A326722 T(n, k) = n! * [x^k] [y^n] sec(z)(x + z*sin(z)/y) where z = y*sqrt(x^2 - 1) for 0 <= k <= n+1 and T(-1, 0) = 1, triangle read by rows.

1, 0, 1, -1, 0, 1, 0, -1, 0, 1, 2, 0, -4, 0, 2, 0, 5, 0, -10, 0, 5, -16, 0, 48, 0, -48, 0, 16, 0, -61, 0, 183, 0, -183, 0, 61, 272, 0, -1088, 0, 1632, 0, -1088, 0, 272, 0, 1385, 0, -5540, 0, 8310, 0, -5540, 0, 1385, -7936, 0, 39680, 0, -79360, 0, 79360, 0, -39680, 0, 7936

Offset

-1,11

Links

Table of n, a(n) for n=-1..64.

Formula

T(n, k) = A178111(n, k)*A000111(n-1) if n > 1 else k^n, assuming 0 based triangle.

Exponential generating functions for the columns (n >= 0) are:

egf_col0(x) = -tanh(x).

egf_col1(x) = sech(x).

egf_col2(x) = (tanh(x) + x*sech(x)^2)/2.

egf_col3(x) = x*tanh(x)*sech(x)/2.

egf_col4(x) = (tanh(x) + x*(2*x*tanh(x) - 1)*sech(x)^2)/8.

egf_col5(x) = x*sech(x)*(x + tanh(x) - 2*x*sech(x)^2)/8.

egf_col6(x) = (3*tanh(x) + x*sech(x)^2*(4*x^2 - 6*x^2*sech(x)^2 - 3))/48.

A recurrence of the row polynomials based on offset 0 is given by the recurrence of the Euler-Bernoulli-Entringer numbers A008281 combined with Paul Barry's A178111. See the Maple script.

Example

[-1]     1;
[ 0]     0,    1;
[ 1]    -1,    0,     1;
[ 2]     0,   -1,     0,     1;
[ 3]     2,    0,    -4,     0,      2;
[ 4]     0,    5,     0,   -10,      0,    5;
[ 5]   -16,    0,    48,     0,    -48,    0,    16;
[ 6]     0,  -61,     0,   183,      0, -183,     0,    61;
[ 7]   272,    0, -1088,     0,   1632,    0, -1088,     0,    272;
[ 8]     0, 1385,     0, -5540,      0, 8310,     0, -5540,      0, 1385;
[ 9] -7936,    0, 39680,     0, -79360,    0, 79360,     0, -39680,    0, 7936;

Maple

z := y*sqrt(x^2 - 1): gf := sec(z)*(x + z*sin(z)/y):
ser := series(gf, y, 16): cy :=  n -> convert(n!*coeff(ser, y, n), polynom):
Trow := n -> `if`(n=-1, [1], PolynomialTools:-CoefficientList(cy(n), x)):
ListTools:-Flatten([seq(Trow(n), n=-1..9)]);
# Alternatively, compute the row polynomials based on offset 0 by recurrence.
RowPoly := proc(n) local E, P, L;
    E := proc(n, k) option remember; if k = 0 then return(`if`(n = 0, 1, 0)) fi;
        E(n, k-1) + E(n-1, n-k) end:
    P := proc(n) option remember; `if`(n < 2, x^n,
        x*P(n-1) - ((1 + (-1)^n)/2)*P(n-2)) end:
    # `if`(n = 0, 1, sort(expand(P(n)*E(n-1, n-1)), x, ascending)):
    L := n -> PolynomialTools:-CoefficientList(P(n), x):
    `if`(n = 0, [1], L(n)*E(n-1, n-1)):
end: for n from 0 to 9 do RowPoly(n) end;
# Alternative:
T := (n, k) -> if n <= 1 then k^n else A178111(n, k)*A000111(n-1) fi:
seq(seq(T(n, k), k=0..n), n=0..10);

Mathematica

z := y Sqrt[x^2 - 1]; gf := Sec[z](x + z Sin[z]/y); ser := Series[gf, {y, 0, 16}];
cy[-1] := {1}; cy[n_] := n! Coefficient[ser, y, n];
row[n_] := CoefficientList[cy[ n], x]; Table[row[n], {n, -1, 9}] // Flatten

Prog

(SageMath)
def A326722(n, k):
    if n == 0: return 1
    if is_odd(n-k): return 0
    b = I^(n-k)*binomial(floor(n/2), floor(k/2))
    if is_odd(n): return b*I^(n-1)*euler_number(n-1)
    return 2*b*psi(n-1, 1/2)/pi^n
for n in range(11): print([A326722(n, k) for k in range(n+1)])

Crossrefs

Cf. A000364, A028296, A000182, A000111, A008281, A178111, A012816, A261042, A326721, A326723, A326724.

T(n, 0) = -A155585(n) for n >= 1.

T(n, 1) = A122045(n) for n >= 0.

|T(2*n-1, 2)| = A024255(n) for n >= 0.

T(n, 3) = A326719(n) for n >= 0.

T(n, 4) = A326718(n) for n >= 0.

Sequence in context: A356771 A053118 A175682 * A349127 A279228 A181481

Adjacent sequences: A326719 A326720 A326721 * A326723 A326724 A326725

Keyword

sign,tabl

Author

Peter Luschny, Aug 08 2019

Status

approved