Real-rooted atlas

A038323 → x ↦ x+1 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(20\,n + 9\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 9\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
199
2360162522
3108009720218722707
428800038880017496026244878004
57200000129600008748000262440029524531827645
61728000003888000003499200001574640003542940031886461107602046
740320000001088640000012247200000734832000024800580004464104403348078337473869223
8921600000002903040000003919104000002939328000001322697600003571283520053569252803443737681241991094248
92073600000000746496000000011757312000000105815808000005952139200000214277011200048212327520061987278240348678440140519959449841
1046080000000000186624000000000335923200000000352719360000000238085568000000107138505600000321415516800006198727824000697356880200348678440101305643137828210
111013760000000000456192000000000092378880000000001108546560000000087298041600000004714094246400000176778534240000045457337376000076709256822000767092568220034519165569941650016096719899
1222118400000000000109486080000000000246343680000000000332563968000000000299307571200000000188563769856000000848536964352000002727440242560000061367405457600009205110818640008284599736776033891544377721317655054696229532
13479232000000000000258785280000000000064049356800000000009607403520000000000972749606400000000070037971660800000003676993512192000000141826892613120000039888813547440000079777627094880000107699796578088008811801538207203304425576827741396329635039877797
14103219200000000000006038323200000000000016303472640000000000026900729856000000000030263321088000000000024513290081280000000014707974048768000000066185883219456000000223377355865664000005584433896641600000100519810139548800012336522153490080092523916151175603202750943694541292839217832783875814
Technical details
Node id
n-8db82a9a849deadd · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-9eb007f6301418ee

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:             9;
  2:           360,           162;
  3:         10800,          9720,           2187;
  4:        288000,        388800,         174960,          26244;
  5:       7200000,      12960000,        8748000,        2624400,        295245;
  6:     172800000,     388800000,      349920000,      157464000,      35429400,       3188646;
  7:    4032000000,   10886400000,    12247200000,     7348320000,    2480058000,     446410440,     33480783;
  8:   92160000000,  290304000000,   391910400000,   293932800000,  132269760000,   35712835200,   5356925280,   344373768;
  9: 2073600000000, 7464960000000, 11757312000000, 10581580800000, 5952139200000, 2142770112000, 482123275200, 61987278240, 3486784401;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (20*n+9*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 9.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 9;
P[n_, x_] := P[n, x] = Expand[Together[((20*n + 9*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] = 9; for(n = 2, N, my(i = n + 1); v[i] = ((20*n + 9*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.