A294746 -id:A294746 - cp's OEIS frontend

A007178 Number of ways to write 1 as ordered sum of n powers of 1/2, allowing repeats.

1, 1, 3, 13, 75, 525, 4347, 41245, 441675, 5259885, 68958747, 986533053, 15292855019, 255321427725, 4567457001915, 87156877087069, 1767115200924299, 37936303950503853, 859663073472084315, 20505904049009202685, 513593410566661282347, 13476082013068430626893

Offset: 1

N. J. A. Sloane, Simon Plouffe, Don Knuth

Also the dimension of the arity n component of the operad of level algebras (see the reference by Chataur-Livernet by definition), and the cardinality of the subset of the free commutative medial magma with n generators that contains each generator exactly once. The linear operad of level algebras is the linearization of the set operad of commutative medial magmas; the statement about commutative medial magmas follows from the description in the paper of Ježek-Kepka. - Vladimir Dotsenko, Mar 12 2022

			For n=3, the 3 sums are 1/2 + 1/4 + 1/4, 1/4 + 1/2 + 1/4, and 1/4 + 1/4 + 1/2.

D. E. Knuth, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..300
D. Chataur and M. Livernet, Adem-Cartan operads, arXiv:math/0209363 [math.AT], 2002-2003; Communications in Algebra 33 (2005), 4337-4360.
A. Giorgilli and G. Molteni, Representation of a 2-power as sum of k 2-powers: a recursive formula, J. Number Theory 133 (2013), no. 4, 1251-1261.
Jia Huang and Erkko Lehtonen, Associative-commutative spectra for some varieties of groupoids, arXiv:2401.15786 [math.CO], 2024. See p. 18.
J. Ježek and T. Kepka, Free entropic groupoids, Commentationes Mathematicae Universitatis Carolinae,Tome 022 (1981), p. 223-233.
D. E. Knuth, Letter to R. E. Tarjan & N. J. A. Sloane, Jul. 1975
Daniel Krenn and Stephan Wagner, Compositions into Powers of b : Asymptotic Enumeration and Parameters, arXiv:1410.4331 [math.NT], 2014.
S. Lehr, J. Shallit and J. Tromp, On the vector space of the automatic reals, Theoret. Comput. Sci. 163 (1996), no. 1-2, 193-210.
G. Molteni, Cancellation in a short exponential sum, J. Number Theory 130 (2010), no. 9, 2011-2027.
G. Molteni, Representation of a 2-power as sum of k 2-powers: the asymptotic behavior, Int. J. Number Theory 8 (2012), no. 8, 1923-1963.

Cf. A002572, A294746, A323840.

b:= proc(n, r, p) option remember; `if`(n b(n, 1, 0):
seq(a(n), n=1..23);  # Alois P. Heinz, Nov 07 2017

f[n_] := Coefficient[Expand[Sum[z^(2^j), {j, n}]^n], z, 2^n]; Array[f, 20] (* Robert G. Wilson v, Apr 08 2012 *)

f(n)={my(M);if(n>1,M=matrix(n,n);M[2,1] = 1;for(k=3,n,for(l=1,k-2,M[k,l] = 0;smx = min(2*l,k-l-1);for(s=1,smx, M[k,l] += binomial(k+l-1,2*l-s)*M[k-l,s]));M[k,k-1] = 1);M[n,1],1)}

a(n) = coefficient of z^(2^n) in (z+z^2+z^4+...+z^(2^n))^n. - Don Knuth.
From Giuseppe Molteni, Dec 14 2012: (Start)
Limit_{n->oo} (a(n)/n!)^(1/n) = 1.192674341213466032221288982528755... (see References: "Representation of a 2-power as sum of k 2-powers: the asymptotic behavior").
a(n) == 4 + (-1)^n (mod 8) for n > 2 (see References: "Representation of a 2-power as sum of k 2-powers: a recursive formula"). (End)
More precise asymptotics: a(n) ~ c * d^n * n!, where d = 1.192674341213466032221288982528755176734489232027131552652821007498903522051783..., c = 0.24849369086953813603231092781945750388624874631949260927875431616785914609... - Vaclav Kotesovec, Sep 20 2019
a(n) = A323840(n,n). - Alois P. Heinz, Mar 31 2021

