Real-rooted atlas

A099097 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-6 + 6\,n\right)\,P_{n}(x) = \left(-36 - 62\,x - 26\,x^{2} + 36\,n + 40\,n x + 13\,n x^{2}\right)\,P_{n-1}(x) + \left(-36\,x - 44\,x^{2} - 13\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2639
336381084
421634818633783
51296280822728141097299
67776211682296812420334836068040
74665615292820822415077661248132361189634257
82799361073088175824015964328676242822765091039275912433
916796167371648141212161542268810504816456980012400081919061297055114668
1010077696498286081092683521394893441142480646226492822581624525555671230242837513769311
1160466176332563968821518848120035520011489230087527660803419289441063280242166430026117741414814789267803
123627970562196937728603653990499350530561088285068883313509764548649824177118315248206260887343704948218446728044644718160
1321767823361439094988843535646720796955397129832449254486134859136549404070402570967235287605100962119906672345812064341449681543321416170266849
141306069401693601640448309143402496623050195968854940220416843483386880615592572288336507905856137783093760417362786889091342416134861407212211157450972433879466556121
157836416409660514548940821666240184324767193626624720171863654479020655073286494599805184406184082521619425820318727069976459521927572140003817724248051926136604341497901683505036163709805636
Technical details
Node id
n-c45ee837fbbe8408 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-f2c232606f089033

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:        1;
   2:        6,        3;
   3:       36,       38,        10;
   4:      216,      348,       186,        33;
   5:     1296,     2808,      2272,       814,       109;
   6:     7776,    21168,     22968,     12420,      3348,      360;
   7:    46656,   152928,    208224,    150776,     61248,    13236,     1189;
   8:   279936,  1073088,   1758240,   1596432,    867624,   282276,    50910,    3927;
   9:  1679616,  7371648,  14121216,  15422688,  10504816,  4569800,  1240008,  191906,  12970;
  10: 10077696, 49828608, 109268352, 139489344, 114248064, 62264928, 22581624, 5255556, 712302, 42837;
%F Formula
The row polynomials satisfy (6*n-6)*P(n,x) = (36*n-36+(40*n-62)*x+(13*n-26)*x^2)*P(n-1,x) - (36*x+44*x^2+13*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((36*n - 36 + (40*n - 62)*x + (13*n - 26)*x^2)*P[n - 1, x] - (36*x + 44*x^2 + 13*x^3)*D[P[n - 1, x], x])/(6*n - 6)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((36*n - 36 + (40*n - 62)*x + (13*n - 26)*x^2)*v[i - 1] - (36*x + 44*x^2 + 13*x^3)*deriv(v[i - 1], x))/(6*n - 6)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741