Real-rooted atlas

A110608 → reverse

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(12 + 14\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(12 + 46\,x + 2\,x^{2} + 14\,n + 64\,n x + 3\,n x^{2} + 4\,n^{2} + 22\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}(x) + \left(13\,x - 11\,x^{2} - 2\,x^{3} + 8\,n x - 7\,n x^{2} - n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1 + 8\,x + 6\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
018615
1118452084
213216822470495
3150450120010502523003
417299044007425475292418564
51981911127403503542042210213432116280
611283360313601274002446082242249152012870735471
71162550868544385560107956815593761145664393822486204686825
81200855013680010174503907008813960093024005668650167960018475630045015
9124212705254100241395012166308344712065627952052762050273581007113106705432193536720
101288182164452805259870336631681243667042741143683640581452876508801294428962995372827041561251677700
1113382535074360010689250846588603950746801128784800201064792522340532501519156210602640480125550100104006008122425444
121392343981192464204954751967565601131350220406362528093336393151382761380013136233110781660152027683797055241902404011660052860229080
131450456751847300374078254279455182971843875131003730003766357237571607285750902251800457456626450039354417375125747842502181340125155117520344867425584
141512595202777600654472008796103687256785536385053926401353705210003208782720005166140179205635789286404109429688001945292160005657227200090515635206010803902254848913647
Technical details
Node id
n-df756dee21ca5713 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-98b5f32915261152

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1,   8,     6;
  1: 1,  18,    45,     20;
  2: 1,  32,   168,    224,      70;
  3: 1,  50,   450,   1200,    1050,      252;
  4: 1,  72,   990,   4400,    7425,     4752,      924;
  5: 1,  98,  1911,  12740,   35035,    42042,    21021,     3432;
  6: 1, 128,  3360,  31360,  127400,   244608,   224224,    91520,    12870;
  7: 1, 162,  5508,  68544,  385560,  1079568,  1559376,  1145664,   393822,    48620;
  8: 1, 200,  8550, 136800, 1017450,  3907008,  8139600,  9302400,  5668650,  1679600,  184756;
  9: 1, 242, 12705, 254100, 2413950, 12166308, 34471206, 56279520, 52762050, 27358100, 7113106, 705432;
%F Formula
The row polynomials satisfy (4*n^2+14*n+12)*P(n,x) = (4*n^2+14*n+12+(22*n^2+64*n+46)*x+(n^2+3*n+2)*x^2)*P(n-1,x) + ((8*n+13)*x-(7*n+11)*x^2-(n+2)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 8*x + 6*x^2.
%t Mathematica
P[0, x_] := 1 + 8*x + 6*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 14*n + 12 + (22*n^2 + 64*n + 46)*x + (n^2 + 3*n + 2)*x^2)*P[n - 1, x] + ((8*n + 13)*x - (7*n + 11)*x^2 - (n + 2)*x^3)*D[P[n - 1, x], x])/(4*n^2 + 14*n + 12)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1 + 8*x + 6*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 14*n + 12 + (22*n^2 + 64*n + 46)*x + (n^2 + 3*n + 2)*x^2)*v[i - 1] + ((8*n + 13)*x - (7*n + 11)*x^2 - (n + 2)*x^3)*deriv(v[i - 1], x))/(4*n^2 + 14*n + 12)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.