A327826 -id:A327826 - cp's OEIS frontend

A327803 Sum T(n,k) of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts that form a set of size k; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

1, 0, 1, 0, 3, 0, 7, 3, 0, 31, 16, 0, 121, 125, 0, 831, 711, 60, 0, 5041, 5915, 525, 0, 42911, 46264, 6328, 0, 364561, 438681, 67788, 0, 3742453, 4371085, 753420, 12600, 0, 39916801, 49321745, 8924685, 166320, 0, 486891175, 588219523, 113501784, 2966040

Offset: 0

Alois P. Heinz, Sep 25 2019

			Triangle T(n,k) begins:
  1;
  0,       1;
  0,       3;
  0,       7,       3;
  0,      31,      16;
  0,     121,     125;
  0,     831,     711,     60;
  0,    5041,    5915,    525;
  0,   42911,   46264,   6328;
  0,  364561,  438681,  67788;
  0, 3742453, 4371085, 753420, 12600;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)

Columns k=0-2 give: A000007, A061095, A327826.
Row sums give A005651.
Cf. A000217, A003056, A022915, A131632 (when the parts are distinct), A226874.

with(combinat):
T:= (n, k)-> add(multinomial(add(i, i=l), l[], 0), l=
             select(x-> nops({x[]})=k, partition(n))):
seq(seq(T(n, k), k=0..floor((sqrt(1+8*n)-1)/2)), n=0..14);
# second Maple program:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add(x^signum(j)*b(n-i*j, i-1)*
      combinat[multinomial](n, n-i*j, i$j), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..14);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i<1, 0, Sum[x^Sign[j]*b[n - i*j, i-1]*multinomial[n, Join[{n-i*j}, Table[i, {j}]]], {j, 0, n/i}]]]];
T[n_] := CoefficientList[b[n, n], x];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, May 06 2020, after 2nd Maple program *)

T(n*(n+1)/2,n) = T(A000217(n),n) = A022915(n).

A331373 Decimal expansion of Sum_{k>=2} 1/(k! - 1).

Original entry on oeis.org

1, 2, 5, 3, 4, 9, 8, 7, 5, 5, 6, 9, 9, 9, 5, 3, 4, 7, 1, 6, 4, 3, 3, 6, 0, 9, 3, 7, 9, 0, 5, 7, 9, 8, 9, 4, 0, 3, 6, 9, 2, 3, 2, 2, 0, 8, 3, 3, 2, 0, 1, 3, 4, 1, 7, 0, 6, 3, 8, 3, 4, 7, 1, 6, 6, 4, 0, 9, 5, 2, 4, 8, 2, 0, 4, 8, 9, 8, 7, 1, 7, 0, 8, 9, 0, 2, 4

Offset: 1

Amiram Eldar, May 03 2020

Erdős was interested in the question whether this constant is irrational.

			1.25349875569995347164336093790579894036923220833201...

Paul Erdős, Some of my favourite unsolved problems, in A. Baker, B. Bollobás and A. Hajnal (eds.), A tribute to Paul Erdős, Cambridge University Press, 1990, p. 470.

Thomas Bloom, Is Sum_{n>=2} 1/(n!-1) irrational?, Erdős Problems.
Paul Erdős, On the irrationality of certain series: problems and results, in Alan Baker (ed.), New Advances in Transcendence Theory, Cambridge University Press, 1988, p. 102.
Paul Erdős and Ronald L. Graham, Old and new problems and results in combinatorial number theory, L'enseignement Mathématique, Université de Genève, 1980, p. 62.
Terence Tao, Erdős problem database, see no. 68.

Cf. A033312, A331372, A327826.

RealDigits[Sum[1/(k! - 1), {k, 2, 300}], 10, 100][[1]]

suminf(k=2, 1/(k!-1)) \\ Michel Marcus, May 03 2020

cp's OEIS Frontend

A327803 Sum T(n,k) of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts that form a set of size k; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

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