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A317533 Regular triangle read rows: T(n,k) = number of non-isomorphic multiset partitions of size n and length k. 60
1, 2, 2, 3, 4, 3, 5, 14, 9, 5, 7, 28, 33, 16, 7, 11, 69, 104, 74, 29, 11, 15, 134, 294, 263, 142, 47, 15, 22, 285, 801, 948, 599, 263, 77, 22, 30, 536, 2081, 3058, 2425, 1214, 453, 118, 30, 42, 1050, 5212, 9769, 9276, 5552, 2322, 761, 181, 42, 56, 1918, 12645, 29538, 34172, 23770, 11545, 4179, 1223, 267, 56 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

EXAMPLE

Non-isomorphic representatives of the T(3,2) = 4 multiset partitions:

  {{1},{1,1}}

  {{1},{1,2}}

  {{1},{2,2}}

  {{1},{2,3}}

Triangle begins:

    1

    2    2

    3    4    3

    5   14    9    5

    7   28   33   16    7

   11   69  104   74   29   11

   15  134  294  263  142   47   15

MATHEMATICA

permcount[v_List] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];

c[p_List, q_List, k_] := SeriesCoefficient[1/Product[(1 - x^LCM[p[[i]], q[[j]]])^GCD[p[[i]], q[[j]]], {j, 1, Length[q]}, {i, 1, Length[p]}], {x, 0, k}];

M[m_, n_, k_] := Module[{s = 0}, Do[Do[s += permcount[p]*permcount[q]*c[p, q, k], {q, IntegerPartitions[n]}], {p, IntegerPartitions[m]}]; s/(m!*n!)];

T[n_, k_] := M[k, n, n] - M[k - 1, n, n];

Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-Fran├žois Alcover, Feb 08 2020, after Andrew Howroyd *)

PROG

(PARI) \\ See A318795 for definition of M.

T(n, k)={M(k, n, n) - M(k-1, n, n)}

for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print) \\ Andrew Howroyd, Dec 28 2019

(PARI) \\ Faster version.

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}

K(q, t, n)={1/prod(j=1, #q, (1-x^lcm(t, q[j]) + O(x*x^n))^gcd(t, q[j]))}

G(m, n)={my(s=0); forpart(q=m, s+=permcount(q)*exp(sum(t=1, n, (K(q, t, n)-1)/t) + O(x*x^n))); s/m!}

A(n, m=n)={my(p=sum(k=0, m, G(k, n)*y^k)*(1-y)); matrix(n, m, n, k, polcoef(polcoef(p, n, x), k, y))}

{ my(T=A(10)); for(n=1, #T, print(T[n, 1..n])) } \\ Andrew Howroyd, Aug 30 2020

CROSSREFS

Row sums are A007716. First and last columns are both A000041.

Cf. A034691, A255397, A255903, A255906, A317532, A334550.

Sequence in context: A316939 A259478 A155706 * A231227 A231441 A284199

Adjacent sequences:  A317530 A317531 A317532 * A317534 A317535 A317536

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, Jul 30 2018

EXTENSIONS

Terms a(29) and beyond from Andrew Howroyd, Dec 28 2019

STATUS

approved

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Last modified November 28 14:27 EST 2020. Contains 338724 sequences. (Running on oeis4.)