OFFSET
1,2
LINKS
Karl Dilcher and Christophe Vignat, An explicit form of the polynomial part of a restricted partition function, Res Number Theory, 2017. [Note a typo in formula (37): 47/12 instead of 47/72.]
Leonid G. Fel and Boris Y. Rubinstein, Sylvester waves in the Coxeter groups, Ramanujan J. 2002, 6(3):307-329, and arXiv:math/0005174 [math.NT], 2000.
G. J. Rieger, Über Partitionen, Mathematische Annalen (1959), Volume: 138, page 356-362.
Boris Y. Rubinstein and Leonid G. Fel, Restricted partition functions as Bernoulli and Eulerian polynomials of higher order, Ramanujan J (2006) 11: 331-347.
A. V. Sills and D. Zeilberger, Formulae for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz), arXiv:1108.4391 [math.CO], 2011.
J. J. Sylvester, On the partition of numbers, Quarterly J. Pure Appl. Math. 1857, 1:141-152.
FORMULA
(1/A375251(n)) * Sum_{k=0..n-1} T(n, k)*x^k = W1([n], x), where W1([n], x) denotes the first Sylvester wave restricted to parts in [n].
T(n, k) = [x^k] p(n) where p(n) = W(n)*denominator(W(n)) and W(n) = [t^(-1)] exp(t*x)/Product_{k=1..n}(1 - exp(-t*k)).
EXAMPLE
Triangle starts:
[1] 1;
[2] 3, 2;
[3] 47, 36, 6;
[4] 175, 135, 30, 2;
[5] 50651, 38250, 9300, 900, 30;
[6] 598731, 439810, 110250, 12320, 630, 12;
[7] 87797891, 62748420, 15840279, 1893360, 116130, 3528, 42;
[8] 706078278, 492161075, 123824862, 15302301, 1031940, 38682, 756, 6;
.
Let A = ((a + b + c)^2 + (b*c) + (a*c) + (a*b))/6; B = a + b + c; C = 1 and W1 = (A + B*x + C*x^2)/(2*a*b*c). If (a, b, c) = (1, 2, 3) then W1([3], x) = (47 + 36*x + 6*x^2)/72. (See formulas (35), (37) and Fig. 2 in Dilcher & Vignat.)
MAPLE
read(PARTITIONS): # See Sills & Zeilberger paper.
FirstWave := proc(n) op(pmnPC(n, x)[1]); %*denom(%) end:
seq(print(seq(coeff(FirstWave(n), x, k), k = 0..n-1)), n = 1..9);
# Or, standalone:
W := proc(n) local k; exp(t*x)/mul(1 - exp(-t*k), k=1..n);
expand(series(%, t, n+1)); coeff(%, t, -1); %*denom(%) end:
Trow := n -> local k; seq(coeff(W(n), x, k), k = 0..n-1):
seq(print(Trow(n)), n = 1..8);
CROSSREFS
KEYWORD
AUTHOR
Peter Luschny, Aug 07 2024
STATUS
approved