Real-rooted atlas

A038226 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(43 + 32\,x\right)\,P_{n-1}(x) + \left(32\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1849 + 3776\,x + 1024\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01849377610246649
17950734236823040032768685043
23418801282218243560857611927552104857680225329
31470084432226038432471321600027974041605609881603355443210478209627
463213630491716571525765755473418245396620902401854459084802476317081610737418241504470768809
52718186111071307657006076867076594457600934305454817284898043723776010961192747008104475079475234359738368234876269129091
61168820027760198944095011302475956458489077761513329684460339211365432223072256379249611925094459627460467097642606075576320109951162777639527980378100065
750259261193684374582093667360032844396216236748800234658892856492032024277544091726643201133570543346253824261484407913709568304566095283486721692148395147264351843720888327121063133809178475

All 15 rows

Technical details
Node id
n-355c148890ae1ebb · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-9de9b7dea182369b

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:               1849,                  3776,                    1024;
  1:              79507,                342368,                  230400,                   32768;
  2:            3418801,              28221824,                35608576,                11927552,                 1048576;
  3:          147008443,            2226038432,              4713216000,              2797404160,               560988160,                33554432;
  4:         6321363049,          171657152576,            575547341824,            539662090240,            185445908480,             24763170816,              1073741824;
  5:       271818611107,        13076570060768,          67076594457600,          93430545481728,          48980437237760,          10961192747008,           1044750794752,            34359738368;
  6:     11688200277601,       989440950113024,        7595645848907776,       15133296844603392,       11365432223072256,        3792496119250944,         596274604670976,         42606075576320,         1099511627776;
  7:    502592611936843,     74582093667360032,      844396216236748800,     2346588928564920320,     2427754409172664320,     1133570543346253824,      261484407913709568,      30456609528348672,      1692148395147264,       35184372088832;
  8:  21611482313284249,   5609739988633981376,    92737022134687642624,   353196539990099886080,   490236849682603048960,   307802961392814784512,    97723093246801870848,   16499415797307801600,   1480563875056189440,    65794775806115840,    1125899906842624;
  9: 929293739471222707, 421422066581573699168, 10102373048047865164800, 52061903766933888729088, 95132790575408317726720, 78171580352584698822656, 32814621677568512753664, 7532483001778842894336, 970669904155650293760, 69156114793622405120, 2514134491979579392, 36028797018963968;
%F Formula
The row polynomials satisfy P(n,x) = (43+32*x)*P(n-1,x) + 32*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1849 + 3776*x + 1024*x^2.
%t Mathematica
P[0, x_] := 1849 + 3776*x + 1024*x^2;
P[n_, x_] := P[n, x] = Expand[(43 + 32*x)*P[n - 1, x] + 32*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] = 1849 + 3776*x + 1024*x^2; for(n = 1, N, my(i = n + 1); v[i] = (43 + 32*x)*v[i - 1] + 32*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

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