Real-rooted atlas

A038271 → x ↦ x/(x+1) → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-5 + 12\,x + 12\,n\right)\,P_{n}(x) = \left(49 + 168\,x + 144\,x^{2} + 84\,n + 144\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1191231
2217456144817
3210778128208172819855
41886510113618748813132820736459553
5160867113190030340803749760196992024883210295359
6132775311582424407484007281792067495680283668482985984225325009
710706059111531252486461808114095520015291763201133927424397135872358318084845725743
88482492910277816645353500096155667778562738292480029360185344181428387845446434816429981696102795249985
96629521159161092332555002098561927260034564203030021125914711756805284833361922799180840967352687001651597803522156912506207
10512490808979554253800549665539920222000839424057817801036801008727205068811829423513600905971433472041987712614409803582668806191736422444853589991281
1139264059611676487867748525058075080024185283756480732602770099201526389947371522219199851151362230691291136001494852865228806158197850112012940729122816743008370688925791004927951
1229857633819356540245839844870712647785625202787603840087067021523328021098959778856963663335873691648456521112236851240152443240448002391764584366080886780490416128169405908516864891610044825618987912200409697
1322561298137634657790875810844101391755075225327705768485129829087165497600271649107152783365485729542502809681640056613699584890216168861859846959756828344320037311527516110848125761596822650882202276810719232106993205379072387330110836578175
141695557682122537902980871218439125443356810722469677938284211210637636422763750433025732876071936076061750002779340813165750902006743041714441188887691264166173684854213836811692391471618457605698487838824202241760662355517112322846019263083315212839184645488647863907907442070033
Technical details
Node id
n-aa4efe654d0b4b2b · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-29ed0a4e339957b1

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:        19,         12;
  2:       217,        456,         144;
  3:      2107,       7812,        8208,         1728;
  4:     18865,     101136,      187488,       131328,        20736;
  5:    160867,    1131900,     3034080,      3749760,      1969920,       248832;
  6:   1327753,   11582424,    40748400,     72817920,     67495680,     28366848,      2985984;
  7:  10706059,  111531252,   486461808,   1140955200,   1529176320,   1133927424,    397135872,     35831808;
  8:  84824929, 1027781664,  5353500096,  15566777856,  27382924800,  29360185344,  18142838784,   5446434816,   429981696;
  9: 662952115, 9161092332, 55500209856, 192726003456, 420303002112, 591471175680, 528483336192, 279918084096, 73526870016, 5159780352;
%F Formula
The row polynomials satisfy (12*n-5+12*x)*P(n,x) = (84*n+49+(144*n+168)*x+144*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[((84*n + 49 + (144*n + 168)*x + 144*x^2)*P[n - 1, x])/(12*n - 5 + 12*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] = ((84*n + 49 + (144*n + 168)*x + 144*x^2)*v[i - 1])/(12*n - 5 + 12*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

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.