Real-rooted atlas

A013611 → basis A007318 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(9 + 4\,x\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
19413
2818816185
37291468624642885
465612200016480384025649137
559049312244368160146560217601024908797
65314414295368750769645504001130240116736409618135977
747829695796554814481230412558918446457600790630460211216384388132405
843046721772684000269367136032166220801663796736415113216514949123014656655368859509217
93874204891021707888448883149120783237491205446140672018693470208335978496031752192014745600262144214658589165
103486784401134371707448871881850576184033132800016748301644807599562629121856467845122518745088018746572807077888010485765497638817945
113138105960917607793344281535947828958442134485290304492320794240002873805228236891661689405441674522820608177610752000106836787203344957444194304148285580562213
12282429536481230156555860002681542482606409462621042547201399339926761216103033183388467241743578416742498622020124672139801321144321191208550400591250063361560281088167772164198676045305297
13254186582832930033324076392446506848427748802094412118239168038768546586049280354769828897003521789670821306368053187578812825609676734971904001095249132257287661959512064319194923008719742566467108864124443863531240413
142287679245496139144995932443288026297529022865645842928420132480010529901493807987201183906690145507328732499302589902848268380874459709440609496449099366408799315083919360813535668994048475651689676801687032954880328833433602684354563851018433469722505
Technical details
Node id
n-28c47f7742a98cf2 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-65e248b1aaac025a

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): no

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,          88,          16;
  3:       729,        1468,         624,          64;
  4:      6561,       22000,       16480,        3840,         256;
  5:     59049,      312244,      368160,      146560,       21760,        1024;
  6:    531441,     4295368,     7507696,     4550400,     1130240,      116736,       4096;
  7:   4782969,    57965548,   144812304,   125589184,    46457600,     7906304,     602112,     16384;
  8:  43046721,   772684000,  2693671360,  3216622080,  1663796736,   415113216,   51494912,   3014656,    65536;
  9: 387420489, 10217078884, 48883149120, 78323749120, 54461406720, 18693470208, 3359784960, 317521920, 14745600, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (9+4*x)*P(n-1,x) + 4*x*d/dx 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[(9 + 4*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], 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] = (9 + 4*x)*v[i - 1] + 4*x*deriv(v[i - 1], 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.