Real-rooted atlas

A038279 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(9 + 2\,n x\right)\,P_{n}(x) = \left(81 + 36\,x + 4\,x^{2} + 18\,n x + 4\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
19413
2817212165
3729972324322057
4656111664583211528025289
55904913122087480259203600192307461
65314411417176118098046656097200103684483704173
7478296914880348148803487348320204120032659228224102444289025
8430467211530550081785641761058158083674160078382081016064737282304526153617
938742048915496819562066242608142851340859521392015872371227433728298598418662451206216407549
103486784401154968195602324522934018366600960892820880028570268166172588808957952083980804608001126473096378421
11313810596091534185136442556975227402272866868801262703816004714094246412221725824221709312027713664022809600111513624576855935011833
122824295364811506290861232276153324559227274402425601704650151600727317398016219991064832478892113927482689280821145600602173442654208532489985908471385
132541865828329146863358970122937267179402431911050837952221604519708001063701694598436769935121929338396221441750949291522401850880023484764161552711686230016114688116121849938677
142287679245496114234448638642430841305383725236552658232563227922169483208014891823724377657912647817024168091131985923676993512192605266421760739770071046521389056392491008144506882457601346385773613309
Technical details
Node id
n-f57da79f6d2b9f7d · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-3042267dbb79a78c

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:         9,          4;
  2:        81,         72,         12;
  3:       729,        972,        324,         32;
  4:      6561,      11664,       5832,       1152,        80;
  5:     59049,     131220,      87480,      25920,      3600,       192;
  6:    531441,    1417176,    1180980,     466560,     97200,     10368,      448;
  7:   4782969,   14880348,   14880348,    7348320,   2041200,    326592,    28224,    1024;
  8:  43046721,  153055008,  178564176,  105815808,  36741600,   7838208,  1016064,   73728,   2304;
  9: 387420489, 1549681956, 2066242608, 1428513408, 595213920, 158723712, 27433728, 2985984, 186624, 5120;
%F Formula
The row polynomials satisfy (9+2*n*x)*P(n,x) = (81+(18*n+36)*x+(4*n+4)*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[((81 + (18*n + 36)*x + (4*n + 4)*x^2)*P[n - 1, x])/(9 + 2*n*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] = ((81 + (18*n + 36)*x + (4*n + 4)*x^2)*v[i - 1])/(9 + 2*n*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.