Real-rooted atlas

A038277 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(7\,n + 11\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 11\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
11111
2154242396
316175082399310692
4150927114811180458564256608
5132055830060195657020497408052555773680
61109262871563027391980430445403382071010629366124711488
79058973854131743355517557030608208286073955208389341364101972618941248
8724717847971896243758179656984285148015467338040145834901527638971032171487104853875362816
9570715299717470661639460886388124019928648243610574130306253293192240627587508108036876024212215292191090976097024
1044388967706277868289039460886388014468991675603410548037820535943263086056146437085203781290660840148550704533025937424601021819521940480
113417950512953710650917037981103148451591589084316043768699818690825352625152441080818913890109705312696156057192021245205199718169427703138428376721432026534421504
122610071300764511694677028354490296052201671168538531805252243978242801155493675213416181577577533536820381156661927601601376594865740838816311596340263627983644564376611405206528483430130458624
13197930406974337324019600868322586169407502168973707784882059744275252511851502141777777440827539265925919748370937051247082323643131753319558525444094785088980119950732558276623427163787379332448795257871103165426887543943168
1414920907602678304812826740422287394950926683616559423362918436650548774971795701840123677777364983855497229628764726058638503702344567140538236506334666233803222346800103918390596903702921679310255815872684398193527136809453164976701653743206736589313359872
Technical details
Node id
n-f8cb36e864fb840a · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-428cb055b0457fc7

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:         0;
  1:        11;
  2:       154,        242;
  3:      1617,       5082,        3993;
  4:     15092,      71148,      111804,        58564;
  5:    132055,     830060,     1956570,      2049740,       805255;
  6:   1109262,    8715630,    27391980,     43044540,     33820710,     10629366;
  7:   9058973,   85413174,   335551755,    703060820,    828607395,    520838934,    136410197;
  8:  72471784,  797189624,  3758179656,   9842851480,  15467338040,  14583490152,   7638971032,   1714871048;
  9: 570715299, 7174706616, 39460886388, 124019928648, 243610574130, 306253293192, 240627587508, 108036876024, 21221529219;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (7*n+11*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 11.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 11;
P[n_, x_] := P[n, x] = Expand[Together[((7*n + 11*n*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 11; for(n = 2, N, my(i = n + 1); v[i] = ((7*n + 11*n*x)*v[i - 1])/(n - 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.