More terms from Hugo van der Sanden

Minor edits from Vaclav Kotesovec, Jul 26 2014

A294775 Number A(n,k) of partitions of 1 into exactly k*n+1 powers of 1/(k+1); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 2, 4, 5, 1, 1, 1, 1, 2, 4, 7, 9, 1, 1, 1, 1, 2, 4, 8, 13, 16, 1, 1, 1, 1, 2, 4, 8, 15, 25, 28, 1, 1, 1, 1, 2, 4, 8, 16, 29, 48, 50, 1, 1, 1, 1, 2, 4, 8, 16, 31, 57, 92, 89, 1, 1, 1, 1, 2, 4, 8, 16, 32, 61, 112, 176, 159, 1

Offset: 0

Alois P. Heinz, Nov 08 2017

			A(4,1) = 3: [1/4,1/4,1/4,1/8,1/8], [1/2,1/8,1/8,1/8,1/8], [1/2,1/4,1/8,1/16,1/16].
A(5,2) = 7: [1/9,1/9,1/9,1/9,1/9,1/9,1/9,1/9,1/27,1/27,1/27], [1/3,1/9,1/9,1/9,1/9,1/27,1/27,1/27,1/27,1/27,1/27], [1/3,1/9,1/9,1/9,1/9,1/9,1/27,1/27,1/81,1/81,1/81], [1/3,1/3,1/27,1/27,1/27,1/27,1/27,1/27,1/27,1/27,1/27], [1/3,1/3,1/9,1/27,1/27,1/27,1/27,1/27,1/81,1/81,1/81], [1/3,1/3,1/9,1/9,1/27,1/81,1/81,1/81,1/81,1/81,1/81], [1/3,1/3,1/9,1/9,1/27,1/27,1/81,1/81,1/243,1/243,1/243].
Square array A(n,k) begins:
  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  2,  2,  2,  2,  2,  2,  2,  2, ...
  1,  3,  4,  4,  4,  4,  4,  4,  4, ...
  1,  5,  7,  8,  8,  8,  8,  8,  8, ...
  1,  9, 13, 15, 16, 16, 16, 16, 16, ...
  1, 16, 25, 29, 31, 32, 32, 32, 32, ...
  1, 28, 48, 57, 61, 63, 64, 64, 64, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
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:1108.5964 [math.CO], Aug 30, 2011. Also IEEE Trans. Information Theory, Vol. 59, No. 2, 2013 pp. 1065-1075.

Columns k=0-10 give (offsets may differ): A000012, A002572, A176485, A176503, A194628, A194629, A194630, A194631, A194632, A194633, A295081.
Main diagonal gives A011782(n-1) for n>0.
Cf. A294746.

b:= proc(n, r, k) option remember;
      `if`(n `if`(k=0, 1, b(k*n+1, 1, k+1)):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, r_, k_] := b[n, r, k] = If[n < r, 0, If[r == 0, If[n == 0, 1, 0], Sum[b[n - j, k*(r - j), k], {j, 0, Min[n, r]}]]];
A[n_, k_] := If[k == 0, 1, b[k*n + 1, 1, k + 1]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Nov 11 2017, after Alois P. Heinz *)

A294747 Number of compositions (ordered partitions) of 1 into exactly n^2+1 powers of 1/(n+1).

