Revision History for A047265
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing entries 1-10
| older changes
|
| |
|
| 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.
(history;
published version)
|
| |
|
|
#38 by N. J. A. Sloane at Fri Sep 22 05:20:56 EDT 2023
| |
| |
| |
|
|
#37 by N. J. A. Sloane at Fri Sep 22 05:20:53 EDT 2023
|
|
| NAME
|
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.
|
|
| STATUS
|
proposed
editing
|
| |
| |
|
|
#36 by Stefano Spezia at Fri Sep 08 09:10:05 EDT 2023
| |
| |
| |
|
|
#35 by Stefano Spezia at Fri Sep 08 09:09:48 EDT 2023
| |
| |
| |
|
|
#34 by G. C. Greubel at Fri Sep 08 01:50:12 EDT 2023
| |
| |
| |
|
|
#33 by G. C. Greubel at Fri Sep 08 01:49:11 EDT 2023
|
|
| FORMULA
|
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)
|
|
| PROG
|
(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 give A010815: A001482 - A001488, A047654A001490, A047655A006665, A001482-A001488A010815, A047649, A047654, A001490A047655, A047938- - A047648, A006665.,
Cf. A000027, A000041, A000217, A000292, A121373, A307059.
|
| |
| |
|
|
#32 by G. C. Greubel at Thu Sep 07 04:08:00 EDT 2023
|
|
| NAME
|
Triangle aT(n,k) (), for n >= 1, 1<= <= k<= <= n) , giving coefficient of x^n in expansion of (Product_{j>=1} (1-(-x)^j) - 1 )^k.
|
|
| COMMENTS
|
f(x) = Product_{j>=1} (1 -(- - (-x)^j) - 1, generating {0, {A121373(n)}_{n>=1}}. - Wolfdieter Lang, Feb 16 2021
|
|
| MATHEMATICA
|
aT[n_, k_] := _]:= SeriesCoefficient[(-1)^n*(Product[(1 - -x^j), {j, 1, , n}] - 1)^k, {x, 0, n}]; Table[a[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Dec 05 2013 *)}];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* Jean-François Alcover, Dec 05 2013 *)
|
|
| PROG
|
(PARI) aT(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
|
|
| STATUS
|
approved
editing
|
| |
| |
|
|
#31 by N. J. A. Sloane at Sun Mar 07 17:32:17 EST 2021
| |
| |
| |
|
|
#30 by Sean A. Irvine at Sun Mar 07 16:29:22 EST 2021
| |
| |
| |
|
|
#29 by Sean A. Irvine at Sun Mar 07 16:29:16 EST 2021
|
|
| FORMULA
|
G.f. column k: (ProdProduct_{j>=1} (1 - (-x)^j) - 1)^k, for k >= 1. See the name and a Riordan triangle comment above. - Wolfdieter Lang, Feb 16 2021
|
|
| STATUS
|
proposed
editing
|
| |
| |
|
|
