Real-rooted atlas

A038279 → basis A013610 → basis A013611

palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(12 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1121224
2144288144576
3172851845184172813824
420736829441244168294420736331776
524883212441602488320248832012441602488327962624
6298598417915904447897605971968044789760179159042985984191102976
73583180825082265675246796812541132801254113280752467968250822656358318084586471424
8429981696343985356812039487488240789749763009871872024078974976120394874883439853568429981696110075314176
95159780352464380231681857520926724334215495686501323243526501323243524334215495681857520926724643802316851597803522641807540224
106191736422461917364224027862813900807430083706880130026464870401560317578444813002646487040743008370688027862813900806191736422406191736422463403380965376
117430083706888173092077568408654603878401225963811635202451927623270403432698672578563432698672578562451927623270401225963811635204086546038784081730920775687430083706881521681143169024
1289161004482561069932053790725884626295848961961542098616320441346972188672070615515550187528238476814188544706155155501875244134697218867201961542098616320588462629584896106993205379072891610044825636520347436056576
13106993205379072139091166992793683454700195676163060005673841459276500141846036480137700255322865664183600340430487552183600340430487552137700255322865664765001418460364803060005673841459283454700195676161390911669927936106993205379072876488338465357824
14128391846454886417974858503684096116836580273946624467346321095786496128520238301341286425704047660268257283855607149040238592440640817033170124838556071490402385922570404766026825728128520238301341286446734632109578649611683658027394662417974858503684096128391846454886421035720123168587776
Technical details
Node id
n-e7ea938b11ca3a81 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-77a7ed3b1d5f257e

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: yes
  • γ-positive (gamma_positive): yes
  • 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:          1;
  1:         12,          12;
  2:        144,         288,          144;
  3:       1728,        5184,         5184,         1728;
  4:      20736,       82944,       124416,        82944,        20736;
  5:     248832,     1244160,      2488320,      2488320,      1244160,       248832;
  6:    2985984,    17915904,     44789760,     59719680,     44789760,     17915904,      2985984;
  7:   35831808,   250822656,    752467968,   1254113280,   1254113280,    752467968,    250822656,     35831808;
  8:  429981696,  3439853568,  12039487488,  24078974976,  30098718720,  24078974976,  12039487488,   3439853568,   429981696;
  9: 5159780352, 46438023168, 185752092672, 433421549568, 650132324352, 650132324352, 433421549568, 185752092672, 46438023168, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = (12+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(12 + 12*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; for(n = 1, N, my(i = n + 1); v[i] = (12 + 12*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

Same fingerprint ?

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.