A047265 Triangle T(n,k), for n >= 1, 1 <= k <= n, read by rows, giving coefficient of x^n in expansion of (Product_{j>=1} (1-(-x)^j) - 1 )^k.

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, -65, 21, 62, -84, 45, -11, 1

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

Columns: A001482 - A001488, A001490, A006665, A010815, A047649, A047654, A047655, A047938 - A047648.

Cf. A341418 (differently signed).

Cf. A000027, A000041, A000217, A000292, A121373, A307059.

Sequence in context: A348951 A369815 A156135 * A341418 A185962 A279928

Adjacent sequences: A047262 A047263 A047264 * A047266 A047267 A047268

Keyword

sign,easy,nice,tabl

Author

N. J. A. Sloane

Status

approved