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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.
7
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
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
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
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, Aug 08 2019
STATUS
approved