Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #30 Dec 12 2023 18:49:31
%S 3,1,8,7,2,1,15,7,13,3,1,1,24,23,22,21,4,3,2,1,35,17,11,8,31,5,2,1,1,
%T 1,48,47,46,45,44,43,6,5,4,3,2,1,63,31,61,15,59,29,57,7,3,5,1,3,1,1,
%U 80,79,26,77,76,25,74,73,8,7,2,5,4,1,2,1
%N Irregular triangle read by rows: numerators of the coefficients of polynomials J(2n-1,z) = Sum_(k=1,2, .. n) ((n+1)^2 - k + (n+1-k)*z^n)*z^(k-1)/k.
%C Irregular triangle of fractions:
%C 3, 1,
%C 8, 7/2, 2, 1/2,
%C 15, 7, 13/3, 3, 1, 1/3,
%C 24, 23/2, 22/3, 21/4, 4, 3/2, 2/3, 1/4,
%C 35, 17, 11, 8, 31/5, 5, 2, 1, 1/2, 1/5,
%C 48, 47/2, 46/3, 45/4, 44/5, 43/6, 6, 5/2, 4/3, 3/4, 2/5, 1/6.
%C etc.
%C First column: A005563; T(n, 1) = A005563(n).
%C Main diagonal: T(n, n) - n = n^2+1 = A002522(n).
%C The first upper diagonal is T(n, n+1) = n.
%C Consider TT(n, k) = k*T(n, k) for k = 1 to n:
%C 3,
%C 8, 7,
%C 15, 14, 13,
%C 24, 23, 22, 21,
%C etc.
%C Row sums: 3, 8+7, ... , are the positive terms of A059270; that is A059270(n).
%H Jean-François Alcover, <a href="/A270861/a270861.pdf">Roots of J(2n-1,z) lie close to two concentric circles (example)</a>
%e Irregular triangle:
%e 3, 1,
%e 8, 7, 2, 1,
%e 15, 7, 13, 3, 1, 1,
%e 24, 23, 22, 21, 4, 3, 2, 1,
%e 35, 17, 11, 8, 31, 5, 2, 1, 1, 1
%e 48, 47, 46, 45, 44, 43, 6, 5, 4, 3, 2, 1
%e etc.
%e Second half part by row: A112543.
%t row[n_] := CoefficientList[Sum[(((n + 1)^2 - k + (n + 1 - k)*z^n))*z^(k - 1)/k, {k, n}], z]; Table[row[n] // Numerator, {n, 1, 9}] // Flatten (* _Jean-François Alcover_, Apr 07 2016 *)
%Y Cf. A002260, A002378, A002522, A005563, A059270, A112543, A122197, A004736.
%K nonn,tabf,frac
%O 1,1
%A _Paul Curtz_, Mar 24 2016