A009843 -id:A009843 - cp's OEIS frontend

A274705 Rectangular array read by ascending antidiagonals. Row n has the exponential generating function 1/M_{n}(z^n) where M_{n}(z) is the n-th Mittag-Leffler function, nonzero coefficients only, for n>=1.

1, 1, -2, 1, -3, 3, 1, -4, 25, -4, 1, -5, 133, -427, 5, 1, -6, 621, -15130, 12465, -6, 1, -7, 2761, -437593, 4101799, -555731, 7, 1, -8, 11999, -12012016, 1026405753, -2177360656, 35135945, -8, 1, -9, 51465, -325204171, 243458990271, -6054175060941, 1999963458217, -2990414715, 9

Offset: 0

Peter Luschny, Jul 03 2016

			Array starts:
n=1: {1, -2,  3, -4, 5, -6, 7, -8,  9, -10,  11,...} [A181983]
n=2: {1, -3, 25, -427, 12465, -555731, 35135945,...} [A009843]
n=3: {1, -4, 133, -15130, 4101799,  -2177360656,...} [A274703]
n=4: {1, -5, 621, -437593, 1026405753, -6054175060941,...} [A274704]
n=5: {1, -6, 2761, -12012016, 243458990271, ...}

L. Carlitz, Some arithmetic properties of the Olivier functions, Math. Ann. 128 (1954), 412-419.
H. J. Haubold, A. M. Mathai, and R. K. Saxena, Mittag-Leffler Functions and Their Applications, Journal of Applied Mathematics, vol. 2011, Article ID 298628, 51 pages.
L. Olivier, Bemerkungen über eine Art von Functionen, welche ähnliche Eigenschaften haben, wie der Cosinus und Sinus, J. Reine Angew. Math. 2 (1827), 243-251.
Eric Weisstein's MathWorld, Generalized hyperbolic functions.

Cf. A009843, A181983, A274703, A274704.

ibn := proc(m, k) local w, om, t;
w := exp(2*Pi*I/m); om := m*x/add(exp(x*w^j), j=0..m-1);
t := series(om, x, k+m); simplify(k!*coeff(t,x,k)) end:
seq(seq(ibn(n-k+2, n*k-n-k^2+3*k-1), k=1..n+1),n=0..8);

A274705Row[m_] := Module[{c}, c = CoefficientList[Series[1/MittagLefflerE[m,z^m],
{z,0,12*m}],z]; Table[Factorial[m*n+1]*c[[m*n+1]], {n,0,9}] ]
Table[Print[A274705Row[n]], {n,1,6}]

def ibn(m, k):
    w = exp(2*pi*I/m)
    om = m*x/sum(exp(x*w^j) for j in range(m))
    t = taylor(om, x, 0, k + m)
    return simplify(factorial(k)*t.list()[k])
def A274705_row(m, size):
    return [ibn(m, k) for k in range(1, m*size, m)]
for n in (1..4): print(A274705_row(n, 8))

Recurrence for the m-th row: R(m, n) = -Sum_{k=0..n-1} binomial(m*n+1, m*k+1)*R(m, k) for n >= 1. See Carlitz (1.3).

A189238 E.g.f. x/cos(x)*exp(x/cos(x)).

Original entry on oeis.org

1, 2, 6, 28, 120, 726, 4424, 31928, 249984, 2131690, 20027392, 199240020, 2162269824, 24676708798, 302660939520, 3897794538864, 53264941301760, 763279034957010, 11499327153704960, 181271619624350860

Offset: 1

Vladimir Kruchinin, Apr 19 2011

nonn

A(x)=A009843(x)*exp(A009843(x)).

Vincenzo Librandi, Table of n, a(n) for n = 1..100

a(n):=sum(binomial(n,k)*k*(1+(-1)^(n-k))*sum(sum(binomial(m,j)/2^(j)*sum((-1)^((n-k)/2-j)*binomial(j,i)*(j-2*i)^(n-k),i,0,floor((j-1)/2)),j,1,m)*binomial(k+m-1,k-1),m,1,n-k),k,1,n-1)+n;

x='x+O('x^66); /* that many terms */
egf=x/cos(x)*exp(x/cos(x)); /* = x + x^2 + x^3 + 7/6*x^4 + x^5 + 121/120*x^6+ ... */
Vec(serlaplace(egf)) /* show terms */ /* Joerg Arndt, Apr 21 2011 */

a(n)=sum(k=1..n-1, binomial(n,k)*k*(1+(-1)^(n-k))*sum(j=1..m, sum(i=0..floor((j-1)/2), binomial(m,j)/2^(j)*sum((-1)^((n-k)/2-j)*binomial(j,i)*(j-2*i)^(n-k)))*binomial(k+m-1,k-1),m,1,n-k))+n.

