Real-rooted atlas

A038323 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(20 + 58\,x + 38\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1203858
2400152014443364
38000456008664054872195112
4160000121600034656004389760208513611316496
532000003040000011552000021948800020851360079235168656356768
66400000072960000034656000008779520000125108160009508220160301093638438068692544
7128000000017024000000970368000003072832000005838380800006655754112004215310937601144155825922207984167552
82560000000038912000000025876480000009833062400000233535232000003549735526400033722487500800183064932147204347792138496128063081718016
95120000000008755200000000665395200000002949918720000008407268352000001597380986880000202334925004800016475843893248007826025849292801652161012628487427658739644928
101024000000000019456000000000016634880000000008428339200000000280242278400000006389523947520000010116746250240000010983895928832000078260258492928000330432202525696006278211847988224430804206899405824
1120480000000000042803200000000004066304000000000023177932800000000088076144640000000023428254474240000004451368350105600000604114276085760000057390856228147200003634754227782656000138120660655740928023857205022355251224986644000165537792
12409600000000000093388800000000000975912960000000000618078208000000000026422843392000000000803254439116800000001780547340042240000002899748525211648000003443451373688832000002907803382226124800001657447927868891136005725729205365260288090657379084949954561449225352009601191936
138192000000000000020234240000000000002306703360000000000016070033408000000000076332658688000000000026105769271296000000006613461548728320000000125655769425838080000001790594714318192640000018900721984469811200000143645487081970565120007443447966974838374400235709185620869881856034449804052280982732884055070416556869132288
14163840000000000000043581440000000000000538230784000000000000409055395840000000000021373144432640000000000812179488440320000000002314711542054912000000005026230777033523200000008356108666818232320000001058440431130309427200000100551840957379395584000069472181025098491494400032999285986921783459840096459451346386751651840130909255398667734384644875194084160298409672704
Technical details
Node id
n-aeb498ce26faf029 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-3a176e70253767ae

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:           20,            38;
  2:          400,          1520,           1444;
  3:         8000,         45600,          86640,           54872;
  4:       160000,       1216000,        3465600,         4389760,         2085136;
  5:      3200000,      30400000,      115520000,       219488000,       208513600,         79235168;
  6:     64000000,     729600000,     3465600000,      8779520000,     12510816000,       9508220160,       3010936384;
  7:   1280000000,   17024000000,    97036800000,    307283200000,    583838080000,     665575411200,     421531093760,     114415582592;
  8:  25600000000,  389120000000,  2587648000000,   9833062400000,  23353523200000,   35497355264000,   33722487500800,   18306493214720,   4347792138496;
  9: 512000000000, 8755200000000, 66539520000000, 294991872000000, 840726835200000, 1597380986880000, 2023349250048000, 1647584389324800, 782602584929280, 165216101262848;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (20+58*x+38*x^2)*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[((20 + 58*x + 38*x^2)*P[n - 1, 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] = ((20 + 58*x + 38*x^2)*v[i - 1])/(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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.