A007576 Number of solutions to k_1 + 2*k_2 + ... + n*k_n = 0, where k_i are from {-1,0,1}, i=1..n.
1, 1, 1, 3, 7, 15, 35, 87, 217, 547, 1417, 3735, 9911, 26513, 71581, 194681, 532481, 1464029, 4045117, 11225159, 31268577, 87404465, 245101771, 689323849, 1943817227, 5494808425, 15568077235, 44200775239, 125739619467
Offset: 0
Examples
For n=4 there are 7 solutions: (-1,-1,1,0), (-1,0,-1,1), (-1,1,1,-1), (0,0,0,0), (1,-1,-1,1), (1,0,1,-1), (1,1,-1,0).
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- P. J. S. Watson, On "LEGO" towers, J. Rec. Math., 12 (No. 1, 1979-1980), 24-27.
Links
- Ray Chandler, Table of n, a(n) for n = 0..2106 (terms < 10^1000; first 101 terms from T. D. Noe)
- D. Andrica and O. Bagdasar, Some remarks on 3-partitions of multisets, Electron. Notes Discrete Math., TCDM'18 (2018).
- Dorin Andrica and Ovidiu Bagdasar, On k-partitions of multisets with equal sums, The Ramanujan J. (2021) Vol. 55, 421-435.
- Andrei Asinowski, Axel Bacher, Cyril Banderier, and Bernhard Gittenberger, Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata, Laboratoire d'Informatique de Paris Nord (LIPN 2019).
- Steven R. Finch, Signum equations and extremal coefficients [Broken link]
- Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]
- P. J. S. Watson, On "LEGO" towers, J. Rec. Math., 12 (No. 1, 1979-1980), 24-27. (Annotated scanned copy)
- Index entry for sequences related to LEGO blocks
Programs
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Mathematica
f[0] = 1; f[n_] := Coefficient[Expand@ Product[1 + x^k + x^(2k), {k, n}], x^(n(n + 1)/2)]; Table[f@n, {n, 0, 28}] (* Robert G. Wilson v, Nov 10 2006 *) -
Maxima
a(n):=coeff(expand(product(1+x^k+x^(2*k),k,1,n)),x,binomial(n+1,2)); makelist(a(n),n,0,24);
Formula
Coefficient of x^(n*(n+1)/2) in Product_{k=1..n} (1+x^k+x^(2*k)).
Equivalently, the coefficient of x^0 in Product_{k=1..n} (1/x^k + 1 + x^k). - Paul D. Hanna, Jul 10 2018
a(n) ~ 3^(n + 1) / (2 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jul 11 2018
a(n) = (1/(2*Pi))*Integral_{t=0..2*Pi} ( Product_{k=1..n} (1+2*cos(k*t)) ) dt. - Ovidiu Bagdasar, Aug 08 2018
Extensions
More terms from David Wasserman, Mar 29 2005
Edited by N. J. A. Sloane, Nov 07 2006. This is a merging of two sequences which, thanks to the work of Søren Eilers, we now know are identical.
Comments