A243963 a(n) = n4^n(-Z(1-n, 1/4)/2 + Z(1-n, 3/4)/2 - Z(1-n, 1)*(1 - 2^(-n))) for n > 0 and a(0) = 0, where Z(n, c) is the Hurwitz zeta function.

Original entry on oeis.org

0, 0, 2, 3, -8, -25, 96, 427, -2176, -12465, 79360, 555731, -4245504, -35135945, 313155584, 2990414715, -30460116992, -329655706465, 3777576173568, 45692713833379, -581777702256640, -7777794952988025, 108932957168730112, 1595024111042171723, -24370173276164456448

Offset: 0

Paul Curtz, Jun 16 2014

sign

Previous name was: 0 followed by -(n+1)*A163747(n).
Difference table of a(n):
0,    0,    2,     3,    -8,   -25,...
0,    2,    1,   -11,   -17,   121,...
2,   -1,  -12,    -6,   138,   210,...
-3, -11,    6,   144,    72, -3144,...
-8,  17,  138,   -72, -3216, -1608,...
25, 121, -210, -3144,  1608,...
a(n) is an autosequence of second kind. Its inverse binomial transform is the signed sequence. Its main diagonal is the first upper diagonal multiplied by 2.

Cf. A132049, A065619.

a := n -> `if`(n=0, 0, n*4^n*(-Zeta(0, 1-n, 1/4)/2 + Zeta(0, 1-n, 3/4)/2 + Zeta(1-n)*(2^(-n)-1))): seq(a(n), n=0..24); # Peter Luschny, Jul 21 2020

a[0] = 0; a[n_] := -n*SeriesCoefficient[(2*E^x*(1 - E^x))/(1 + E^(2*x)), {x, 0, n-1}]*(n-1)!; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jun 17 2014 *)

a(n) = 0, 0, followed by (period 4: repeat 1, 1, -1, -1)*A065619(n+2).

a(2n) = (-1)^(n+1)A009752(n). a(2n+1) = (-1)^n*A009843(n+1).

New name by Peter Luschny, Jul 21 2020

A371688 Triangle read by rows: T(n, k) = (2n + 1)! [y^(2k)] [x^(2n+1)] arctan(sec(xy)sinh(x)).

Original entry on oeis.org

1, -1, 3, 5, -50, 25, -61, 1281, -2135, 427, 1385, -49860, 174510, -116340, 12465, -50521, 2778655, -16671930, 23340702, -8335965, 555731, 2702765, -210815670, 1932476975, -4637944740, 3478458555, -772990790, 35135945

Offset: 0

Peter Luschny, Apr 03 2024

Expansion of the exponential generating function arctan(sec(x*y)*sinh(x)), nonzero terms only.

			Triangle starts:
  [0]      1;
  [1]     -1,       3;
  [2]      5,     -50,        25;
  [3]    -61,    1281,     -2135,      427;
  [4]   1385,  -49860,    174510,  -116340,    12465;
  [5] -50521, 2778655, -16671930, 23340702, -8335965, 555731;

Cf. A000364 (column 0), A009843 (main diagonal), A012816 (row sums), A002436 (alternating row sums).

egf := arctan(sec(x*y)*sinh(x)):
serx := simplify(series(egf, x, 26)): coeffx := n -> n!*coeff(serx, x, n):
seq(lprint(seq(coeff(coeffx(2*n + 1), y, 2*k), k = 0..n)), n = 0..7);

T[n_,k_]:=(-1)^k*Binomial[2*n+1,2*k]*EulerE[2*n];Flatten[Table[T[n,k],{n,0,6},{k,0,n}]] (* Detlef Meya, Apr 07 2024 *)

T(n, k) = (-1)^k*binomial(2*n + 1, 2*k)*Euler(2*n). - Detlef Meya, Apr 07 2024

A245683 Array T(n,k) read by antidiagonals, where T(0,k) = -A226158(k) and T(n+1,k) = 2*T(n,k+1) - T(n,k).

