Real-rooted atlas

A038284 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(7 + 18\,x + 9\,n x\right)\,P_{n}(x) = \left(98 + 504\,x + 486\,x^{2} + 126\,n x + 162\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 196 + 1008\,x + 972\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
019610089722176
1274421168408242332888064
238416395136114307213063685248803407872
3537824691488026671680457228803674160011337408127926272
47529536116169984560105280128024064015431472009523422722380855684697620480
5105413504189744307210978063488313658956805040947520046664771328233323856644897760256169651208192
6147578905630359089152204923851776702596063232141146530560017421514629121306613597184548549148672991796451846047313952768
72066104678447815565414436886293319681475451732787235568925701120548777710817285487777108172834558596366336124966352931841983592903680213305255788544
8289254654976743797684224064551013309440295090346557440829941599692800153657759028838419207219878604801612734497095680874764470522880277703006515200392751394928647459086882832384
94049565169664114544843370496110453956107264056804891712307201825871519324160039438824817401856591582372261027846209027813818368044904576153507840213831315016704006048371481901056771220920950784258956978573803520
10566939123752961749412153294848185562646260203521060357978629734403834330190580736009465317956176445441656430642330877952208623334544297164818859921984473292801197455364093542400508063204479688704129565114719731712150388079585402888935141660703064064
117937147732541442653275099163852830702183290324582419298515211061166087753867718729932800215335983503014133764306719670060282675263282411478436806656686501160234827857925448421906625617920030822501071767781376117904254394955857922737063048454332416291521508119396352306677120225421295616
12111120068255580164000322457200885765014689937419681792343864452851635322881519758072871066828804689539196287863357441055146319164769255424177190752139623058636822425704567671043358722135781387397242224640151030255251662128742477030779538037827174426823217874852457676857138215591401684992562220051373121536010477750633867025317888
13155568095557812224600048368580132864081006529758317936640601762792490361815040290135632093567303680098480323122045130506242462008078051128262656046512572436651052892160672771137030131300761607475234855890347786240063432707205698094071808404411592574698592665601877625251239672037376059995126370971769241601180662107883555225600107946249863639334912356538669456658213634048
142177953337809371136896072230412998410241296104476133086986240103688358090646958899205415865132413256335360020054174890307372030361655148980948345273083494411576462473122039830937601883759183684367642132480239207515388491129159680023681544023460621786808321811763934734649695141888105147014069421634093056044796361023658921033728013223415608295818526720024179959969455211020288206447202864210228019212089258196146291747061760
Technical details
Node id
n-b7253df716371604 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-8e2fe519da2299bf

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           196,            1008,              972;
  1:          2744,           21168,            40824,            23328;
  2:         38416,          395136,          1143072,          1306368,            524880;
  3:        537824,         6914880,         26671680,         45722880,          36741600,          11337408;
  4:       7529536,       116169984,        560105280,       1280240640,        1543147200,         952342272,         238085568;
  5:     105413504,      1897443072,      10978063488,      31365895680,       50409475200,       46664771328,       23332385664,        4897760256;
  6:    1475789056,     30359089152,     204923851776,     702596063232,     1411465305600,     1742151462912,     1306613597184,      548549148672,       99179645184;
  7:   20661046784,    478155654144,    3688629331968,   14754517327872,    35568925701120,    54877771081728,    54877771081728,    34558596366336,    12496635293184,     1983592903680;
  8:  289254654976,   7437976842240,   64551013309440,  295090346557440,   829941599692800,  1536577590288384,  1920721987860480,  1612734497095680,   874764470522880,   277703006515200,   39275139492864;
  9: 4049565169664, 114544843370496, 1104539561072640, 5680489171230720, 18258715193241600, 39438824817401856, 59158237226102784, 62090278138183680, 44904576153507840, 21383131501670400, 6048371481901056, 771220920950784;
%F Formula
The row polynomials satisfy (7+(9*n+18)*x)*P(n,x) = (98+(126*n+504)*x+(162*n+486)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 1008*x + 972*x^2.
%t Mathematica
P[0, x_] := 196 + 1008*x + 972*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((98 + (126*n + 504)*x + (162*n + 486)*x^2)*P[n - 1, x])/(7 + (9*n + 18)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 196 + 1008*x + 972*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((98 + (126*n + 504)*x + (162*n + 486)*x^2)*v[i - 1])/(7 + (9*n + 18)*x)); 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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.