OFFSET
1,4
COMMENTS
This is similar to a question about Egyptian fractions, except that there the denominators of the fractions must be distinct. - N. J. A. Sloane, Jan 13 2021
Math. Rev. 22 #11020, Minc, H. A problem in partitions ... 1959: v(c, d) is the number of partitions of d into positive integers of the form d = c + c_1 + c_2 + ... + c_n, where c_1 <= 2*c, c_{i+1} <= 2*c_i.
Top row of Table 1 of Elsholtz. [Jonathan Vos Post, Aug 30 2011]
a(n+1) is the number of compositions n = p(1) + p(2) + ... + p(m) with p(1)=1 and p(k) <= 2*p(k+1), see example. [Joerg Arndt, Dec 18 2012]
Over an algebraically closed field of characteristic 2, a(n) gives dimensions of the generic cohomology groups H^i_gen(SL_2,L(1)) which are isomorphic to algebraic group cohomology groups H^i(SL_2,L(1)^[i]), where ^[i] denotes i-th Frobenius twist. [David I. Stewart, Oct 22 2013]
REFERENCES
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 192-194, 307.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..2000 (first 201 terms from T. D. Noe)
Christian Elsholtz, Clemens Heuberger, and Daniel Krenn, Algorithmic counting of nonequivalent compact Huffman codes, arXiv:1901.11343 [math.CO], 2019.
Christian Elsholtz, Clemens Heuberger, and Helmut Prodinger, The number of Huffman codes, compact trees, and sums of unit fractions, arXiv:1108.5964v1 [math.CO], Aug 30, 2011. Also IEEE Trans. Information Theory, Vol. 59, No. 2, 2013 pp. 1065-1075.
Shimon Even and Abraham Lempel, Generation and enumeration of all solutions of the characteristic sum condition, Information and Control 21 (1972), 476-482.
P. Flajolet and H. Prodinger, Level number sequences for trees, Discrete Math., 65 (1987), 149-156.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 200
E. N. Gilbert, Codes based on inaccurate source probabilities, IEEE Trans. Inform. Theory, 17 (1971), 304-315.
R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission]
H. Minc, A problem in partitions: Enumeration of elements of a given degree in the free commutative entropic cyclic groupoid, Proc. Edinburgh Math. Soc. (2) 11 1958/1959 223-224.
E. Norwood, The Number of Different Possible Compact Codes, IEEE Transactions on Information Theory, Vol. 13, P. 614, 1967.
J. Paschke et al., Computing and estimating the number of n-ary Huffman sequences of a specified length, Discrete Math., 311 (2011), 1-7. (See p. 3.)
Helmut Prodinger, Philippe Flajolet's early work in combinatorics, arXiv:2103.15791 [math.CO], 2021.
N. J. A. Sloane, Richard Guy and the Encyclopedia of Integer Sequences: A Fifty-Year Friendship, Slides of talk at Conference "Celebrating Richard Guy", University of Calgary, October 2, 2020.
D. I. Stewart, Unbounding Ext, J. Algebra, 365 (2012), 1-11. (See p. 7)
Paul R. Stein, Letter to N. J. A. Sloane, Jul 20 1971
Paul R. Stein, Table of first 127 terms
FORMULA
From Jon E. Schoenfield, Dec 18 2016: (Start)
Numerically, it appears that
lim_{n->infinity} a(n)/c0^n = c1
and
lim_{n->infinity} (a(n)/c0^n - c1) / c2^n = c3
where
c0 = 1.79414718754168546349846498809380776370136441826513
55647141291458811011534167435879115275875728251544
55034381754309507738861994388752350104180891093803
27324310643547892073673907996758374498542252887021
90... = A102375
c1 = 0.14185320208540933707157739062733520381135377764439
00938624762999524081108574037129602775796177848175
96757823284956317508884467180902882086032012675483
68631687927534330190816399081295424373415296405657
19...
c2 = 0.71317957835995615685267138702642988919007297942329
35...
c3 = 0.06124104103121269745282188448763211918477582400104
06... (End)
a(n) = A294775(n-1,1). - Alois P. Heinz, Nov 08 2017
EXAMPLE
{1}; {1/2 + 1/2}; { 1/2 + 1/4 + 1/4 }; { 1/2 + 1/4 + 1/8 + 1/8, 1/4 + 1/4 + 1/4 + 1/4 }; ...
