n,k),", ")); print()); \\ Michel Marcus, Nov 16 2015

from sage.combinat.combinat import eulerian_number
def A130850(n, k):
    return add(eulerian_number(n, j)*binomial(n-j, k) for j in (0..n))
for n in (0..7): [A130850(n, k) for k in (0..n)] # Peter Luschny, May 21 2013

T(n,k) = (-1)^k*A075263(n,k).
T(n,k) = (n-k)!*A008278(n+1,k+1).
T(n,n-1) = 2^n - 1 for n > 0. - Derek Orr, Dec 31 2015
E.g.f.: x/(e^(-x*t)*(1+x)-1). - Tom Copeland, Nov 14 2016
Sum_{k=1..floor(n/2)} T(n,2k) = Sum_{k=0..floor(n/2)} T(n,2k+1) = A000670(n). - Jacob Sprittulla, Oct 03 2021

New name from Peter Luschny, May 21 2013

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

A162660 Triangle read by rows: coefficients of the complementary Swiss-Knife polynomials.

Original entry on oeis.org

0, 1, 0, 0, 2, 0, -2, 0, 3, 0, 0, -8, 0, 4, 0, 16, 0, -20, 0, 5, 0, 0, 96, 0, -40, 0, 6, 0, -272, 0, 336, 0, -70, 0, 7, 0, 0, -2176, 0, 896, 0, -112, 0, 8, 0, 7936, 0, -9792, 0, 2016, 0, -168, 0, 9, 0, 0, 79360, 0, -32640, 0, 4032, 0, -240, 0, 10, 0

Offset: 0

Peter Luschny, Jul 09 2009

Definition. V_n(x) = (skp(n, x+1) - skp(n, x-1))/2 where skp(n,x) are the Swiss-Knife polynomials A153641. - Peter Luschny, Jul 23 2012
Equivalently, let the polynomials V_n(x) (n>=0) defined by V_n(x) = Sum_{k=0..n} Sum_{v=0..k} (-1)^v*C(k,v)*L(k)*(x+v+1)^n; the sequence L(k) = -1 - H(k-1)*(-1)^floor((k-1)/4) / 2^floor(k/2) if k > 0 and L(0)=0; H(k) = 1 if k mod 4 <> 0, otherwise 0.
(1) V_n(0) = 2^n * Euler(n,1) for n > 0, A155585.
(2) V_n(1) = 1 - Euler(n).
(3) V_{n-1}(0) n / (4^n - 2^n) = B_n gives for n > 1 the Bernoulli numbers A027641/A027642.
(4) V_{n-1}(0) n (2/2^n-2)/(2^n-1) = G_n the Genocchi number A036968 for n > 1.
(5) V_n(1/2)2^{n} - 1 is a signed version of the generalized Euler (Springer) numbers, see A001586.
The Swiss-Knife polynomials (A153641) are complementary to the polynomials defined here. Adding both gives polynomials with e.g.f. exp(x*t)*(sech(t)+tanh(t)), the coefficients of which are a signed variant of A109449.
The Swiss-Knife polynomials as well as the complementary Swiss-Knife polynomials are closely related to the Bernoulli and Euler polynomials. Let F be a sequence and
P_{F}[n](x) = Sum_{k=0..n} Sum_{v=0..k} (-1)^v*C(k,v)*F(k)*(x+v+1)^n.
V_n(x) = P_{F}[n](x) with F(k)=L(k) defined above, are the Co-Swiss-Knife polynomials,
W_n(x) = P_{F}[n](x) with F(k)=c(k) the Chen sequence defined in A153641 are the Swiss-Knife polynomials.
B_n(x) = P_{F}[n](x-1) with F(k)=1/(k+1) are the Bernoulli polynomials,
E_n(x) = P_{F}[n](x-1) with F(k)=2^(-k) are the Euler polynomials.
The most striking formal difference between the Swiss-Knife-type polynomials and the Bernoulli-Euler type polynomials is: The SK-type polynomials have integer coefficients whereas the BE-type polynomials have rational coefficients.
Let R be the exponential Riordan array (exp(x)*sech(x), x) = P * A119879 = 2*P(I + P^2)^(-1) where P denotes Pascal's triangle A007318. Then T = R - I. - Peter Bala, Mar 07 2024

