A375251 -id:A375251 - cp's OEIS frontend

A375250 a(n) = A375251(n) / A010790(n) = denominator(W1([n], x)) / (n!*(n - 1)!), where W1([n], x) is the first Sylvester wave for parts in [n].

1, 2, 6, 2, 30, 12, 42, 6, 30, 20, 44, 12, 910, 420, 30, 6, 102, 12, 7980, 420, 13860, 1320, 4140, 180, 2730, 1092, 84, 28, 58, 60, 2046, 66, 117810, 7140, 420, 12, 36556, 9880, 780, 20, 189420, 9240, 397320, 9240, 48300, 19320, 19740, 1260, 46410, 39780, 87516, 1716, 6996, 264

Offset: 1

Peter Luschny, Aug 09 2024

nonn

J. S. Dowker, Relations between the Ehrhart polynomial, the heat kernel and Sylvester waves, arXiv:1108.1760 [math.NT], 2011. (See the factor in formula (27).)
G. J. Rieger, Über Partitionen, Mathematische Annalen (1959), Volume: 138, page 356-362. (See Satz 1.)
J. J. Sylvester, On the partition of numbers, Quarterly J. Pure Appl. Math. 1857, 1:141-152.

Cf. A375251, A375252, A010790, A008284.

read(PARTITIONS):  # From the paper of Sills & Zeilberger cited in A375252.
a := n -> denom(op(pmnPC(n, x)[1])) / (n!*(n - 1)!):
seq(a(n), n = 1..54);
# 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) end:
a := n -> n*denom(W(n))/(n!^2): seq(a(n), n = 1..24);

a(n) = denominator(W(n))/(n!*(n - 1)!) where W(n) = [t^(-1)] exp(t*x)/ Product_{k=1..n}(1 - exp(-t*k)).

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.)

Original entry on oeis.org

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

Peter Luschny, Aug 07 2024

			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.)

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.

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

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);

(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)).

cp's OEIS Frontend

A375250 a(n) = A375251(n) / A010790(n) = denominator(W1([n], x)) / (n!*(n - 1)!), where W1([n], x) is the first Sylvester wave for parts in [n].

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