Original entry on oeis.org

0, 2, 1, 0, 1, 1, -6, -3, -1, 0, 0, -3, -3, -2, -1, 50, 25, 11, 4, 1, 0, 0, 25, 25, 18, 11, 6, 3, -854, -427, -201, -88, -35, -12, -3, 0, 0, -427, -427, -314, -201, -118, -65, -34, -17, 24930, 12465, 6019, 2796, 1241, 520, 201, 68, 17, 0

Offset: 0

Paul Curtz, Jul 29 2014

Take T(n,k) = -A226158(k) and its transform via T(n+1,k) = 2*T(n,k+1) - T(n,k):
0,       1,    1,    0,   -1,    0,   3,   0, -17, ...
2,       1,   -1,   -2,    1,    6,  -3, -34, ...     = A230324
0,      -3,   -3,    4,   11,  -12, -65, ...
-6,     -3,   11,   18,  -35, -118, ...
0,      25,   25,  -88, -201, ...
50,     25, -201, -314, ...
0,    -427, -427, ...
-854, -427, ...
0, ...
Every row is alternatively an autosequence of the first kind, see A226158, and of the second kind, see A190339.
The second column is twice 1, -3, 25, -427, 12465, ... = (-1)^n*A009843(n) which is in the third column. See A132049(n), numerators of Euler's formula for Pi from the Bernoulli numbers, A243963 and A245244. Hence a link between the Genocchi numbers and Pi.
a(n) is the triangle of the increasing antidiagonals.

			Triangle a(n):
   0,
   2,  1,
   0,  1,  1,
  -6, -3, -1,  0,
   0, -3, -3, -2, -1,
  50, 25, 11,  4,  1,  0,
  etc.

Cf. A226158, A230324, A009843, A132049, A243963, A245244.

t[0, 0] = 0; t[0, 1] = 1; t[0, k_] := -k*EulerE[k-1, 0]; t[n_, k_] := t[n, k] = -t[n-1, k] + 2*t[n-1, k+1]; Table[t[n-k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 04 2014 *)

A371687 Triangle read by rows: T(n, k) = (-1)^(n-k) * (2n + 1)! [y^(2k)] [x^(2n+1)] arctan(sec(xy)tanh(x)).

Original entry on oeis.org

1, 4, 3, 80, 80, 25, 3904, 5376, 2660, 427, 354560, 626688, 433440, 131712, 12465, 51733504, 111738880, 99242880, 43804992, 9021540, 555731, 11070525440, 28258074624, 30647302400, 17666508288, 5509286640, 816337808, 35135945

Offset: 0

Peter Luschny, Apr 03 2024

Expansion of the exponential generating function arctan(sec(x*y)*tanh(x)), nonzero terms only.

			Triangle starts:
  [0]        1;
  [1]        4,         3;
  [2]       80,        80,       25;
  [3]     3904,      5376,     2660,      427;
  [4]   354560,    626688,   433440,   131712,   12465;
  [5] 51733504, 111738880, 99242880, 43804992, 9021540, 555731;

Cf. A002436 (column 0), A009843 (main diagonal), A012798 (row sums), A012835 (alternating row sums).

Cf. A371688.

egf := arctan(sec(x*y)*tanh(x)):
serx := simplify(series(egf, x, 26)): coeffx := n -> n!*coeff(serx, x, n):
seq(print(seq((-1)^(n-k)*coeff(coeffx(2*n+1), y, 2*k), k = 0..n)), n = 0..6);

cp's OEIS Frontend

A274705 Rectangular array read by ascending antidiagonals. Row n has the exponential generating function 1/M_{n}(z^n) where M_{n}(z) is the n-th Mittag-Leffler function, nonzero coefficients only, for n>=1.

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A189238 E.g.f. x/cos(x)*exp(x/cos(x)).

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A243963 a(n) = n*4^n*(-Z(1-n, 1/4)/2 + Z(1-n, 3/4)/2 - Z(1-n, 1)*(1 - 2^(-n))) for n > 0 and a(0) = 0, where Z(n, c) is the Hurwitz zeta function.

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A371688 Triangle read by rows: T(n, k) = (2*n + 1)! * [y^(2*k)] [x^(2*n+1)] arctan(sec(x*y)*sinh(x)).

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A245683 Array T(n,k) read by antidiagonals, where T(0,k) = -A226158(k) and T(n+1,k) = 2*T(n,k+1) - T(n,k).

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Mathematica

A371687 Triangle read by rows: T(n, k) = (-1)^(n-k) * (2*n + 1)! * [y^(2*k)] [x^(2*n+1)] arctan(sec(x*y)*tanh(x)).

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A243963 a(n) = n4^n(-Z(1-n, 1/4)/2 + Z(1-n, 3/4)/2 - Z(1-n, 1)*(1 - 2^(-n))) for n > 0 and a(0) = 0, where Z(n, c) is the Hurwitz zeta function.

A371688 Triangle read by rows: T(n, k) = (2n + 1)! [y^(2k)] [x^(2n+1)] arctan(sec(xy)sinh(x)).

A371687 Triangle read by rows: T(n, k) = (-1)^(n-k) * (2n + 1)! [y^(2k)] [x^(2n+1)] arctan(sec(xy)tanh(x)).