Original entry on oeis.org

1, 1, 10, 4245, 216456376, 2713420774885145, 14138484434475011392912026, 46050764886573707269872023694736134925, 131223281654667714701311635640432890136981994039662720, 435699237793484726791774188056400878106883117166142375354233228879800569

Offset: 0

Alois P. Heinz, Nov 07 2017

nonn

			a(0) = 1: [1].
a(1) = 1: [1/2,1/2].
a(2) = 10 = binomial(5,2): [1/3,1/3,1/9,1/9,1/9], [1/3,1/9,1/3,1/9,1/9], [1/3,1/9,1/9,1/3,1/9], [1/3,1/9,1/9,1/9,1/3], [1/9,1/3,1/3,1/9,1/9], [1/9,1/3,1/9,1/3,1/9], [1/9,1/3,1/9,1/9,1/3], [1/9,1/9,1/3,1/3,1/9], [1/9,1/9,1/3,1/9,1/3], [1/9,1/9,1/9,1/3,1/3].

Alois P. Heinz, Table of n, a(n) for n = 0..26

Main diagonal of A294746.

Cf. A002522.

b:= proc(n, r, p, k) option remember;
      `if`(n `if`(n=0, 1, b(n^2+1, 1, 0, n+1)):
seq(a(n), n=0..10);

b[n_, r_, p_, k_] := b[n, r, p, k] = If[n < r, 0, If[r == 0, If[n == 0, p!, 0], Sum[b[n - j, k*(r - j), p + j, k]/j!, {j, 0, Min[n, r]}]]];
a[n_] := If[n == 0, 1, b[n^2 + 1, 1, 0, n + 1]];
Table[a[n], {n, 0, 10}] (* Jean-François Alcover, May 21 2018, translated from Maple *)

a(n) = [x^((n+1)^n)] (Sum_{j=0..n^2+1} x^((n+1)^j))^(n^2+1) for n>0, a(0) = 1.
a(n) = A294746(n,n).
a(n) ~ exp(-1/12) * n^(n^2 - n/2 + 2) / (2*Pi)^((n-1)/2). - Vaclav Kotesovec, Sep 20 2019

A294850 Number of compositions (ordered partitions) of 1 into exactly 2*n+1 powers of 1/3.

Original entry on oeis.org

1, 1, 10, 217, 8317, 487630, 40647178, 4561368175, 663134389930, 121218250616173, 27212315953140892, 7359774260167595035, 2360287411461166320775, 885627663284464131142801, 384376149675044501884907410, 191068288010770323577312291141

Offset: 0

Alois P. Heinz, Nov 09 2017

nonn

			a(0) = 1: [1].
a(1) = 1: [1/3,1/3,1/3].
a(2) = 10: [1/3,1/3,1/9,1/9,1/9], [1/3,1/9,1/3,1/9,1/9], [1/3,1/9,1/9,1/3,1/9], [1/3,1/9,1/9,1/9,1/3], [1/9,1/3,1/3,1/9,1/9], [1/9,1/3,1/9,1/3,1/9], [1/9,1/3,1/9,1/9,1/3], [1/9,1/9,1/3,1/3,1/9], [1/9,1/9,1/3,1/9,1/3], [1/9,1/9,1/9,1/3,1/3].

Alois P. Heinz, Table of n, a(n) for n = 0..200

Column k=2 of A294746.

a(n) = [x^(3^n)] (Sum_{j=0..2*n+1} x^(3^j))^(2*n+1).

a(n) ~ c * d^n * n^(2*n + 3/2), where d = 0.28934785344292228780991..., c = 1.984098413887380996408... - Vaclav Kotesovec, Sep 20 2019

A294851 Number of compositions (ordered partitions) of 1 into exactly 3*n+1 powers of 1/4.

Original entry on oeis.org

1, 1, 35, 4245, 1239823, 709097481, 701954099115, 1104353764428365, 2594884910993019575, 8684483842898500680225, 39880061006390454401626995, 243797643642188511890409843525, 1935230187172759446730224649089055, 19533122859042054951154895127392582265

Offset: 0

Alois P. Heinz, Nov 09 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..165

Column k=3 of A294746.

a(n) = [x^(4^n)] (Sum_{j=0..3*n+1} x^(4^j))^(3*n+1).

a(n) ~ c * d^n * n^(3*n + 3/2), where d = 0.228881755274644937676549309..., c = 3.08458888791535695636629... - Vaclav Kotesovec, Sep 20 2019

A294852 Number of compositions (ordered partitions) of 1 into exactly 4*n+1 powers of 1/5.

