A375252 First Sylvester wave. Triangle read by rows: Coefficients of the numerator of the polynomial part of the partition function restricted to partitions of the integer x with parts in (1,2,...,n). (The denominators are A375251.)

1, 3, 2, 47, 36, 6, 175, 135, 30, 2, 50651, 38250, 9300, 900, 30, 598731, 439810, 110250, 12320, 630, 12, 87797891, 62748420, 15840279, 1893360, 116130, 3528, 42, 706078278, 492161075, 123824862, 15302301, 1031940, 38682, 756, 6

Offset: 1

Author

Peter Luschny, Aug 07 2024

Keywords

• nonn
• tabl
• frac

Examples

			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.)
		

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.

Crossrefs

Cf. A375251 (denominators), A375250 (main diagonal).

Programs

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);

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)).