A395553 Number of no-overhang stacked partitions (teeth) of base 4 and total size n.

1, 4, 7, 12, 20, 30, 42, 58, 77, 100, 127, 158, 194, 236, 282, 334, 393, 458, 529, 608, 694, 788, 890, 1000, 1119, 1248, 1385, 1532, 1690, 1858, 2036, 2226, 2427, 2640, 2865, 3102, 3352, 3616, 3892, 4182, 4487, 4806, 5139, 5488, 5852, 6232, 6628

Offset

4,2

Comments

A tooth of base r is a finite stack of rows of unit cubes with bottom row of length r, weakly decreasing row lengths, and no row overhanging the row beneath it. This sequence is column c=4 of the triangle A072704.

References

Paul M. King, Generating Functions for Temperley's No-Overhang Stacked Partitions, Journal of Integer Sequences (submitted).

Links

Table of n, a(n) for n=4..50.

Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-2,0,0,1,1,-1).

Formula

G.f.: h_4(q) = q^4*(1+q)*(1+2q+q^3)/((1-q)*(1-q^2)*(1-q^3)*(1-q^4)).

T(4,N) = [q^N] h_4(q) where h_r(q) satisfies (1-q^r)*h_r(q) = 2q*h_{r-1}(q) - q^2*h_{r-2}(q), with h_0(q)=1 and h_1(q)=q/(1-q).

Crossrefs

Cf. A000012 (c=1), A001477 (c=2), A000212 (c=3, with offset), A395554 (c=5).

Column 4 of A072704.

Sequence in context: A344421 A022809 A297554 * A188554 A020732 A339891

Adjacent sequences: A395550 A395551 A395552 * A395554 A395557 A395558

Keyword

nonn,easy

Author

Paul M. King, Apr 28 2026

Status

approved