cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A151751 Triangle of coefficients of generalized Bernoulli polynomials associated with a Dirichlet character modulus 8.

Original entry on oeis.org

2, 0, 6, -44, 0, 12, 0, -220, 0, 20, 2166, 0, -660, 0, 30, 0, 15162, 0, -1540, 0, 42, -196888, 0, 60648, 0, -3080, 0, 56, 0, -1771992, 0, 181944, 0, -5544, 0, 72, 28730410, 0, -8859960, 0, 454860, 0, -9240, 0, 90
Offset: 2

Views

Author

Peter Bala, Jun 17 2009

Keywords

Comments

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..inf} B_n(X,x)*t^n/n!.
For the theory and properties of these polynomials 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.
Cf. A153641.

Examples

			The triangle begins
n\k|........0.......1........2.......3......4.......5.......6
=============================================================
.2.|........2
.3.|........0.......6
.4.|......-44.......0.......12
.5.|........0....-220........0......20
.6.|.....2166.......0.....-660.......0......30
.7.|........0...15162........0...-1540.......0.....42
.8.|..-196888.......0....60648.......0...-3080......0......56
...

References

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

Crossrefs

Cf. A000111, A000464, A000828, A001586, A109572, A153641, A161722.

Programs

  • Maple
    with(gfun):
for n from 2 to 10 do
Genbernoulli(n,x) := 8^(n-1)*(bernoulli(n,(x+1)/8)-bernoulli(n,(x+3)/8)-bernoulli(n,(x+5)/8)+bernoulli(n,(x+7)/8));
seriestolist(series(Genbernoulli(n,x),x,10))
end do;

Formula

TABLE ENTRIES
(1)... T(2*n,2*k+1) = 0, T(2*n+1,2*k) = 0;
(2)... T(2*n,2*k) = (-1)^(n-k-1)*C(2*n,2*k)*2*(n-k)*A000464(n-k-1);
(3)... T(2*n+1,2*k+1) = (-1)^(n-k-1)*C(2*n+1,2*k+1)*2*(n-k)*A000464(n-k-1);
where C(n,k) = binomial(n,k).
GENERATING FUNCTION
The e.g.f. for these generalized Bernoulli polynomials is
(4)... t*exp(x*t)*(exp(t)-exp(3*t)-exp(5*t)+exp(7*t))/(exp(8*t)-1)
= sum {n = 2..inf} B_n(X,x)*t^n/n! = 2*t^2/2! + 6*x*t^3/3! + (12*x^2 - 44)*t^4/4! + ....
In terms of the ordinary Bernoulli polynomials B_n(x)
(5)... B_n(X,x) = 8^(n-1)*{B_n((x+1)/8) - B_n((x+3)/8) - B_n((x+5)/8) + B_n((x+7)/8)}.
The B_n(X,x) are Appell polynomials of the form
(6)... B_n(X,x) = sum {j = 0..n} binomial(n,j)*B_j(X,0)*x*(n-j).
The sequence of generalized Bernoulli numbers
(7)... [B_n(X,0)]n>=2 = [2,0,-44,0,2166,0,...]
has the e.g.f.
(8)... t*(exp(t)-exp(3*t)-exp(5*t)+exp(7*t))/(exp(8*t)-1),
which simplifies to
(9)... t*sinh(t)/cosh(2*t).
Hence
(10)... B_(2*n)(X,0) = (-1)^(n+1)*2*n*A000464(n-1); B_(2*n+1)(X,0) = 0.
The sequence {B_(2*n)(X,0)}n>=2 is A161722.
RELATION WITH TWISTED SUMS OF POWERS
The generalized Bernoulli polynomials may be used to evaluate sums of k-th powers twisted by the function X(n). For the present case the result is
(11)... sum{n = 0..8*N-1} X(n)*n^k = 1^k-3^k-5^k+7^k- ... +(8*N-1)^k
= [B_(k+1)(X,8*N) - B_(k+1)(X,0)]/(k+1)
For the proof, apply [Cohen, Corollary 9.4.17 with m = 8 and x = 0].
MISCELLANEOUS
(12)... Row sums [2, 6, -32, ...] = (-1)^(1+binomial(n,2))*A109572(n)
= (-1)^(1+binomial(n,2))*n*A000828(n-1) = (-1)^(1+binomial(n,2))*n* 2^(n-2)*A000111(n-1).

