A252654 -id:A252654 - 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

A292805 Number of sets of nonempty words with a total of n letters over n-ary alphabet.

Original entry on oeis.org

1, 1, 5, 55, 729, 12376, 250735, 5904746, 158210353, 4747112731, 157545928646, 5726207734545, 226093266070501, 9632339536696943, 440262935648935344, 21482974431740480311, 1114363790702406540897, 61219233429920494716931, 3550130647865299090804375

Offset: 0

Alois P. Heinz, Sep 23 2017

nonn

			a(0) = 1: {}.
a(1) = 1: {a}.
a(2) = 5: {aa}, {ab}, {ba}, {bb}, {a,b}.

Alois P. Heinz, Table of n, a(n) for n = 0..381

Main diagonal of A292804.

Cf. A252654, A292845.

h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1, k)*binomial(k^i, j), j=0..n/i)))
    end:
a:= n-> h(n$3):
seq(a(n), n=0..20);

h[n_, i_, k_] := h[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[h[n - i*j, i - 1, k]*Binomial[k^i, j], {j, 0, n/i}]]];
a[n_] := h[n, n, n];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 04 2018, from Maple *)

a(n) = [x^n] Product_{j=1..n} (1+x^j)^(n^j).

a(n) ~ n^(n - 3/4) * exp(2*sqrt(n) - 1/2) / (2*sqrt(Pi)). - Vaclav Kotesovec, Aug 26 2019

A075197 Number of partitions of n balls of n colors.

Original entry on oeis.org

1, 1, 6, 38, 305, 2777, 28784, 330262, 4152852, 56601345, 829656124, 12992213830, 216182349617, 3804599096781, 70540645679070, 1373192662197632, 27982783451615363, 595355578447896291, 13193917702518844859, 303931339674133588444, 7263814501407389465610

Offset: 0

Christian G. Bower, Sep 07 2002

nonn

For each integer partition of n, consider each part of size k to be a box containing k balls of up to n color. Order among parts and especially among parts of the same size does not matter. - Olivier Gérard, Aug 26 2016

			Illustration of first terms, ordered by number of parts, size of parts and smallest color of parts, etc.
a(1) = 1:
  {{1}}
a(2) = 6 = 3+3:
  {{1,1}},{{1,2}},{{2,2}},
  {{1},{1}},{{1},{2}},{{2},{2}}
a(3) = 38 = 10+18+10:
  {{1,1,1}},{{1,1,2}},{{1,1,3}},{{1,2,2}},{{1,2,3}},{{1,3,3}},
  {{2,2,2}},{{2,2,3}},{{2,3,3}},{{3,3,3}},
  {{1},{1,1}},{{1},{1,2}},{{1},{1,3}},{{1},{2,2}},{{1},{2,3}},{{1},{3,3}},
  {{2},{1,1}},{{2},{1,2}},{{2},{1,3}},{{2},{2,2}},{{2},{2,3}},{{2},{3,3}},
  {{3},{1,1}},{{3},{1,2}},{{3},{1,3}},{{3},{2,2}},{{3},{2,3}},{{3},{3,3}},
  {{1},{1},{1}},{{1},{1},{2}},{{1},{1},{3}},{{1},{2},{2}},{{1},{2},{3}},{{1},{3},{3}},
  {{2},{2},{2}},{{2},{2},{3}},{{2},{3},{3}},{{3},{3},{3}}

Alois P. Heinz, Table of n, a(n) for n = 0..400

Main diagonal of A075196.
Cf. A001700 (n balls of one color in n unlabeled boxes).
Cf. A209668 (boxes are ordered by size but not by content among a given size: order among boxes of the same size matters.),
Cf. A261783 (compositions of balls of n colors: boxes are labeled)
Cf. A252654 (lists instead of boxes : order of balls matter)
Cf. A000262 (lists instead of boxes and all n colors are used)
Cf. A255906 (the c colors used form the interval [1,c])
Cf. A255951 (the n-1 colors used form the interval [1,n-1])
Cf. A255942 (0/1 binary coloring)
Cf. A066186 (only 1 color among n = n * p(n))
Cf. A000110 (the n possible colors are used : set partitions of [n])
Cf. A005651 (the n possible colors are used and order of parts of the same size matters)
Cf. A000670 (the n possible colors are used and order of all parts matters)

with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*
      binomial(d+k-1, k-1), d=divisors(j))*A(n-j, k), j=1..n)/n)
    end:
a:= n-> A(n, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 26 2012

