A375251
Denominators of the polynomials A375252 (polynomial part of the partition function restricted to partitions of the integer x with parts in (1,2,...,n)).
Original entry on oeis.org
1, 4, 72, 288, 86400, 1036800, 152409600, 1219276800, 438939648000, 26336378880000, 6373403688960000, 229442532802560000, 2714305163054284800000, 228001633696559923200000, 3420024505448398848000000, 164161176261523144704000000, 759081279033283021111296000000
Offset: 1
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read(PARTITIONS): # See the Sills & Zeilberger paper cited in A375252.
seq(denom(op(pmnPC(n,x)[1])), n = 1..17);
# 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 -> denom(W(n)): seq(a(n), n = 1..17);
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
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.
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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);
Showing 1-2 of 2 results.