			Triangle begins:
  [0]    0;
  [1]    1,     0;
  [2]    0,     2,     0;
  [3]   -2,     0,     3,   0;
  [4]    0,    -8,     0,   4,    0;
  [5]   16,     0,   -20,   0,    5,    0;
  [6]    0,    96,     0, -40,    0,    6,    0;
  [7] -272,     0,   336,   0,  -70,    0,    7,  0;
  [8]    0, -2176,     0, 896,    0, -112,    0,  8,  0;
  [9] 7936,     0, -9792,   0, 2016,    0, -168,  0,  9,  0;

Ayse Yilmaz Ceylan and Yilmaz Simsek, Formulae for Generalization of Touchard Polynomials with Their Generating Functions, Symmetry (2025) Vol. 17, Issue 7, Art. No. 1126. See Eq. 28 and after.
Leonhard Euler (1735), De summis serierum reciprocarum, Opera Omnia I.14, E 41, 73-86; On the sums of series of reciprocals, arXiv:math/0506415 [math.HO], 2005-2008.
Peter Luschny, The Swiss-Knife polynomials.
Peter Luschny, Swiss-Knife Polynomials and Euler Numbers.
Wikipedia, Bernoulli number.
J. Worpitzky, Studien über die Bernoullischen und Eulerschen Zahlen, Journal für die reine und angewandte Mathematik, 94 (1883), 203-232.

V_n(k), n=0, 1, ..., k=0: A155585, k=1: A009832,
V_n(k), k=0, 1, ..., V_0: A000004, V_1: A000012, V_2: A005843, V_3: A100536.
Cf. A153641, A154341, A154342, A154343, A154344, A154345.

# Polynomials V_n(x):
V := proc(n,x) local k,pow; pow := (n,k) -> `if`(n=0 and k=0,1,n^k); add(binomial(n,k)*euler(k)*pow(x+1,n-k),k=0..n) - pow(x,n) end:
# Coefficients a(n):
seq(print(seq(coeff(n!*coeff(series(exp(x*t)*tanh(t),t,16),t,n),x,k),k=0..n)),n=0..8);

skp[n_, x_] := Sum[Binomial[n, k]*EulerE[k]*x^(n-k), {k, 0, n}]; v[n_, x_] := (skp[n, x+1]-skp[n, x-1])/2; t[n_, k_] := Coefficient[v[n, x], x, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 09 2014 *)

R = PolynomialRing(QQ, 'x')
@CachedFunction
def skp(n, x) : # Swiss-Knife polynomials A153641.
    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 A162660(n,k) : return 0 if k > n else R((skp(n, x+1)-skp(n, x-1))/2)[k]
matrix(ZZ, 9, A162660) # Peter Luschny, Jul 23 2012

T(n, k) = [x^(n-k)](skp(n,x+1)-skp(n,x-1))/2 where skp(n,x) are the Swiss-Knife polynomials A153641. - Peter Luschny, Jul 23 2012
E.g.f. exp(x*t)*tanh(t) = 0*(t^0/0!) + 1*(t^1/1!) + (2*x)*(t^2/2!) + (3*x^2-2)*(t^3/3!) + ...
V_n(x) = -x^n + Sum_{k=0..n} C(n,k)*Euler(k)*(x+1)^(n-k).

A119881 Expansion of e.g.f. exp(3x)sech(x).

Original entry on oeis.org

1, 3, 8, 18, 32, 48, 128, 528, 512, -6912, 2048, 357888, 8192, -22351872, 32768, 1903822848, 131072, -209865080832, 524288, 29088886161408, 2097152, -4951498048929792, 8388608, 1015423886523629568, 33554432, -246921480190140874752, 134217728, 70251601603944228323328

Offset: 0

Paul Barry, May 26 2006

Transform of 3^n under the matrix A119879.

Also the Swiss-Knife polynomials A153641 evaluated at x=3. - Peter Luschny, Nov 23 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A027641, A027642, A119879, A119880, A153641.

