Real-rooted atlas

A038221 → basis A013610 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(6\,n + 9\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 9\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
199
2108162270
3972291621876075
47776349925248826244121500
5583203499207873207873202952452278125
6419904314928094478401417176010628820318864641006250
72939328264539529920232019840464022320522013392313233480783717609375
820155392211631616952342272238085568035712835203214155168160707758434437376812301875000
91360488961632586752857108044825713241344482123275205785479302443391094768185961834723486784401207594140625
109069926401224440064073466403840257132413440578547930240867821895360867821895360557885504160209207064060348678440103459902343750
11598615142489792271360606097831680242439132672063640272326401145524901875214319061273440122734810915206903833113980230127770466034519165569957088388671875
12391820820486465043537924848782653440218195219404806545856582144013746298822502420619448233753622092265964736016569199473552082845997367760248537992103283389154437772934173632812500
13254683533312458430359961637820504696832189102523484160638221016759040153173044022169626805282703879683446393490498816323099389734264021539959315617609692981692027922643540461462163304425576827715180321533203125
1416456474460163209012519731228881112677580815884611972669445956729489751040160831696223278083216633924465561648249508866983424542806974753563524523391456279696027140348737678176111028699381410722775717484535268320275094369454245220578613281250
Technical details
Node id
n-046fa60d0b9012ed · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-f06e51c139cfae05

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:       108,        162;
  3:       972,       2916,       2187;
  4:      7776,      34992,      52488,       26244;
  5:     58320,     349920,     787320,      787320,      295245;
  6:    419904,    3149280,    9447840,    14171760,    10628820,     3188646;
  7:   2939328,   26453952,   99202320,   198404640,   223205220,   133923132,    33480783;
  8:  20155392,  211631616,  952342272,  2380855680,  3571283520,  3214155168,  1607077584,   344373768;
  9: 136048896, 1632586752, 8571080448, 25713241344, 48212327520, 57854793024, 43391094768, 18596183472, 3486784401;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (6*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[((6*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] = ((6*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.