Real-rooted atlas

A130123 → basis A013609 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(6 + 16\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
161622
236192256484
321617284608409610648
4129613824552969830465536234256
577761036805529601474560196608010485765153632
646656746496497664017694720353894403774873616777216113379904
72799365225472418037761857945604954521607927234567046430722684354562494357888
816796163583180833443020817836277765945425920126835752961691143372812884901888429496729654875873536
91007769624186470425798901761605264998464210599936171228266496304405807104347892350976231928233984687194767361207269217792
106046617616124313601934917632013759414272064210599936020547391979524566087106560695784701952069578470195204123168604160109951162777626559922791424
11362797056106420469761418939596801135151677440605414227968022602131177472602723498065921148044758220801530726344294401360645639372807256776743321617592186044416584318301411328
1221767823366965703475210216365096969081213419520544872805171202324790635397127232681976791041653184451837952275530741972992032655495344947202612439627595776126663739519795228147497671065612855002631049216
1313060694016452770725888724433161420870833464672256472223097815040226667086951219280592742027100162149139787389337642982795747786752636782159226470406792343031749017649398858412720128219550481834311684503599627370496282810057883082752
147836416409629255954595845071032129945654091009386086439666740216463362115559478211379284622379128455168257896774486720512601759140469014528106979402750047027214263920366672936961383168035556163584922112023704109056378302368699121664720575940379279366221821273427820544
Technical details
Node id
n-2a3b89b3afdb558e · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-10cba46fcb583bff

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:        6,        16;
  2:       36,       192,        256;
  3:      216,      1728,       4608,        4096;
  4:     1296,     13824,      55296,       98304,       65536;
  5:     7776,    103680,     552960,     1474560,     1966080,      1048576;
  6:    46656,    746496,    4976640,    17694720,    35389440,     37748736,     16777216;
  7:   279936,   5225472,   41803776,   185794560,   495452160,    792723456,    704643072,    268435456;
  8:  1679616,  35831808,  334430208,  1783627776,  5945425920,  12683575296,  16911433728,  12884901888,   4294967296;
  9: 10077696, 241864704, 2579890176, 16052649984, 64210599936, 171228266496, 304405807104, 347892350976, 231928233984, 68719476736;
%F Formula
The row polynomials satisfy P(n,x) = (6+16*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[(6 + 16*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] = (6 + 16*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.