Real-rooted atlas

A193685 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(5 + 12\,x + 8\,x^{2} + x^{3} - n x - 2\,n x^{2} - n x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
15611
2256137123
31254665752351401
4625317159804973154016289
531252027652044661514168410427193707
615625124781409511708324682081347124730132360459
7781257486863022389668073187506296802916291022152903229522729
8390625441399121358534580208999704737110256063966986788247549833967195379501025
919531252570269614638473847598632397735873113177861211169150296659245759214659755307857475019013291
109765625148356801981412345374865375491994943821520873692717198362448131621572886534829276190308575524712645068341981051
11488281258510783066475516563286540188238244145818616264477052022523744219821948910240514778090367665574680018172804387682050937445958529175033
12244140625486232921142238271648214245850961712347493430164453992917427137455094763234662701126277036645844316656790510686683347107261608722206041758549779713849724164289
131220703125277091315162732888971921576004643071598874472439015941881619951307305803734924358664596615045609592511911348498577937521893170343993469336043266241536239570761155666739742206142740441611
14610351562515770986722117586074821391145566087480049316520007327149647514493190331963326448755549157600501652682917161803449636864465922827439276024770727217650382770118733250272824951505086536504514214284843373159968399589707
Technical details
Node id
n-9b32624866673305 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-ce3aa60852070e39

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:       5,        6;
  2:      25,       61,        37;
  3:     125,      466,       575,       235;
  4:     625,     3171,      5980,      4973,      1540;
  5:    3125,    20276,     52044,     66151,     41684,      10427;
  6:   15625,   124781,    409511,    708324,    682081,     347124,      73013;
  7:   78125,   748686,   3022389,   6680731,   8750629,    6802916,    2910221,    529032;
  8:  390625,  4413991,  21358534,  58020899,  97047371,  102560639,   66986788,  24754983,   3967195;
  9: 1953125, 25702696, 146384738, 475986323, 977358731, 1317786121, 1169150296, 659245759, 214659755, 30785747;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (5-(n-12)*x-(2*n-8)*x^2-(n-1)*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*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[Together[((5 - (n - 12)*x - (2*n - 8)*x^2 - (n - 1)*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], x])/(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] = ((5 - (n - 12)*x - (2*n - 8)*x^2 - (n - 1)*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], x))/(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

Proof chains ? JSON

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.