Real-rooted atlas

A038221 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2 + 3\,n x\right)\,P_{n}(x) = \left(12 + 36\,x + 27\,x^{2} + 18\,n x + 27\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
161824
236216243495
32161944437429169450
4129615552524886998432805172125
5777611664052488010497609841503542943037500
6466568398084723920125971201771470012754584372008752396875
727993658786563968092813226976024800580026784626415624365438263752888468750
81679616403107843174474241269789696297606960042855402243749847696183666009638742048914864765625
91007769627209779224488801281142810726432141551680578547930246749725852849589822592209207064063874204890246037500000
1060466176181398528018366600960979552051203214155168006942575162881012458877920991796451840627621192180232452293400383546284114036552734375
1136279705611972302848134688407040808130442240303048915840076368326791681336445718854416364641455360138076662279607670925682200253140547512637657271530865738144531250
1221767823367836416409696975653068864650435379202727440242560078550278985728160373486262528235650836957184248537992103280184102216372800911305971045362711323550217636715839742531063919970703125
13130606940165093670666246876455399424504273395957762363781543552007658652201108481787018846925312306346088044339238771926768111683589993219269600236939552471793610574161845848642863835499917343558612159660617126516601562500
147836416409632912948920324813518779596838508150236774419855764965836807148075387701248187636978927157763676153056532070454280697475356352603118860837292804975730601907665629607653168376192120281090996528282989234214114904343151886824415274413504638671875
Technical details
Node id
n-c3bc7e5d0848af75 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-311883d116e971c2

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:        6,        18;
  2:       36,       216,        243;
  3:      216,      1944,       4374,        2916;
  4:     1296,     15552,      52488,       69984,       32805;
  5:     7776,    116640,     524880,     1049760,      984150,      354294;
  6:    46656,    839808,    4723920,    12597120,    17714700,    12754584,     3720087;
  7:   279936,   5878656,   39680928,   132269760,   248005800,   267846264,   156243654,    38263752;
  8:  1679616,  40310784,  317447424,  1269789696,  2976069600,  4285540224,  3749847696,  1836660096,   387420489;
  9: 10077696, 272097792, 2448880128, 11428107264, 32141551680, 57854793024, 67497258528, 49589822592, 20920706406, 3874204890;
%F Formula
The row polynomials satisfy (2+3*n*x)*P(n,x) = (12+(18*n+36)*x+(27*n+27)*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[((12 + (18*n + 36)*x + (27*n + 27)*x^2)*P[n - 1, x])/(2 + 3*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] = ((12 + (18*n + 36)*x + (27*n + 27)*x^2)*v[i - 1])/(2 + 3*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

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.