A092932 -id:A092932 - cp's OEIS frontend

A092931 Number of ways of factorizing n into parts whose sum divides n.

0, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 2

Offset: 1

Amarnath Murthy, Mar 20 2004

nonn

Most of the terms are 1. But there are infinitely many terms for which a(n) >1. Example: a(n^n) >= 2, two such factorizations being n^n and n*n*n... n times, e.g. a(27) = 2 from 27, 3*3*3.

For any prime p the only factorization of p is p, which sums to p, which divides p, hence a(p) = 1. For the square of any positive even number e = 2*k we have e^2 = (2*k)^2 = 4*k^2; since we can factor e^2 as (2*k)*(2*k) whose factors sum to 4*k and 4*k | 4*k^2, we have a((2*k)^2) >= 2. For any odd semiprime s = p*q, s in A046315, we have p+q is even, hence p+q cannot divide p*q, hence a(p*q) = 1. For any even semiprime s > 4, s in A100484, we have s = 2*p for an odd prime p, hence 2+p is odd an cannot divide either 2 nor p, so a(2*p) = 1. See also: A016742 Even squares: (2n)^2. - Jonathan Vos Post, Mar 21 2006

			a(1) = 0. The only factorization of 1 is the empty multiset, whose sum is 0 and that does not divide 1.
a(16) = 4, the factorizations of 16 are 16, 8*2, 4*4, 4*2*2, 2*2*2*2. In four of them, all except 8*2, the sum of the parts divides 16.
a(30) = 2 because (besides 30 itself) we have 30 = 2 * 3 * 5 and 2 + 3 + 5 = 10 which divides 30.
a(100) = 3 from 100 = 5*20 = 10*10.

Amarnath Murthy, "Generalization of partition function, introducing Smarandache Factor partition", Smarandache Notions Journal, Vol. 11, 1-2-3, 2000.

Richard J Mathar, Table of n, a(n) for n = 1..10000
Richard J Mathar, Maple program

Cf. A000040, A016742, A046315, A100484, A092932.

More terms from Jonathan Vos Post, Mar 21 2006
More terms from Franklin T. Adams-Watters, Jun 12 2006
a(100) corrected by N. J. A. Sloane, Nov 23 2007

A257635 Triangle with n-th row polynomial equal to Product_{k = 1..n} (x + n + k).

Original entry on oeis.org

1, 2, 1, 12, 7, 1, 120, 74, 15, 1, 1680, 1066, 251, 26, 1, 30240, 19524, 5000, 635, 40, 1, 665280, 434568, 117454, 16815, 1345, 57, 1, 17297280, 11393808, 3197348, 495544, 45815, 2527, 77, 1, 518918400, 343976400, 99236556, 16275700, 1659889, 107800, 4354, 100, 1

Offset: 0

Peter Bala, Nov 05 2015

The row polynomials are a Sheffer sequence. For the associated polynomial sequence of binomial type see A038455.

			Triangle begins:
[0]       1;
[1]       2,      1;
[2]      12,      7,      1;
[3]     120,     74,     15,     1;
[4]    1680,   1066,    251,    26,    1;
[5]   30240,  19524,   5000,   635,   40,  1;
[6]  665280, 434568, 117454, 16815, 1345, 57, 1;
  ...

R. Sprugnoli, An Introduction to Mathematical Methods in Combinatorics, Section 5.6 CreateSpace Independent Publishing Platform 2006, ISBN-13: 978-1502925244.
Wikipedia, Sheffer sequence

Cf. A001813 (column 0), A005449 (first subdiagonal), A098118 (column 1).
Cf. A006963 (row sums), A000407 (alternating row sum).
Cf. A054649, A000108, A000984, A038455, A092932.

seq(seq(coeff(product(n + x + k, k = 1 .. n), x, i), i = 0..n), n = 0..8);
# Alternative:
p := n -> n!*hypergeom([-n, -x + n], [-n], 1):
seq(seq((-1)^k*coeff(simplify(p(n)), x, k), k=0..n), n=0..6); # Peter Luschny, Nov 27 2021

p[n_, x_] := FunctionExpand[Gamma[2*n + x + 1] / Gamma[n + x + 1]];
Table[CoefficientList[p[n, x], x], {n,0,8}] // Flatten (* Peter Luschny, Mar 21 2022 *)

E.g.f.: A(x,t) = B(t)*C(t)^x = 1 + (2 + x)*t + (3 + x)*(4 + x)*t^2/2! + (4 + x)*(5 + x)*(6 + x)*t^3/3! + ..., where B(t) = 1/sqrt(1 - 4*t) is the o.g.f. for A000984 and C(t) = (1 - sqrt(1 - 4*t))/(2*t) is the o.g.f. for A000108.
n-th row polynomial: n!*binomial(2*n + x,n).
T(n, k) = (-1)^k*n!*[x^k] hypergeom([-n, -x + n], [-n], 1). - Peter Luschny, Nov 27 2021
T(n, k) = [x^k] Gamma(2*n + x + 1) / Gamma(n + x + 1). - Peter Luschny, Mar 21 2022

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A092931 Number of ways of factorizing n into parts whose sum divides n.

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A257635 Triangle with n-th row polynomial equal to Product_{k = 1..n} (x + n + k).

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