A144067 -id:A144067 - cp's OEIS frontend

A144074 Number A(n,k) of multisets of nonempty words with a total of n letters over k-ary alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 7, 3, 0, 1, 4, 15, 20, 5, 0, 1, 5, 26, 64, 59, 7, 0, 1, 6, 40, 148, 276, 162, 11, 0, 1, 7, 57, 285, 843, 1137, 449, 15, 0, 1, 8, 77, 488, 2020, 4632, 4648, 1200, 22, 0, 1, 9, 100, 770, 4140, 13876, 25124, 18585, 3194, 30, 0, 1, 10, 126

Offset: 0

Alois P. Heinz, Sep 09 2008

Column k > 1 is asymptotic to k^n * exp(2*sqrt(n) - 1/2 + c(k)) / (2 * sqrt(Pi) * n^(3/4)), where c(k) = Sum_{m>=2} 1/(m*(k^(m-1)-1)). - Vaclav Kotesovec, Mar 14 2015

			A(4,1) = 5: {aaaa}, {aaa,a}, {aa,aa}, {aa,a,a}, {a,a,a,a}.
A(2,2) = 7: {aa}, {a,a}, {bb}, {b,b}, {ab}, {ba}, {a,b}.
A(2,3) = 15: {aa}, {a,a}, {bb}, {b,b}, {cc}, {c,c}, {ab}, {ba}, {a,b}, {ac}, {ca}, {a,c}, {bc}, {cb}, {b,c}.
A(3,2) = 20: {aaa}, {a,aa}, {a,a,a}, {bbb}, {b,bb}, {b,b,b}, {aab}, {aba}, {baa}, {a,ab}, {a,ba}, {aa,b}, {a,a,b}, {bba}, {bab}, {abb}, {b,ba}, {b,ab}, {bb,a}, {b,b,a}.
Square array begins:
  1, 1,   1,    1,    1,     1, ...
  0, 1,   2,    3,    4,     5, ...
  0, 2,   7,   15,   26,    40, ...
  0, 3,  20,   64,  148,   285, ...
  0, 5,  59,  276,  843,  2020, ...
  0, 7, 162, 1137, 4632, 13876, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.
N. J. A. Sloane, Transforms

Columns k=0-10 give: A000007, A000041, A034899, A144067, A144068, A144069, A144070, A144071, A144072, A144073, A292837.
Rows n=0-2 give: A000012, A001477, A005449.
Main diagonal gives A252654.
Cf. A004248, A257740, A256142, A292804.

with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: A:= (n,k)-> etr(j->k^j)(n); seq(seq(A(n, d-n), n=0..d), d=0..14);

a[n_, k_] := SeriesCoefficient[ Product[1/(1-x^j)^(k^j), {j, 1, n}], {x, 0, n}]; a[0, ] = 1; a[?Positive, 0] = 0;
Table[a[n-k, k], {n, 0, 14}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jan 15 2014 *)
etr[p_] := Module[{b}, b[n_] := b[n] = If[n==0, 1, Sum[Sum[d p[d], {d, Divisors[j]}] b[n-j], {j, 1, n}]/n]; b];
A[n_, k_] := etr[k^#&][n];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 30 2020, after Alois P. Heinz *)

G.f. of column k: Product_{j>=1} 1/(1-x^j)^(k^j).
Column k is Euler transform of the powers of k.
T(n,k) = Sum_{i=0..k} C(k,i) * A257740(n,k-i). - Alois P. Heinz, May 08 2015

Name changed by Alois P. Heinz, Sep 21 2018

A256142 G.f.: Product_{j>=1} (1+x^j)^(3^j).

Original entry on oeis.org

1, 3, 12, 55, 225, 927, 3729, 14787, 57888, 224220, 860022, 3270744, 12343899, 46264257, 172305837, 638039136, 2350109736, 8613851832, 31428857611, 114187160631, 413222547846, 1489829356657, 5352683946903, 19167988920930, 68427472477338, 243559693397025

Offset: 0

Vaclav Kotesovec, Mar 16 2015

nonn

In general, if g.f. = Product_{j>=1} (1+x^j)^(k^j), then a(n) ~ k^n * exp(2*sqrt(n) - 1/2 - c(k)) / (2 * sqrt(Pi) * n^(3/4)), where c(k) = Sum_{m>=2} (-1)^m/(m*(k^(m-1)-1)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotics of sequence A034691
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.

Cf. A102866, A144067, A144074.

Column k=3 of A292804.

nmax=30; CoefficientList[Series[Product[(1+x^k)^(3^k),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ 3^n * exp(2*sqrt(n) - 1/2 - c) / (2 * sqrt(Pi) * n^(3/4)), where c = Sum_{m>=2} (-1)^m/(m*(3^(m-1)-1)) = 0.215985336303958581708278160877115129... .

A261520 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(3^k).

Original entry on oeis.org

1, 6, 36, 200, 1038, 5160, 24776, 115632, 527172, 2355998, 10349448, 44783064, 191211512, 806737800, 3367294320, 13918479872, 57020736942, 231697484304, 934399998412, 3742041461976, 14888854356840, 58881590423856, 231542984619720, 905666813058384

Offset: 0

Vaclav Kotesovec, Aug 23 2015

nonn

Convolution of A144067 and A256142.