A201594 E.g.f. satisfies: A(x) = 1/(1 - tan( x*A(x) )).

Original entry on oeis.org

1, 1, 4, 32, 384, 6176, 124928, 3049472, 87265280, 2865848320, 106258440192, 4391008927744, 200131590356992, 9973976451383296, 539604322034384896, 31496226303081709568, 1972926888464596598784, 132015791534989604028416, 9398128264859870497341440, 709248762402156849800413184
Offset: 0

Views

Author

Paul D. Hanna, Dec 02 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 32*x^3/3! + 384*x^4/4! + 6176*x^5/5! +...
The coefficients in the initial powers of G(x) = 1/(1 - tan(x)) begin:
G^1: [(1), 1, 2, 8, 40, 256, 1952, 17408, ..., A000828(n), ...];
G^2: [1,(2), 6, 28, 168, 1232, 10656, 106048, ...];
G^3: [1, 3,(12), 66, 456, 3768, 36192, 395616, ...];
G^4: [1, 4, 20,(128), 1000, 9184, 96800, 1150208, ...];
G^5: [1, 5, 30, 220,(1920), 19400, 222480, 2852320, ...];
G^6: [1, 6, 42, 348, 3360,(37056), 459312, 6317088, ...];
G^7: [1, 7, 56, 518, 5488, 65632, (874496), 12841808, ...];
G^8: [1, 8, 72, 736, 8496, 109568, 1562112, (24395776), ...]; ...
where coefficients in parenthesis form initial terms of this sequence:
[1/1, 2/2, 12/3, 128/4, 1920/5, 37056/6, 874496/7, 24395776/8, ...].

Crossrefs

Cf. A214224, A201595, A201128, A000828.

Programs

  • Mathematica
    CoefficientList[1/x*InverseSeries[Series[x*(1-Tan[x]), {x, 0, 21}], x],x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 12 2014 *)
  • PARI
    {a(n)=n!*polcoeff(1/x*serreverse(x-x*tan(x+x^2*O(x^n))),n)}
  • PARI
    {a(n)=n!*polcoeff(1/(1-tan(x+x*O(x^n)))^(n+1)/(n+1), n)}

Formula

E.g.f. A(x) satisfies: A( x*(1-tan(x)) ) = 1/(1-tan(x)).
E.g.f.: (1/x)*Series_Reversion( x*(1-tan(x)) ).
a(n) = [x^n/n!] 1/(1 - tan(x))^(n+1) / (n+1).
a(n) = A214224(n+1)/(n+1).
a(n) ~ n^(n-1) * ((t^2+1)/(t-1)^2)^(n+1/2) / (sqrt(2*(t+1)) * exp(n)), where t = 0.46733877379062994365... is the root of the equation t = tan((1-t)/(1+t^2)). - Vaclav Kotesovec, Jan 12 2014

A201923 E.g.f. satisfies: A(x) = 1/(cos(x*A(x)) - sin(x*A(x))).

Original entry on oeis.org

1, 1, 5, 44, 581, 10256, 227529, 6088256, 190930729, 6870227200, 279066777613, 12632667642880, 630670054092525, 34426087332253696, 2039903110075608977, 130404672744539242496, 8946117466489960168913, 655585000075494566199296, 51111210765059412626238741
Offset: 0

Views

Author

Paul D. Hanna, Dec 06 2011

Keywords

Comments

Compare e.g.f. to: LambertW(-x)/(-x) = (1/x)*Series_Reversion(x*(cosh(x) - sinh(x))).
The radius of convergence r of e.g.f. A(x) is given by:
r = t*(cos(t) - sin(t)) where tan(t) = (1-t)/(1+t), which evaluates as:
r = 0.21266685344074710045360679397024815598865409988038...
t = 0.40262817418811160981993252391123072456350647779608...
Further, A(r) = 1/(cos(t) - sin(t)), thus
A(r) = 1.89323426605496483543109751303457163422769666683274...

