Real-rooted atlas

A038255 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(11 + 4\,x\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
111415
212110416241
313312044720644159
4146413598421856435225677089
51610515983245592001875202432010241531439
6177156196190641301809665497601406720129024409632498321
71948717115137220428582008020271686464180480962662465945616384733879263
821435888123485317446036070336580576819225437404165551472646158745632768006553617568546625
92357947691360854116841240794633601576769497609190406400027384526848437615001637414502415925248262144444254845775
1025937424601550712966024250185145057641228876979203112117527040121653658828826270335795232096256000218136576076021760104857611827101704497
1128531167061183644441887644973802942504010483382285446410051872402176050161104345088140607638814722302567415808222183751680122984857603575644164194304330499611799167
123138428376721126607909513904978480335830816261013004335283231333408400053761957069130784768692771153231872146043184742400187641709854721466763837440674297282561660944384167772169667881068111457
133452271214393119116723562154441909755801884112063946912340438400950407228535564807320250641434931232075266886254592846676881788108813910320913448961439945843015689305971490816361070854144763363328067108864295320631818413935
1437974983358324128813176191807384370500291782843056154716921590544768028218871664077785602649440590259054592141544436667630796845850505144238080093681455199354880123318738275532801050582883237888570827829411841894667780096347623915522684354569399262532890897617
Technical details
Node id
n-c3b6a504c8c99d5f · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-ee32e1d26c1bc310

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:         11,           4;
  2:        121,         104,           16;
  3:       1331,        2044,          720,           64;
  4:      14641,       35984,        21856,         4352,         256;
  5:     161051,      598324,       559200,       187520,       24320,        1024;
  6:    1771561,     9619064,     13018096,      6549760,     1406720,      129024,       4096;
  7:   19487171,   151372204,    285820080,    202716864,    64180480,     9626624,     659456,     16384;
  8:  214358881,  2348531744,   6036070336,   5805768192,  2543740416,   555147264,   61587456,   3276800,    65536;
  9: 2357947691, 36085411684, 124079463360, 157676949760, 91904064000, 27384526848, 4376150016, 374145024, 15925248, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (11+4*x)*P(n-1,x) + 4*x*d/dx 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[(11 + 4*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], 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] = (11 + 4*x)*v[i - 1] + 4*x*deriv(v[i - 1], 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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. [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
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.