Original entry on oeis.org

1, 1, 126, 90376, 216456376, 1303699790001, 16596702491586251, 396695587555058190126, 16336038155342083651640376, 1085776473843765315524916060126, 110656003660578876500875377620844376, 16592064238583710535773294961607940956876

Offset: 0

Alois P. Heinz, Nov 09 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..128

Column k=4 of A294746.

a(n) = [x^(5^n)] (Sum_{j=0..4*n+1} x^(5^j))^(4*n+1).

A294853 Number of compositions (ordered partitions) of 1 into exactly 5*n+1 powers of 1/6.

Original entry on oeis.org

1, 1, 462, 2019836, 41175714454, 2713420774885145, 461871979542736134676, 174436242482643190451211853, 130958058407369286623026190867082, 179835209135492330050411858875313971595, 423205992807070499591372608204571223421862945

Offset: 0

Alois P. Heinz, Nov 09 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..106

Column k=5 of A294746.

a(n) = [x^(6^n)] (Sum_{j=0..5*n+1} x^(6^j))^(5*n+1).

A294854 Number of compositions (ordered partitions) of 1 into exactly 6*n+1 powers of 1/7.

Original entry on oeis.org

1, 1, 1716, 46570140, 8251690444250, 6078597035484932995, 14138484434475011392912026, 86237678200608256132213084584295, 1206534243283932582765850205674424343577, 34994508245963099403565066291175900528344592700

Offset: 0

Alois P. Heinz, Nov 09 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..90

Column k=6 of A294746.

a(n) = [x^(7^n)] (Sum_{j=0..6*n+1} x^(7^j))^(6*n+1).

A294855 Number of compositions (ordered partitions) of 1 into exactly 7*n+1 powers of 1/8.

Original entry on oeis.org

1, 1, 6435, 1097525253, 1713228373452375, 14303426764164190428105, 460977928965130046448503507051, 46050764886573707269872023694736134925, 12176825528022093702548525617184407475359333407, 7565469782615095731665958935875509379368611893407583633

Offset: 0

Alois P. Heinz, Nov 09 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..79

Column k=7 of A294746.

a(n) = [x^(8^n)] (Sum_{j=0..7*n+1} x^(8^j))^(7*n+1).

A294856 Number of compositions (ordered partitions) of 1 into exactly 8*n+1 powers of 1/9.

Original entry on oeis.org

1, 1, 24310, 26293568950, 365077361327242168, 34883776613634643730481238, 15732393344641740454307566725567376, 25997337847684377365651388718120083246723460, 131223281654667714701311635640432890136981994039662720

Offset: 0

Alois P. Heinz, Nov 09 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..71

Column k=8 of A294746.

a(n) = [x^(9^n)] (Sum_{j=0..8*n+1} x^(9^j))^(8*n+1).

cp's OEIS Frontend

A007178 Number of ways to write 1 as ordered sum of n powers of 1/2, allowing repeats.

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A294775 Number A(n,k) of partitions of 1 into exactly k*n+1 powers of 1/(k+1); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A294747 Number of compositions (ordered partitions) of 1 into exactly n^2+1 powers of 1/(n+1).

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A294850 Number of compositions (ordered partitions) of 1 into exactly 2*n+1 powers of 1/3.

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A294851 Number of compositions (ordered partitions) of 1 into exactly 3*n+1 powers of 1/4.

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A294852 Number of compositions (ordered partitions) of 1 into exactly 4*n+1 powers of 1/5.

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A294853 Number of compositions (ordered partitions) of 1 into exactly 5*n+1 powers of 1/6.

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A294854 Number of compositions (ordered partitions) of 1 into exactly 6*n+1 powers of 1/7.

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A294855 Number of compositions (ordered partitions) of 1 into exactly 7*n+1 powers of 1/8.

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