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 44*x^3/3! + 581*x^4/4! + 10256*x^5/5! +...
where
1/(cos(x)-sin(x)) = 1 + x + 3*x^2/2! + 11*x^3/3! + 57*x^4/4! + 361*x^5/5! + 2763*x^6/6! +...+ A001586(n)*x^n/n! +...
The coefficient of x^n/n! in powers of G(x) = 1/(cos(x)-sin(x)) begins:
G^1: [(1), 1, 3, 11, 57, 361, 2763, 24611, ..., A001586(n), ...];
G^2: [1,(2), 8, 40, 256, 1952, 17408, 177280, ..., A000828(n+1), ...];
G^3: [1, 3,(15), 93, 705, 6243, 63375, 724413, ...];
G^4: [1, 4, 24,(176), 1536, 15424, 175104, 2214656, ...];
G^5: [1, 5, 35, 295,(2905), 32525, 407435, 5638495, ...];
G^6: [1, 6, 48, 456, 4992, (61536), 841728, 12633216, ...];
G^7: [1, 7, 63, 665, 8001, 107527, (1592703), 25738265, ...];
G^8: [1, 8, 80, 928, 12160, 176768, 2816000, (48706048), ...]; ...
where coefficients in parenthesis form the initial terms of this sequence:
[1/1, 2/2, 15/3, 176/4, 2905/5, 61536/6, 1592703/7, 48706048/8, ...].

Crossrefs

Cf. A001586.

Programs

  • Mathematica
    CoefficientList[1/x*InverseSeries[Series[x*(Cos[x] - Sin[x]), {x, 0, 21}], x],x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 12 2014 *)
  • PARI
    {a(n)=local(X=x+x*O(x^n));n!*polcoeff(1/x*serreverse(x*(cos(X)-sin(X) )),n)}
  • PARI
    {a(n)=local(A=1+x,X=x+x*O(x^n));for(i=1,n,A=1/(cos(X*A) - sin(X*A)));n!*polcoeff(A,n)}

Formula

E.g.f. satisfies: A( x*(cos(x) - sin(x)) ) = 1/(cos(x) - sin(x)).
E.g.f: (1/x) * Series_Reversion( x*(cos(x) - sin(x)) ).
a(n) = [x^n/n!] 1/(cos(x)-sin(x))^(n+1) / (n+1).
a(n) ~ n^(n-1) * sqrt((t*cos(2*t))/(3+sin(2*t))) / (exp(n) * r^(n+1)), where r and t were described above. - Vaclav Kotesovec, Jan 12 2014

A235131 E.g.f. 1/(1 - tan(2*x))^(1/2).

Original entry on oeis.org

1, 1, 3, 23, 201, 2401, 33723, 564983, 10832721, 235620481, 5715989043, 153231400343, 4495861836441, 143343873560161, 4934418832685163, 182409363179578103, 7206898465033427361, 303073359560984509441, 13516205633151976330083, 637174194752117499594263
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 03 2014

Keywords

Comments

Generally, for e.g.f. 1/(1-tan(p*x))^(1/p) is a(n) ~ n! * 2^(2*n+1/p) * p^n / (Gamma(1/p) * Pi^(n+1/p) * n^(1-1/p)).

Crossrefs

Cf. A000828 (p=1), A235132 (p=3).

Programs

  • Mathematica
    CoefficientList[Series[1/(1 - Tan[2*x])^(1/2), {x, 0, 20}], x] * Range[0, 20]!

Formula

a(n) ~ n! * 2^(3*n+1/2) / (Pi^(n+1) * sqrt(n)).

A320957 a(n) = (1/6)*n!*[x^n] (2 + sec(3*x) + tan(3*x) + 3*sec(x) + 3*tan(x)).

Original entry on oeis.org

1, 1, 2, 10, 70, 656, 7442, 99280, 1515190, 26038016, 497227682, 10445708800, 239394707110, 5943715352576, 158922998335922, 4552807055288320, 139123511874743830, 4517007538261262336, 155283277843358756162, 5634815061983539363840, 215234080472925069593350
Offset: 0

Views

Author

Peter Luschny, Nov 08 2018

Keywords

Comments

See A320956 for motivation and definitions.

