Real-rooted atlas

A013610 → basis A007318 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(7 + 3\,x\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
17310
249519109
3343657270271297
424017599548112428116804
5168078319394050363155265243235873
61176498823511472229863190208980211417293566269
7823543917645721786030182277276560190109204281648218757749824
85764801942351993107477613570017221793267914370549453173263061806561996411835
94035360795964639343224264906644270835447821419514995012412640496322452501811219311968318234129025
1028247524997175247515907048354911927561283010501888221046423669887110997445231478849400108354915403501559049352479691378
11197732674398022673257797068860390208562125592723531856004801336377384144416764622736747070167697795550565462255255142898581771477171997013271
12138412872019861587127991065596320484135761146676002509673901769013645985925260814428247720832334209033774046578311782239103330365191549051149955454531441153121549794517
139688901040799031109895931414859978013306041462364305551075663853389125955019074088079470085770778624136863272619216244655476657022726862585876188183140002774468969317271832515943233420652717500850
1467822307284999321776927151186902730438606910090797776292870222500519236850402423741119010511314617201491729837524242894567392011690217954944101661099709168901514336393770287433700772630660957054591493833478296979758099356463433
Technical details
Node id
n-61db030885b2d224 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-72277c8e823c26dc

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:        7,         3;
  2:       49,        51,          9;
  3:      343,       657,        270,         27;
  4:     2401,      7599,       5481,       1242,         81;
  5:    16807,     83193,      94050,      36315,       5265,        243;
  6:   117649,    882351,    1472229,     863190,     208980,      21141,       729;
  7:   823543,   9176457,   21786030,   18227727,    6560190,    1092042,     81648,     2187;
  8:  5764801,  94235199,  310747761,  357001722,  179326791,   43705494,   5317326,   306180,    6561;
  9: 40353607, 959646393, 4322426490, 6644270835, 4478214195, 1499501241, 264049632, 24525018, 1121931, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (7+3*x)*P(n-1,x) + 3*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[(7 + 3*x)*P[n - 1, x] + 3*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] = (7 + 3*x)*v[i - 1] + 3*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.