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 A002572 Number of partitions of 1 into n powers of 1/2; or (according to one definition of "binary") the number of binary rooted trees. (Formerly M0710 N0261) 38
 1, 1, 1, 2, 3, 5, 9, 16, 28, 50, 89, 159, 285, 510, 914, 1639, 2938, 5269, 9451, 16952, 30410, 54555, 97871, 175586, 315016, 565168, 1013976, 1819198, 3263875, 5855833, 10506175, 18849555, 33818794, 60675786, 108861148, 195312750, 350419594, 628704034, 1127987211, 2023774607, 3630948907 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS 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, Daniel Krenn, Algorithmic counting of nonequivalent compact Huffman codes, arXiv:1901.11343 [math.CO], 2019. Christian Elsholtz, Clemens Heuberger, 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 & 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.) 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->inf} a(n)/c0^n = c1   and      lim_{n->inf} (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 ]   [ 1 2 1 1 1 1 ]   [ 1 2 1 1 2 ]   [ 1 2 1 2 1 ]   [ 1 2 2 1 1 ]   [ 1 2 2 2 ]   [ 1 2 3 1 ]   [ 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   [ . . 3 1 1 1 2 . ]     + 3/4 + 1/8 + 1/16 + 1/32 + 2/64   [ . . 3 1 . 4 . . ]     + 3/4 + 1/8 + 4/32   [ . . 3 . 3 2 . . ]     + 3/4 + 3/16 + 2/32   [ . . 2 3 1 2 . . ]     + 2/4 + 3/8 + 1/16 + 2/32   [ . . 2 2 4 . . . ]     + 2/4 + 2/8 + 4/16   [ . . 1 5 2 . . . ]     + 1/4 + 5/8 + 2/16   [ . . . 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; 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); t=2;  /* t-ary: t=2 for A002572, t=3 for A176485, t=4 for A176503  */ 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

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Last modified July 9 09:40 EDT 2020. Contains 335542 sequences. (Running on oeis4.)