In general, for m > 1, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^(m^k), then a(n) ~ m^n * exp(2*sqrt(2*n) - 1 + c) / (sqrt(Pi) * 2^(3/4) * n^(3/4)), where c = 2 * Sum_{j>=1} 1/((2*j+1)*(m^(2*j)-1)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO]
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.

Cf. A156616, A261519, A260916, A001934, A015128.

nmax = 40; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(3^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ 3^n * exp(2*sqrt(2*n) - 1 + c) / (sqrt(Pi) * 2^(3/4) * n^(3/4)), where c = 2 * Sum_{j>=1} 1/((2*j+1)*(3^(2*j)-1)) = 0.0887630729103166089354170592729856346...

A297568 Number of nonisomorphic proper colorings of partition star graph using four colors.

Original entry on oeis.org

4, 12, 24, 36, 40, 108, 108, 60, 216, 180, 324, 324, 84, 360, 540, 648, 972, 972, 972, 112, 540, 1080, 660, 1080, 2916, 1512, 1944, 2916, 2916, 2916, 144, 756, 1800, 1980, 1620, 5832, 4860, 4536, 3240, 8748, 8748, 5832, 8748, 8748, 8748, 180, 1008, 2700, 3960, 1980, 2268, 9720, 14580, 9072, 13608, 4860, 17496, 14580, 26244, 13284, 9720, 26244, 26244, 17496, 26244, 26244, 26244, 220, 1296, 3780, 6600, 5940, 3024, 14580

Offset: 0

Marko Riedel, Dec 31 2017

A partition star graph consists of a multiset of paths with lengths given by the elements of the partition attached to a distinguished root node. The ordering of the partitions is by traversing antichains in Young's lattice bottom to top, left to right. Isomorphism refers to the automorphisms of the star graph corresponding to the partition.

			Rows are:
   4;
  12;
  24,  36;
  40, 108, 108;
  60, 216, 180, 324, 324;
  84, 360, 540, 648, 972, 972, 972;

Marko Riedel et al., Orbital chromatic polynomials

Cf. A297567, A297569, A297570.
Row sums give 4*A144067.
Row lengths give A000041.

b:= (n, i)-> `if`(n=0, [4], `if`(i<1, [], [seq(map(x-> x*
     binomial(3^i+j-1, j), b(n-i*j, i-1))[], j=0..n/i)])):
T:= n-> b(n$2)[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Jan 14 2018

b[n_, i_] := If[n == 0, {4}, If[i<1, {}, Table[Map[Function[x, x*Binomial[ 3^i + j - 1, j]], b[n - i*j, i - 1]], {j, 0, n/i}]] // Flatten];
T[n_] := b[n, n];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 17 2018, after Alois P. Heinz *)

For a partition lambda we have the OCP: k Product_{p^v in lambda} C((k-1)^p+v-1, v). Here we have k=4.

A275414 Triangle read by rows: T(n,k) is the number of multisets of k ternary words with a total of n letters.

Original entry on oeis.org

3, 9, 6, 27, 27, 10, 81, 126, 54, 15, 243, 486, 297, 90, 21, 729, 1836, 1380, 540, 135, 28, 2187, 6561, 5994, 2763, 855, 189, 36, 6561, 23004, 24543, 13212, 4635, 1242, 252, 45, 19683, 78732, 96723, 59130, 23490, 6996, 1701, 324, 55, 59049, 265842, 368874, 253719

Offset: 1

R. J. Mathar, Jul 27 2016

Ternary analog of A209406. Multiset transformation of A000244.

			      3
      9       6
     27      27      10
     81     126      54      15
    243     486     297      90      21
    729    1836    1380     540     135      28
   2187    6561    5994    2763     855     189      36
   6561   23004   24543   13212    4635    1242     252      45
  19683   78732   96723   59130   23490    6996    1701     324      55
  59049  265842  368874  253719  111609   36828    9846    2232     405      66

Alois P. Heinz, Rows n = 1..141, flattened
Index entries for triangles generated by the Multiset Transformation

Cf. A144067 (row sums), A000244 (column 1), A027468 (subdiagonal ?).

b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j)*
       binomial(3^i+j-1, j), j=0..min(n/i, p)))))
    end:
T:= (n, k)-> b(n$2, k):
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Apr 13 2017

b[n_, i_, p_] := b[n, i, p] = If[p > n, 0, If[n == 0, 1, If[Min[i, p] < 1, 0, Sum[b[n - i*j, i-1, p - j]*Binomial[3^i + j - 1, j], {j, 0, Min[n/i, p]}]]]];
T[n_, k_] := b[n, n, k];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 19 2018, after Alois P. Heinz *)

T(n,1) = A000244(n).

T(n,k) = Sum_{c_i*N_i=n,i=1..k} binomial(T(N_i,1)+c_i-1,c_i) for 1

G.f.: Product_{j>=1} (1-y*x^j)^(-3^j). - Alois P. Heinz, Apr 13 2017

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A144074 Number A(n,k) of multisets of nonempty words with a total of n letters over k-ary alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A256142 G.f.: Product_{j>=1} (1+x^j)^(3^j).

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A261520 Expansion of Product_{k>=1} ((1+x^k)/(1-x^k))^(3^k).

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A297568 Number of nonisomorphic proper colorings of partition star graph using four colors.

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A275414 Triangle read by rows: T(n,k) is the number of multisets of k ternary words with a total of n letters.

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