From Joerg Arndt, Dec 18 2012: (Start)
There are a(7+1)=16 compositions 7=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 2*p(k+1):
[ 1] [ 1 1 1 1 1 1 1 ]
[ 2] [ 1 1 1 1 1 2 ]
[ 3] [ 1 1 1 1 2 1 ]
[ 4] [ 1 1 1 2 1 1 ]
[ 5] [ 1 1 1 2 2 ]
[ 6] [ 1 1 2 1 1 1 ]
[ 7] [ 1 1 2 1 2 ]
[ 8] [ 1 1 2 2 1 ]
[ 9] [ 1 1 2 3 ]
[10] [ 1 2 1 1 1 1 ]
[11] [ 1 2 1 1 2 ]
[12] [ 1 2 1 2 1 ]
[13] [ 1 2 2 1 1 ]
[14] [ 1 2 2 2 ]
[15] [ 1 2 3 1 ]
[16] [ 1 2 4 ]
(End)
From Joerg Arndt, Dec 26 2012: (Start)
There are a(8)=16 partitions of 1 into 8 powers of 1/2 (dots denote zeros in the tables of multiplicities in the left column)
[ 1] [ . 1 1 1 1 1 1 2 ] + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 2/128
[ 2] [ . 1 1 1 1 . 4 . ] + 1/2 + 1/4 + 1/8 + 1/16 + 4/64
[ 3] [ . 1 1 1 . 3 2 . ] + 1/2 + 1/4 + 1/8 + 3/32 + 2/64
[ 4] [ . 1 1 . 3 1 2 . ] + 1/2 + 1/4 + 3/16 + 1/32 + 2/64
[ 5] [ . 1 1 . 2 4 . . ] + 1/2 + 1/4 + 2/16 + 4/32
[ 6] [ . 1 . 3 1 1 2 . ] + 1/2 + 3/8 + 1/16 + 1/32 + 2/64
[ 7] [ . 1 . 3 . 4 . . ] + 1/2 + 3/8 + 4/32
[ 8] [ . 1 . 2 3 2 . . ] + 1/2 + 2/8 + 3/16 + 2/32
[ 9] [ . 1 . 1 6 . . . ] + 1/2 + 1/8 + 6/16
[10] [ . . 3 1 1 1 2 . ] + 3/4 + 1/8 + 1/16 + 1/32 + 2/64
[11] [ . . 3 1 . 4 . . ] + 3/4 + 1/8 + 4/32
[12] [ . . 3 . 3 2 . . ] + 3/4 + 3/16 + 2/32
[13] [ . . 2 3 1 2 . . ] + 2/4 + 3/8 + 1/16 + 2/32
[14] [ . . 2 2 4 . . . ] + 2/4 + 2/8 + 4/16
[15] [ . . 1 5 2 . . . ] + 1/4 + 5/8 + 2/16
[16] [ . . . 8 . . . . ] + 8/8
(End)
MAPLE
v := proc(c, d) option remember; local i; if d < 0 or c < 0 then 0 elif d = c then 1 else add(v(i, d-c), i=1..2*c); fi; end; [ seq(v(1, n), n=1..50) ];
MATHEMATICA
v[c_, d_] := v[c, d] = If[d < 0 || c < 0, 0, If[d == c, 1, Sum[v[i, d - c], {i, 1, 2*c}]]]; a[n_] := v[1, n-1]; a[1] = 1; Table[a[n], {n, 1, 36}] (* Jean-François Alcover, Oct 19 2011, after Maple *)
PROG
(PARI) v(c, d) = if ( d<0 || c<0, 0, if ( d==c, 1, sum(i=1, 2*c, v(i, d-c) ) ) )
(PARI)
/* g.f. as given in the Elsholtz/Heuberger/Prodinger reference */
N=66; q='q+O('q^N);
L=2 + 2*ceil( log(N) / log(t) );
f(k)=(1-t^k)/(1-t);
la(j)=prod(i=1, j, q^f(i) / ( 1 - q^f(i) ) );
nm=sum(j=0, L, (-1)^j * q^f(j) * la(j) );
dn=sum(j=0, L, (-1)^j * la(j) );
gf=nm / dn;
Vec( gf )
/* Joerg Arndt, Dec 27 2012 */
(PARI) {a(n, k=2) = if( n<2 && k==2, n>=0, n<k || k<1, 0, n==k, 1, sum(i=2, min(n-k+1, 2*k-1), a(n-k+1, i)))}; /* Michael Somos, Dec 20 2016 */
CROSSREFS
KEYWORD
core,nonn,nice,easy
AUTHOR
STATUS
approved