Crossrefs

Cf. A000111 (n=1), A000828 (n=2), this sequence (n=3), A321394 (n=4), A320956.

Programs

  • Maple
    egf := 2 + sec(3*x) + tan(3*x) + 3*sec(x) + 3*tan(x):
ser := series(egf, x, 22): seq((1/6)*n!*coeff(ser, x, n), n=0..20);
  • Mathematica
    m = 20;
egf = 2 + Sec[3x] + Tan[3x] + 3 Sec[x] + 3 Tan[x];
(1/6) CoefficientList[egf + O[x]^(m+1), x] Range[0, m]! (* Jean-François Alcover, Aug 19 2021 *)
  • PARI
    sectan(x) = 1/cos(x) + tan(x);
my(x='x+O('x^25)); Vec(serlaplace(2 + sectan(3*x) + 3*sectan(x)))/6 \\ Michel Marcus, Aug 19 2021

A321394 a(n) = (1/24)*n!*[x^n] (9 + sectan(4*x) + 6*sectan(2*x) + 8*sectan(x)) where sectan(x) = sec(x) + tan(x).

Original entry on oeis.org

1, 1, 2, 10, 75, 816, 11407, 194480, 3871075, 87700736, 2220246387, 62010892800, 1892138207375, 62591994720256, 2230631475837767, 85188256574494720, 3470563987113896475, 150234341045137637376, 6886077311552162511547, 333165973379285030666240, 16967906593223743786978375
Offset: 0

Views

Author

Peter Luschny, Nov 08 2018

Keywords

Comments

See A320956 for motivation and definitions.

Crossrefs

Cf. A000111 (n=1), A000828 (n=2), A320957 (n=3), this sequence (n=4), A320956.

Programs

  • Maple
    sectan := x -> sec(x) + tan(x): # sin(Pi/4 + x/2)*csc(Pi/4 - x/2)
egf := 9 + sectan(4*x) + 6*sectan(2*x) + 8*sectan(x):
ser := series(egf, x, 22): seq((1/24)*n!*coeff(ser, x, n), n=0..20);
  • Mathematica
    m = 20;
sectan[x_] := Sec[x] + Tan[x];
egf = 9 + sectan[4x] + 6 sectan[2x] + 8 sectan[x];
(1/24) CoefficientList[egf + O[x]^(m+1), x] Range[0, m]! (* Jean-François Alcover, Aug 19 2021 *)
  • PARI
    sectan(x) = 1/cos(x) + tan(x);
my(x='x+O('x^25)); Vec(serlaplace(9 + sectan(4*x) + 6*sectan(2*x) + 8*sectan(x)))/24 \\ Michel Marcus, Aug 19 2021

A193474 Table read by rows: The coefficients of the polynomials P(n, x) = Sum{k=0..n} Sum{j=0..k} (-1)^j * 2^(-k) * binomial(k, j) * (k-2*j)^n * x^(n-k).

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 6, 0, 1, 0, 24, 0, 8, 0, 0, 120, 0, 60, 0, 1, 0, 720, 0, 480, 0, 32, 0, 0, 5040, 0, 4200, 0, 546, 0, 1, 0, 40320, 0, 40320, 0, 8064, 0, 128, 0, 0, 362880, 0, 423360, 0, 115920, 0, 4920, 0, 1, 0, 3628800, 0, 4838400, 0, 1693440, 0, 130560, 0, 512, 0, 0
Offset: 1

Views

Author

Peter Luschny, Aug 01 2011

Keywords

Comments

See A196776 for a row reversed form of this triangle. - Peter Bala, Oct 06 2011

Examples

			The sequence of polynomials P(n, x) begins:
[0]    1;
[1]    1;
[2]    2;
[3]    6 +      x^2;
[4]   24 +    8*x^2;
[5]  120 +   60*x^2 +     x^4;
[6]  720 +  480*x^2 +  32*x^4;
[7] 5040 + 4200*x^2 + 546*x^4 + x^6.

