cp's OEIS Frontend

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.)

%I A375252 #25 Aug 10 2024 08:52:45
%S A375252 1,3,2,47,36,6,175,135,30,2,50651,38250,9300,900,30,598731,439810,
%T A375252 110250,12320,630,12,87797891,62748420,15840279,1893360,116130,3528,
%U A375252 42,706078278,492161075,123824862,15302301,1031940,38682,756,6
%N 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.)
%H A375252 Karl Dilcher and Christophe Vignat, <a href="https://pubmed.ncbi.nlm.nih.gov/32355872/">An explicit form of the polynomial part of a restricted partition function</a>, Res Number Theory, 2017. [Note a typo in formula (37): 47/12 instead of 47/72.]
%H A375252 Leonid G. Fel and Boris Y. Rubinstein, <a href="https://doi.org/10.48550/arXiv.math/0005174">Sylvester waves in the Coxeter groups</a>, Ramanujan J. 2002, 6(3):307-329, and arXiv:math/0005174 [math.NT], 2000.
%H A375252 G. J. Rieger, <a href="https://eudml.org/doc/160721">Über Partitionen</a>, Mathematische Annalen (1959), Volume: 138, page 356-362.
%H A375252 Boris Y. Rubinstein and Leonid G. Fel, <a href="https://www.researchgate.net/publication/226298733_Restricted_partition_functions_as_Bernoulli_and_Eulerian_polynomials_of_higher_order">Restricted partition functions as Bernoulli and Eulerian polynomials of higher order</a>, Ramanujan J (2006) 11: 331-347.
%H A375252 A. V. Sills and D. Zeilberger, <a href="http://arxiv.org/abs/1108.4391">Formulae for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz)</a>, arXiv:1108.4391 [math.CO], 2011.
%H A375252 J. J. Sylvester, <a href="https://babel.hathitrust.org/cgi/pt?id=uc1.$b417523&amp;seq=161">On the partition of numbers</a>, Quarterly J. Pure Appl. Math. 1857, 1:141-152.
%F A375252 (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].
%F A375252 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)).
%e A375252 Triangle starts:
%e A375252   [1]         1;
%e A375252   [2]         3,         2;
%e A375252   [3]        47,        36,         6;
%e A375252   [4]       175,       135,        30,        2;
%e A375252   [5]     50651,     38250,      9300,      900,      30;
%e A375252   [6]    598731,    439810,    110250,    12320,     630,    12;
%e A375252   [7]  87797891,  62748420,  15840279,  1893360,  116130,  3528,  42;
%e A375252   [8] 706078278, 492161075, 123824862, 15302301, 1031940, 38682, 756, 6;
%e A375252 .
%e A375252 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.)
%p A375252 read(PARTITIONS):  # See Sills & Zeilberger paper.
%p A375252 FirstWave := proc(n) op(pmnPC(n, x)[1]); %*denom(%) end:
%p A375252 seq(print(seq(coeff(FirstWave(n), x, k), k = 0..n-1)), n = 1..9);
%p A375252 # Or, standalone:
%p A375252 W := proc(n) local k; exp(t*x)/mul(1 - exp(-t*k), k=1..n);
%p A375252 expand(series(%, t, n+1)); coeff(%, t, -1); %*denom(%) end:
%p A375252 Trow := n -> local k; seq(coeff(W(n), x, k), k = 0..n-1):
%p A375252 seq(print(Trow(n)), n = 1..8);
%Y A375252 Cf. A375251 (denominators), A375250 (main diagonal).
%K A375252 nonn,tabl,frac
%O A375252 1,2
%A A375252 _Peter Luschny_, Aug 07 2024