EulerPoly:= func< n,x | (&+[ (&+[ (-1)^j*Binomial(k,j)*(x+j)^n : j in [0..k]])/2^k: k in [0..n]]) >;
A119881:= func< n| (-2)^n*EulerPoly(n,-1) >;
[A119881(n): n in [0..40]]; // G. C. Greubel, Jun 07 2023

a := proc(n) add(binomial(n,k)*bernoulli(k,1)*2^(n+k)/(n-k+1),k=0..n) end: # Peter Luschny, Dec 14 2008
a := n -> 2^n*abs(euler(n,-1)):  # Peter Luschny, Jan 25 2009
P := proc(n,x) option remember; if n = 0 then 1 else
   (n*x+2*(1-x))*P(n-1,x)+x*(1-x)*diff(P(n-1,x),x);
   expand(%) fi end:
A119881 := n -> subs(x=-1,P(n,x)):
seq(A119881(n), n=0..27);  # Peter Luschny, Mar 07 2014

Table[2^(n+1) (Zeta[-n] (2^(n+1)-1)+1), {n,0,27}] (* Peter Luschny, Jul 16 2013 *)
Range[0, 30]! CoefficientList[Series[Exp[3 x] Sech[x], {x, 0, 30}], x] (* Vincenzo Librandi, Mar 08 2014 *)

my(x='x+O('x^66)); Vec(serlaplace(exp(3*x)/cosh(x))) \\ Joerg Arndt, Apr 20 2013

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))
A119881 = lambda n: skp(n,3)
[A119881(n) for n in (0..27)]  # Peter Luschny, Nov 23 2012

a(n) = Sum_{k=0..n} A119879(n,k)*3^k.
a(n) = Sum_{k=0..n} binomial(n,k)*B(k,1)*2^(n+k)/(n-k+1). Here B(k,1) are the Bernoulli number A027641(k)/A027642(k) with the exception B(1,1)=1/2. - Peter Luschny, Dec 14 2008
a(n) = 2^n |E(n,-1)| where E(n,x) are the Euler polynomials. - Peter Luschny, Jan 25 2009
The odd part of a(n) = numerator(Euler(n,2)/2) = 1, 3, 1, 9, 1, 3, 1, 33, 1, -27, 1, 699, ... (compare A143074). - Peter Luschny, Nov 23 2012
G.f.: 1/Q(0), where Q(k) = 1 - 2*x - x*(k+1)/(1+x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 19 2013
G.f.: 1/Q(0), where Q(k) = 1 - 4*x + x*(k+1)/(1-x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 20 2013
a(n) = 2^(n+1)*(zeta[-n]*(2^(n+1)-1)+1). - Peter Luschny, Jul 16 2013
E.g.f.: 2/Q(0), where Q(k) = 1 + 2^k/( 1 - 2*x/( 2*x - 2^k*(k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 16 2013
a(n) = 2^(n+1)*(1+(-1)^n*(2^(n+1)-1)*Bernoulli(n+1)/(n+1)). - Vladimir Reshetnikov, Oct 21 2015

A119882 Expansion of e.g.f.: (1+x)*sech(x).

Original entry on oeis.org

1, 1, -1, -3, 5, 25, -61, -427, 1385, 12465, -50521, -555731, 2702765, 35135945, -199360981, -2990414715, 19391512145, 329655706465, -2404879675441, -45692713833379, 370371188237525, 7777794952988025, -69348874393137901, -1595024111042171723, 15514534163557086905

Offset: 0

Paul Barry, May 26 2006

Transform of binomial(1,n) under the matrix A119879.

Robert Israel, Table of n, a(n) for n = 0..480

Cf. A119879, A122045.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( (1+x)/Cosh(x) ))); // G. C. Greubel, Jun 07 2023

seq(`if`(n::odd, n*euler(n-1), euler(n)), n=0..24); # Peter Luschny, May 30 2016

Table[EulerE[n] + n*EulerE[n-1], {n,20}] (* Benedict W. J. Irwin, May 30 2016 *)
With[{nn=30},CoefficientList[Series[(1+x)Sech[x],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 22 2025 *)

Vec(serlaplace((1+x)/cosh(x + O(x^30)))) \\ Andrew Howroyd, Feb 27 2018

def A119882(n): return n*euler_number(n-1) if n%2==1 else euler_number(n)
[A119882(n) for n in range(41)] # G. C. Greubel, Jun 07 2023

a(n) = Sum_{k=0..n} A119879(n,k)*C(1,k).
E.g.f.: (1+x)/sech(x) = (1+x)*(1 - x^2/Q(0)), where Q(k) = (2*k+1)*(2*k+2) + x^2 - (2*k+1)*(2*k+2)*x^2/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 06 2013
a(n) = EulerE[n] + n*EulerE[n-1], n>0. - Benedict W. J. Irwin, May 30 2016

A119883 Expansion of E.g.f. (1 + 2x + x^2/2) sech(x).