Crossrefs

Cf. A196776, A000142, A006154, A191277, A000111, A000828, A000557, A107403, A007289, A005990.

Programs

  • Maple
    A193474_polynom := proc(n,x) local k, j;
add(add((-1)^j*2^(-k)*binomial(k,j)*(k-2*j)^n*x^(n-k),j=0..k),k=0..n) end: seq(seq(coeff(A193474_polynom(n,x),x,i),i=0..n),n=0..10);
  • Mathematica
    p[n_, x_] := Sum[(-1)^j*2^(-k)*Binomial[k, j]*(k-2*j)^n*x^(n-k), {k, 0, n}, {j, 0, k}]; t[n_, k_] := Coefficient[p[n, x], x, k]; t[0, 0] = 1; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 27 2014 *)

Formula

P(n, 0) = A000142(n).
P(n, 1) = A006154(n).
P(n, 2) = A191277(n).
P(n, i) = A000111(n+1), where i is the imaginary unit.
P(n, i)*2^n = A000828(n+1).
P(n, 1/2)*2^n = A000557(n).
P(n, 1/3)*3^n = A107403(n).
P(n, i/2)*2^n = A007289(n).
G(m, x) = 1/(1 - m*sinh(x)) is the generating function of m^n*P(n, 1/m).
GI(m, x) = 1/(1 - m*sin(x)) is the generating function of m^n*P(n, i/m).
[x^2] P(n+1, x) = A005990(n).

A352151 Expansion of e.g.f. 1/(cos(x) - tan(x)).

Original entry on oeis.org

1, 1, 3, 14, 81, 616, 5523, 58064, 697281, 9417856, 141368643, 2334020864, 42039523281, 820296426496, 17237259945363, 388087200241664, 9320064293358081, 237814050877505536, 6425096888209255683, 183232685725482942464, 5500505587921088841681
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2022

Keywords

Crossrefs

Cf. A000182, A000828, A001586, A175288, A352164.

Programs

  • Mathematica
    m = 20; Range[0, m]! * CoefficientList[Series[1/(Cos[x] - Tan[x]), {x, 0, m}], x] (* Amiram Eldar, Mar 06 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(cos(x)-tan(x))))
  • PARI
    c(n) = ((-4)^n-(-16)^n)*bernfrac(2*n)/(2*n);
b(n) = if(n%2==1, c((n+1)/2), (-1)^(n/2+1));
a(n) = if(n==0, 1, sum(k=1, n, b(k)*binomial(n, k)*a(n-k)));

Formula

a(0) = 1; a(n) = Sum_{k=1..n} b(k) * binomial(n,k) * a(n-k), where b(k) = A000182((k+1)/2) if k is odd, otherwise (-1)^(k/2+1).
From Vaclav Kotesovec, Mar 06 2022: (Start)
a(n) ~ n! / (sqrt(5) * (arctan(sqrt((sqrt(5) - 1)/2)))^(n+1)).
a(n) ~ n! / (sqrt(5) * A175288^(n+1)). (End)

A253165 a(n) = (-1)^n*2^(6*n+3)*(zeta(-2*n-1,1/2) - zeta(-2*n-1,1)), where zeta(a,z) is the generalized Riemann zeta function.

Original entry on oeis.org

1, 8, 256, 17408, 2031616, 362283008, 91620376576, 31191159799808, 13753735117275136, 7625476699018231808, 5192022022552652087296, 4258996468871236847403008, 4142655008190840426050093056, 4714505177821257067736657297408, 6206008749802659037752564348092416
Offset: 0

Views

Author

Peter Luschny, Mar 11 2015

Keywords

Crossrefs

Cf. A000825, A000828, A000831, A012393.

