Real-rooted atlas

A008278 → x ↦ x+1 (shift +1) → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(-3 - 17\,x - 34\,x^{2} - 29\,x^{3} - 9\,x^{4} + 6\,n + 27\,n x + 45\,n x^{2} + 33\,n x^{3} + 9\,n x^{4}\right)\,P_{n-1}(x) + \left(-4 - 24\,x - 57\,x^{2} - 67\,x^{3} - 39\,x^{4} - 9\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1347
211291959
349193252109603
42571350264722957427291
51539101222652734617224875815101107
61029981439267454466844456704237366511931571299
7759057015712771243606350379355886211190269164549811826948763
86094416449260297843867839819812862710113468467887884437326695495296321504403371
9528445163019696333278490102578401020248010492657920800232003748112983926714227035226098781710212209987
104913492365211830538869187891370067094831624106287499433162915464889785840907526444200463591455806725861754940848222020715699
11487026929712098816547240308718187680850701496128776969916219784275120609986801111316660847237416575325753233117572978436979721999838452615151966620843
1251209054418179745400259784312854526436455752537876883379809166591528335482564283233410628943479989834237741302397841385846289656854415774951721292178502440140329768789126957434503291
1356878092067985551358902786955041435338339968654475127149656681602303099889390109534291758464962705129976594726696688138680582508945643709002267186432952404954516256845522079742235010320871822443083308079675272147
1466492002181912423175335139107611225105992572783420506880209286760381460155529053835180011103249607225325016673440691814944192607538747069321692331816344996011132822944356006531681169821024417425343102437953507937327393613272531507213590804952531149059
Technical details
Node id
n-a815fbeef14d9e19 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-f45f4085bd3a2e4d

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:       3,        4;
  2:      11,       29,        19;
  3:      49,      193,       252,        109;
  4:     257,     1350,      2647,       2295,        742;
  5:    1539,    10122,     26527,      34617,      22487,       5815;
  6:   10299,    81439,    267454,     466844,     456704,     237366,      51193;
  7:   75905,   701571,   2771243,    6063503,    7935588,    6211190,    2691645,     498118;
  8:  609441,  6449260,  29784386,   78398198,  128627101,  134684678,   87884437,   32669549,   5296321;
  9: 5284451, 63019696, 333278490, 1025784010, 2024801049, 2657920800, 2320037481, 1298392671, 422703522, 60987817;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (6*n-3+(27*n-17)*x+(45*n-34)*x^2+(33*n-29)*x^3+(9*n-9)*x^4)*P(n-1,x) - (4+24*x+57*x^2+67*x^3+39*x^4+9*x^5)*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[((6*n - 3 + (27*n - 17)*x + (45*n - 34)*x^2 + (33*n - 29)*x^3 + (9*n - 9)*x^4)*P[n - 1, x] - (4 + 24*x + 57*x^2 + 67*x^3 + 39*x^4 + 9*x^5)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
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*n - 3 + (27*n - 17)*x + (45*n - 34)*x^2 + (33*n - 29)*x^3 + (9*n - 9)*x^4)*v[i - 1] - (4 + 24*x + 57*x^2 + 67*x^3 + 39*x^4 + 9*x^5)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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.