A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[d*Binomial[d+k-1, k-1], {d, Divisors[j]}]*A[n-j, k], {j, 1, n}]/n]; a[n_] := A[n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

a(n) = [x^n] Product_{k>=1} 1 / (1 - x^k)^binomial(k+n-1,n-1). - Ilya Gutkovskiy, May 09 2021

A292873 Total number of words beginning with the first letter of an n-ary alphabet in all multisets of nonempty words with a total of n letters.

Original entry on oeis.org

0, 1, 5, 37, 415, 6051, 109476, 2348767, 58191451, 1631827927, 51029454163, 1758883278967, 66200568699170, 2699977173047181, 118561410689195358, 5574984887552288475, 279398986674750754195, 14863338415349068099348, 836304620387823727353480

Offset: 0

Alois P. Heinz, Sep 25 2017

nonn

			For n=2 and alphabet {a,b} we have 7 multisets:  {aa}, {ab}, {ba}, {bb}, {a,a}, {a,b}, {b,b}. There is a total of 5 words beginning with the first alphabet letter, thus a(2) = 5.

Alois P. Heinz, Table of n, a(n) for n = 0..382

Cf. A252654, A292845.

h:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, 0, add(
     (p-> p+[0, p[1]*j])(binomial(k^i+j-1, j)*h(n-i*j, i-1, k)), j=0..n/i)))
    end:
a:= n-> `if`(n=0, 0, h(n$3)[2]/n):
seq(a(n), n=0..22);

h[n_, i_, k_] := h[n, i, k] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[ Function[p, p + {0, p[[1]]*j}][Binomial[k^i + j - 1, j]*h[n - i*j, i - 1, k]], {j, 0, n/i}]]];
a[n_] := If[n == 0, 0, h[n, n, n][[2]]/n];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Mar 19 2022, after Alois P. Heinz *)

A300456 a(n) = [x^n] Product_{k=1..n} ((1 + x^k)/(1 - x^k))^(n^k).

Original entry on oeis.org

1, 2, 16, 200, 3264, 65752, 1565744, 42878432, 1324344832, 45464289482, 1715228012048, 70471268834936, 3129746696619072, 149318596196238328, 7612660420021177200, 412865831480749700928, 23725813528034949148672, 1439701175150489313314864, 91967625580609006328344400, 6167733266497532499924699672

Offset: 0

Ilya Gutkovskiy, Mar 06 2018

nonn

			The table of coefficients of x^k in expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(n^k) begins:
n = 0: (1),  0,    0,    0,     0,       0,  ...
n = 1:  1,  (2),   4,    8,    14,      24,  ...
n = 2:  1,   4,  (16),  60,   208,     692,  ...
n = 3:  1,   6,   36, (200), 1038,    5160   ...
n = 4:  1,   8,   64,  472, (3264),  21608,  ...
n = 5:  1,  10,  100,  920,  7950,  (65752), ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..377

Cf. A015128, A252654, A261519, A261520, A270919, A270923, A270924, A292805, A300457, A300458.

Table[SeriesCoefficient[Product[((1 + x^k)/(1 - x^k))^(n^k), {k, 1, n}], {x, 0, n}], {n, 0, 19}]

a(n) ~ exp(2*sqrt(2*n) - 1) * n^(n - 3/4) / (2^(3/4)*sqrt(Pi)). - Vaclav Kotesovec, Aug 26 2019

A300457 a(n) = [x^n] Product_{k=1..n} (1 - x^k)^(n^k).

