Real-rooted atlas

A013610 → basis A013611 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(16 + 24\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1162440
22567685761600
3409618432276481382464000
4655363932168847368847363317762560000
5104857678643202359296035389440265420807962624102400000
616777216150994944566231040113246208012740198407644119041911029764096000000
7268435456281857228812683575296317089382404756340736042807066624214035333124586471424163840000000
84294967296515396075522705829396488117488189441522029035520182643484262413698261319685870683422721100753141766553600000000
96871947673692771293593655662776156161948197165465643834436222976657516543344646575165433446442268920643584158508452413442641807540224262144000000000
10109951162777616492674416640111325552312320445302209249280116891829927936021040529387028482630066173378560225434243432448012680676193075204226892064358406340338096537610485760000000000
111759218604441629027106973286421770330229964809796648603484160293899458104524806171888620195020892578329302925312991910671102771207439330033270784037196650166353920111589950499061761521681143169024419430400000000000
122814749767106565066549580791808417990340415324162089951702076620807053586994508595201692860878682062848296250653769360998438089369770346414083570878415969976320238058561064665088010712635247909928962921627794884526083652034743605657616777216000000000000
13450359962737049687820192733724672790381734603522048434709954031937126416301623276197642240440143828457336340488802876569146726809613204314853720090214414854854210435101491212379045175362584576074274271052175507456303849290667990712327596232266699767808876488338465357824671088640000000000000
14720575940379279361513209474796486656147537923792657448968852275427559446937636515636138682718617610954690841604815585282464805439361083506688422538075319042886860855458122385624378900485545812238562437890048415935917892182841753622687413703209973186568507780138703739944961963333878162401525762103572012316858777626843545600000000000000
Technical details
Node id
n-f5bcaca8318904f4 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-f06e51c139cfae05

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           1;
  1:          16,           24;
  2:         256,          768,           576;
  3:        4096,        18432,         27648,          13824;
  4:       65536,       393216,        884736,         884736,         331776;
  5:     1048576,      7864320,      23592960,       35389440,       26542080,        7962624;
  6:    16777216,    150994944,     566231040,     1132462080,     1274019840,      764411904,      191102976;
  7:   268435456,   2818572288,   12683575296,    31708938240,    47563407360,    42807066624,    21403533312,     4586471424;
  8:  4294967296,  51539607552,  270582939648,   811748818944,  1522029035520,  1826434842624,  1369826131968,   587068342272,   110075314176;
  9: 68719476736, 927712935936, 5566277615616, 19481971654656, 43834436222976, 65751654334464, 65751654334464, 42268920643584, 15850845241344, 2641807540224;
%F Formula
The row polynomials satisfy P(n,x) = (16+24*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[(16 + 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; for(n = 1, N, my(i = n + 1); v[i] = (16 + 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

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.