Original entry on oeis.org

1, 2, 0, -6, -1, 50, 14, -854, -323, 24930, 11804, -1111462, -631621, 70271890, 46590634, -5980829430, -4531805575, 659311412930, 562021682744, -91385427666758, -86555950096265, 15555589905976050, 16206870089730374, -3190048222084343446, -3625755168948973771

Offset: 0

Paul Barry, May 26 2006

Transform of binomial(2,n) under the matrix A119879.

G. C. Greubel, Table of n, a(n) for n = 0..480

Cf. A119879, A122045.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( (1+2*x+x^2/2) /Cosh(x) ))); // G. C. Greubel, Jun 07 2023

Table[If[n<2, n+1, EulerE[n] +2*n*EulerE[n-1] +n*(n-1)*EulerE[n-2]/2], {n,0,30}] (* Benedict W. J. Irwin, May 30 2016 *)
With[{nn=30},CoefficientList[Series[(1+2x+x^2/2)Sech[x],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jul 01 2018 *)

my(x='x+O('x^44)); Vec(serlaplace((1 + 2*x + x^2/2) / cosh(x))) \\ Joerg Arndt, Jun 01 2016

E=euler_number
def A119883(n):
    if n<2: return n+1
    else: return E(n) +2*n*E(n-1) +binomial(n,2)*E(n-2)
[A119883(n) for n in range(41)] # G. C. Greubel, Jun 07 2023

E.g.f.: (1 + 2*x + x^2/2) * sech(x).
a(n) = Sum_{k=0..n} A119879(n,k)*C(2,k).
a(n) = EulerE(n) + 2*n*EulerE(n-1) + n*(n-1)*EulerE(n-2)/2, n>1. - Benedict W. J. Irwin, May 30 2016

A119884 Expansion of e.g.f. sech(x)/(1-x).

Original entry on oeis.org

1, 1, 1, 3, 17, 85, 449, 3143, 26529, 238761, 2337089, 25707979, 311198513, 4045580669, 56438768385, 846581525775, 13564695924545, 230599830717265, 4148392073235329, 78819449391471251, 1576759359017662545, 33111946539370913445, 728393474991766957889

Offset: 0

Paul Barry, May 26 2006

Transform of n! under the matrix A119879.

G. C. Greubel, Table of n, a(n) for n = 0..445

Cf. A119879, A122045.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( 1/((1-x)*Cosh(x)) ))); // G. C. Greubel, Jun 07 2023

restart: G(x):= sech(x)/(1-x): f[0]:=G(x): for n from 1 to 21 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..20); # Zerinvary Lajos, Apr 03 2009

CoefficientList[Series[1/((1-x)*(E^x/2+E^(-x)/2)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 25 2013 *)
Table[n!*Sum[EulerE[j]/j!, {j,0,n}], {n,0,40}] (* G. C. Greubel, Jun 07 2023 *)

def A119884(n): return factorial(n)*sum(euler_number(j)/factorial(j) for j in range(n+1))
[A119884(n) for n in range(41)] # G. C. Greubel, Jun 07 2023

a(n) = Sum_{k=0..n} A119879(n,k) * k!.
E.g.f.: 1/U(0) where U(k) = 1 - x/(1 - x/(x - (2*k+1)*(2*k+2)/U(k+1))); (continued fraction). - Sergei N. Gladkovskii, Oct 17 2012
a(n) ~ n! * 2*exp(1)/(exp(2)+1). - Vaclav Kotesovec, Sep 25 2013
a(n) = n! * Sum_{j=0..n} A122045(j)/j!. - G. C. Greubel, Jun 07 2023

cp's OEIS Frontend

A130850 Triangle read by rows, 0 <= k <= n, T(n,k) = Sum_{j=0..n} A(n,j)*binomial(n-j,k) where A(n,j) are the Eulerian numbers A173018.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Python

Sage

Formula

Extensions

A162660 Triangle read by rows: coefficients of the complementary Swiss-Knife polynomials.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A119881 Expansion of e.g.f. exp(3*x)*sech(x).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A119882 Expansion of e.g.f.: (1+x)*sech(x).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

A119883 Expansion of E.g.f. (1 + 2*x + x^2/2) * sech(x).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

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

A119881 Expansion of e.g.f. exp(3x)sech(x).

A119883 Expansion of E.g.f. (1 + 2x + x^2/2) sech(x).