Original entry on oeis.org

1, -1, -3, -1, 25, 624, 9871, 170470, 3027249, 55077245, 979330606, 15079702923, 94670678245, -7958168036625, -626145997536240, -34564907982551791, -1733699815491494303, -84294315853736719077, -4067859614343931897505, -196552300464314521511610, -9519733465269825759734169

Offset: 0

Ilya Gutkovskiy, Mar 06 2018

sign

			The table of coefficients of x^k in expansion of Product_{k>=1} (1 - x^k)^(n^k) begins:
n = 0: (1),  0,    0,    0,   0,     0,  ...
n = 1:  1, (-1),  -1,    0,   0,     1,  ...
n = 2:  1,  -2,  (-3),   0,   2,    12,  ...
n = 3:  1,  -3,   -6,  (-1),  9,    63,  ...
n = 4:  1,  -4,  -10,   -4, (25),  224,  ...
n = 5:  1,  -5,  -15,  -10,  55,  (624), ...

Cf. A010815, A008705, A252654, A252782, A255672, A270917, A270922, A281266, A281267, A281268, A283333, A292805, A300456, A300458.

Table[SeriesCoefficient[Product[(1 - x^k)^(n^k), {k, 1, n}], {x, 0, n}], {n, 0, 20}]

A300458 a(n) = [x^n] Product_{k=1..n} 1/(1 + x^k)^(n^k).

Original entry on oeis.org

1, -1, -1, -10, 11, 374, 9792, 183847, 3469427, 65038049, 1195396233, 19667738452, 189089161562, -6219720781782, -606316892131934, -35104997710496175, -1795953382595105853, -88223902016631657740, -4283800987347611165184, -207864171877269042498096, -10102590396625592962089500

Offset: 0

Ilya Gutkovskiy, Mar 06 2018

sign

			The table of coefficients of x^k in expansion of Product_{k>=1} 1/(1 + x^k)^(n^k) begins:
n = 0: (1),  0,    0,    0,   0,     0,  ...
n = 1:  1, (-1),   0,   -1,   1,    -1,  ...
n = 2:  1,  -2,  (-1),  -4,   3,    -2,  ...
n = 3:  1,  -3,   -3, (-10),  6,    15,  ...
n = 4:  1,  -4,   -6,  -20, (11),  104,  ...
n = 5:  1,  -5,  -10,  -35,  20,  (374), ...

Cf. A081362, A252654, A255526, A252782, A255672, A270917, A270922, A281266, A281267, A281268, A283333, A292805, A300456, A300457.

Table[SeriesCoefficient[Product[1/(1 + x^k)^(n^k), {k, 1, n}], {x, 0, n}], {n, 0, 20}]

A305207 a(n) = [x^n] exp(Sum_{k>=1} x^k/(k(1 - nx^k))).

Original entry on oeis.org

1, 1, 3, 13, 95, 921, 11586, 176324, 3162447, 65233120, 1521743103, 39609506223, 1138093049808, 35779807446670, 1221719353617885, 45025117385882889, 1781345658408660655, 75304205654268663567, 3387556543611248410593, 161575661076504392490150, 8144909167115962980271095

Offset: 0

Ilya Gutkovskiy, May 27 2018

nonn

N. J. A. Sloane, Transforms

Cf. A034691, A104460, A252654, A305209.

Table[SeriesCoefficient[Exp[Sum[x^k/(k (1 - n x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[Product[1/(1 - x^k)^(n^(k - 1)), {k, 1, n}], {x, 0, n}], {n, 0, 20}]

a(n) = [x^n] Product_{k>=1} 1/(1 - x^k)^(n^(k-1)).

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.

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A292805 Number of sets of nonempty words with a total of n letters over n-ary alphabet.

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A075197 Number of partitions of n balls of n colors.

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A292873 Total number of words beginning with the first letter of an n-ary alphabet in all multisets of nonempty words with a total of n letters.

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A300456 a(n) = [x^n] Product_{k=1..n} ((1 + x^k)/(1 - x^k))^(n^k).

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A300457 a(n) = [x^n] Product_{k=1..n} (1 - x^k)^(n^k).

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A300458 a(n) = [x^n] Product_{k=1..n} 1/(1 + x^k)^(n^k).

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A305207 a(n) = [x^n] exp(Sum_{k>=1} x^k/(k*(1 - n*x^k))).

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A305207 a(n) = [x^n] exp(Sum_{k>=1} x^k/(k(1 - nx^k))).