Real-rooted atlas

A099040 → basis A013611 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(10 + 24\,x\right)\,P_{n-2}(x) + \left(10 + 24\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 10 + 24\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1102434
21105045761190
312008160184321382441616
4131001180803980165944323317761455404
5143000160560071942401607731217915904796262450898680
61561000209832001172908803489315845825986565175705601911029761780038856
71704000026678400017869824006637731840147653591041966768128014523826176458647142462251876224
81860100003324096000259491456001155691929603211594997765702035046406315953356803990230138881100753141762177085112720
92030500000407820000003635424000001887736320000629221478400013960908472320206181000806401954295473766410787380789248264180754022476137457624096
1022165100000494257200000495346176000029380852224000114213075148800304032102580224561243641610240709412467507200587582027071488287957021884416634033809653762662694453051744
1124195600000059310864000006601098240000044029398528000019554990243840006072057068912640134504496821698562125423602302976023478624212484096172668540829040647608405715845120152168114316902493120284962978560
12264121100000070592342400000863852688000000639989503488000031969317095424000113433982103715840293143073353629696555917124676091904767789626168442880753137060797218816498033557483028480199340229755142144365203474360565763256621300145030336
132883167000000083443029600000011135198995200000907186182912000005034126976819200002009255978601676800593408017050107904013129958159850405888217647951605881896962669447715794347622423546113589111685120141440262257560780805185889335920033792876488338465357824113891213893672302464
1431472881000000098055755280000001417110601536000001259162373657600000768466446759936000034076067961710182400113216791495476510720286312110697777987584553786854696202272768815258177069566132224899184212715723816960720493196073385328640396464891765830189056134102715785199747072210357201231685877763983026396589789315200
Technical details
Node id
n-3caed5f0a2fcd6d2 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-3f33e9af1c977e6a

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 0)
  • 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;
  1:         10,          24;
  2:        110,         504,          576;
  3:       1200,        8160,        18432,         13824;
  4:      13100,      118080,       398016,        594432,        331776;
  5:     143000,     1605600,      7194240,      16077312,      17915904,        7962624;
  6:    1561000,    20983200,    117290880,     348931584,     582598656,      517570560,      191102976;
  7:   17040000,   266784000,   1786982400,    6637731840,   14765359104,    19667681280,    14523826176,     4586471424;
  8:  186010000,  3324096000,  25949145600,  115569192960,  321159499776,   570203504640,   631595335680,   399023013888,   110075314176;
  9: 2030500000, 40782000000, 363542400000, 1887736320000, 6292214784000, 13960908472320, 20618100080640, 19542954737664, 10787380789248, 2641807540224;
%F Formula
The row polynomials satisfy P(n,x) = (10+24*x)*P(n-2,x) + (10+24*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 10 + 24*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 10 + 24*x;
P[n_, x_] := P[n, x] = Expand[(10 + 24*x)*P[n - 2, x] + (10 + 24*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 10 + 24*x; for(n = 2, N, my(i = n + 1); v[i] = (10 + 24*x)*v[i - 2] + (10 + 24*x)*v[i - 1]); 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

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