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. (Formerly M2656)

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

0,4

Comments

Also, number of maximally stable towers of 2 X 2 LEGO blocks.

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

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

Example

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).

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 *)

Prog

(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);

Crossrefs

Cf. A007575, A063865, A039826.

Sequence in context: A124696 A081669 A086821 * A322913 A167539 A223167

Adjacent sequences: A007573 A007574 A007575 * A007577 A007578 A007579

Keyword

easy,nonn

Author

Simon Plouffe, Robert G. Wilson v and Vladeta Jovovic

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.

Status

approved