A047265 - OEIS

%I #38 Sep 22 2023 05:20:56

%S 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,

%T 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,

%U -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

%N 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.

%C 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

%C   f(x) = Product_{j>=1} (1 - (-x)^j) - 1, generating {0, {A121373(n)}_{n>=1}}. - _Wolfdieter Lang_, Feb 16 2021

%H Alois P. Heinz, <a href="/A047265/b047265.txt">Rows n = 1..200, flattened</a>

%H H. Gupta, <a href="https://doi.org/10.1112/jlms/s1-39.1.433">On the coefficients of the powers of Dedekind's modular form</a>, J. London Math. Soc., 39 (1964), 433-440.

%H H. Gupta, <a href="/A001482/a001482.pdf">On the coefficients of the powers of Dedekind's modular form</a> (annotated and scanned copy)

%F 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

%F From _G. C. Greubel_, Sep 07 2023: (Start)

%F T(n, n) = 1.

%F T(n, n-1) = -A000027(n-1).

%F T(n, n-2) = A000217(n-3).

%F T(n, n-3) = -A000292(n-5).

%F Sum_{k=1..n} T(n, k) = (-1)^n * A307059(n).

%F Sum_{k=1..n} (-1)^k * T(n, k) = (-1)^n * A000041(n). (End)

%e Triangle starts:

%e    1,

%e   -1,   1,

%e    0,  -2,   1,

%e    0,   1,  -3,   1,

%e   -1,   0,   3,  -4,   1,

%e    0,  -2,  -1,   6,  -5,   1,

%e   -1,   2,  -3,  -4,  10,  -6,   1,

%e    0,  -2,   6,  -3, -10,  15,  -7,   1,

%e    0,   2,  -6,  12,   0, -20,  21,  -8,   1,

%e    0,   1,   6, -16,  19,   9, -35,  28,  -9,   1,

%e    0,   0,   0,  16, -35,  24,  28, -56,  36, -10,   1,

%e   -1,   2,  -3,  -6,  40, ...

%p g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]

%p       [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)

%p     end:

%p T:= proc(n, k) option remember;

%p      `if`(k=0, `if`(n=0, 1, 0), `if`(k=1, `if`(n=0, 0, g(n)),

%p          (q-> add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))

%p     end:

%p seq(seq(T(n, k), k=1..n), n=1..12);  # _Alois P. Heinz_, Feb 07 2021

%t T[n_, k_]:= SeriesCoefficient[(-1)^n*(Product[(1-x^j), {j,n}] - 1)^k, {x, 0, n}];

%t Table[T[n, k], {n,12}, {k,n}]//Flatten (* _Jean-François Alcover_, Dec 05 2013 *)

%o (PARI) T(n,k) = polcoeff((-1)^n*(Ser(prod(i=1,n,1-x^i)-1)^k), n) \\ _Ralf Stephan_, Dec 08 2013

%o (Magma)

%o R<x>:=PowerSeriesRing(Integers(), 40);

%o T:= func< n,k | Coefficient(R!( (-1)^n*(-1 + (&*[1 - x^j: j in [1..n]]) )^k ), n) >;

%o [T(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Sep 07 2023

%o (SageMath)

%o from sage.combinat.q_analogues import q_pochhammer

%o P.<x> = PowerSeriesRing(ZZ, 50)

%o def T(n,k): return P( (-1)^n*(-1 + q_pochhammer(n,x,x) )^k ).list()[n]

%o flatten([[T(n,k) for k in range(1,n+1)] for n in range(1,13)]) # _G. C. Greubel_, Sep 07 2023

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

%Y Cf. A341418 (differently signed).

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

%K sign,easy,nice,tabl

%O 1,5

%A _N. J. A. Sloane_