Programs

  • Maple
    a := n -> (-1)^n*2^(6*n+3)*(Zeta(0,-2*n-1,1/2)-Zeta(0,-2*n-1, 1)):
seq(a(n), n=0..14);
  • Mathematica
    f[n_] := (-1)^n*2^(6 n + 3) (Zeta[-2 n - 1, 1/2] - Zeta[-2 n - 1, 1]); Array[f, 15, 0] (* Robert G. Wilson v, Mar 11 2015 *)
max = 20; Clear[g]; g[max + 2] = 1; g[k_] := g[k] = 1 - 4*x*(k+1)*(k+2)/(4*x*(k+1)*(k+2) - 1/g[k+1]); gf = g[0]; CoefficientList[Series[gf, {x, 0, max}], x] (* Vaclav Kotesovec, Jun 01 2015, after Sergei N. Gladkovskii *)

Formula

a(n) = (-1)^n*2^(4*n+1)*(E(2*n+1,1/2)-E(2*n+1,0)), where E(n,x) are the Euler polynomials.
a(n) = A000825(2*n+1).
a(n) = A000828(2*n+1).
a(n) = A000831(2*n+1)/2.
a(n) = A012393(n+1)/2.
G.f.: S(0), where S(k)= 1 - 4*x*(k+1)*(k+2)/(4*x*(k+1)*(k+2) - 1/S(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 28 2015
a(n) ~ (2*n+1)! * 2^(4*n+3) / Pi^(2*n+2). - Vaclav Kotesovec, Jun 01 2015

A321400 A family of sequences converging to the exponential limit of sec + tan (A320956). Square array A(n, k) for n >= 0 and k >= 0, read by descending antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 2, 1, 1, 0, 5, 8, 2, 1, 1, 0, 16, 40, 10, 2, 1, 1, 0, 61, 256, 70, 10, 2, 1, 1, 0, 272, 1952, 656, 75, 10, 2, 1, 1, 0, 1385, 17408, 7442, 816, 75, 10, 2, 1, 1, 0, 7936, 177280, 99280, 11407, 832, 75, 10, 2, 1, 1
Offset: 0

Views

Author

Peter Luschny, Nov 08 2018

Keywords

Comments

See the comments and definitions in A320956. Note also the corresponding construction for the exp function in A320955.

Examples

			Array starts:
n\k   0  1  2   3   4    5      6       7        8  ...
-------------------------------------------------------
  [0] 1, 0, 0,  0,  0,   0,     0,      0,       0, ... A000007
  [1] 1, 1, 1,  2,  5,  16,    61,    272,    1385, ... A000111
  [2] 1, 1, 2,  8, 40, 256,  1952,  17408,  177280, ... A000828
  [3] 1, 1, 2, 10, 70, 656,  7442,  99280, 1515190, ... A320957
  [4] 1, 1, 2, 10, 75, 816, 11407, 194480, 3871075, ... A321394
  [5] 1, 1, 2, 10, 75, 832, 12322, 232560, 5325325, ...
  [6] 1, 1, 2, 10, 75, 832, 12383, 238272, 5693735, ...
  [7] 1, 1, 2, 10, 75, 832, 12383, 238544, 5732515, ...
  [8] 1, 1, 2, 10, 75, 832, 12383, 238544, 5733900, ...
-------------------------------------------------------
Seen as a triangle given by descending antidiagonals:
  [0] 1
  [1] 0,  1
  [2] 0,  1,   1
  [3] 0,  1,   1,  1
  [4] 0,  2,   2,  1,  1
  [5] 0,  5,   8,  2,  1, 1
  [6] 0, 16,  40, 10,  2, 1, 1
  [7] 0, 61, 256, 70, 10, 2, 1, 1

Crossrefs

Cf. A000111 (n=1), A000828 (n=2), A320957 (n=3), A321394 (n=4), A320956 (limit).
Antidiagonal sums (and row sums of the triangle): A321399.

Programs

  • Maple
    sf := proc(n) option remember; `if`(n <= 1, 1-n, (n-1)*(sf(n-1) + sf(n-2))) end:
kernel := proc(n, k) option remember; binomial(n, k)*sf(k) end:
egf := n -> add(kernel(n, k)*((tan + sec)(x*(n - k))), k=0..n):
A321400Row := proc(n, len) series(egf(n), x, len + 2):
seq(coeff(%, x, k)*k!/n!, k=0..len) end:
seq(lprint(A321400Row(n, 9)), n=0..9);
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