OFFSET
1,5
COMMENTS
This is an ordinary convolution triangle. If a column k=0 starting at n=0 is added, then this is the Riordan triangle R(1, f(x)), with
f(x) = Product_{j>=1} (1 - (-x)^j) - 1, generating {0, {A121373(n)}_{n>=1}}. - Wolfdieter Lang, Feb 16 2021
LINKS
Alois P. Heinz, Rows n = 1..200, flattened
H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)
FORMULA
G.f. column k: (Product_{j>=1} (1 - (-x)^j) - 1)^k, for k >= 1. See the name and a Riordan triangle comment above. - Wolfdieter Lang, Feb 16 2021
From G. C. Greubel, Sep 07 2023: (Start)
T(n, n) = 1.
T(n, n-1) = -A000027(n-1).
T(n, n-2) = A000217(n-3).
T(n, n-3) = -A000292(n-5).
Sum_{k=1..n} T(n, k) = (-1)^n * A307059(n).
Sum_{k=1..n} (-1)^k * T(n, k) = (-1)^n * A000041(n). (End)
EXAMPLE
Triangle starts:
1,
-1, 1,
0, -2, 1,
0, 1, -3, 1,
-1, 0, 3, -4, 1,
0, -2, -1, 6, -5, 1,
-1, 2, -3, -4, 10, -6, 1,
0, -2, 6, -3, -10, 15, -7, 1,
0, 2, -6, 12, 0, -20, 21, -8, 1,
0, 1, 6, -16, 19, 9, -35, 28, -9, 1,
0, 0, 0, 16, -35, 24, 28, -56, 36, -10, 1,
-1, 2, -3, -6, 40, ...
MAPLE
g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
[1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
end:
T:= proc(n, k) option remember;
`if`(k=0, `if`(n=0, 1, 0), `if`(k=1, `if`(n=0, 0, g(n)),
(q-> add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
end:
seq(seq(T(n, k), k=1..n), n=1..12); # Alois P. Heinz, Feb 07 2021
MATHEMATICA
T[n_, k_]:= SeriesCoefficient[(-1)^n*(Product[(1-x^j), {j, n}] - 1)^k, {x, 0, n}];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* Jean-François Alcover, Dec 05 2013 *)
PROG
(PARI) T(n, k) = polcoeff((-1)^n*(Ser(prod(i=1, n, 1-x^i)-1)^k), n) \\ Ralf Stephan, Dec 08 2013
(Magma)
R<x>:=PowerSeriesRing(Integers(), 40);
T:= func< n, k | Coefficient(R!( (-1)^n*(-1 + (&*[1 - x^j: j in [1..n]]) )^k ), n) >;
[T(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Sep 07 2023
(SageMath)
from sage.combinat.q_analogues import q_pochhammer
P.<x> = PowerSeriesRing(ZZ, 50)
def T(n, k): return P( (-1)^n*(-1 + q_pochhammer(n, x, x) )^k ).list()[n]
flatten([[T(n, k) for k in range(1, n+1)] for n in range(1, 13)]) # G. C. Greubel, Sep 07 2023
CROSSREFS
KEYWORD
AUTHOR
STATUS
approved