Real-rooted atlas

A038255 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(6 + n x\right)\,P_{n}(x) = \left(36 + 12\,x + x^{2} + 6\,n x + n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1628
23624363
3216216544490
4129617286489653773
577761296064801440150628812
64665693312583201728027002167218491
727993665318448988818144037800453629481647086
81679616447897639191041741824453600725767056384912353145
9100776963023308830233088156764164898880979776127008103684861092236816
1060466176201553920226748160134369280489888001175731219051202073601458060011686011319
1136279705613302558721662819840110854656046189440012933043225147584342144032076019800726125084554482
122176782336870712934411972302848886837248041570496001330255872301771008492687365773680475200261368641337569208117
13130606940165659634073684894511104691733053443602776320012969994752336259123264049356890069408926640067953633696101414276825744020
1478364164096365699432448594261577728528232513536302633210880121053284352353072079367685922816126097171215567552014270256943488425881176152034669218547
Technical details
Node id
n-c2ec0a851ffaa528 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-7fe168c40dd27d29

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,        2;
  2:       36,       24,        3;
  3:      216,      216,       54,        4;
  4:     1296,     1728,      648,       96,       5;
  5:     7776,    12960,     6480,     1440,     150,      6;
  6:    46656,    93312,    58320,    17280,    2700,    216,      7;
  7:   279936,   653184,   489888,   181440,   37800,   4536,    294,     8;
  8:  1679616,  4478976,  3919104,  1741824,  453600,  72576,   7056,   384,   9;
  9: 10077696, 30233088, 30233088, 15676416, 4898880, 979776, 127008, 10368, 486, 10;
%F Formula
The row polynomials satisfy (6+n*x)*P(n,x) = (36+(6*n+12)*x+(n+1)*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[((36 + (6*n + 12)*x + (n + 1)*x^2)*P[n - 1, x])/(6 + 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] = ((36 + (6*n + 12)*x + (n + 1)*x^2)*v[i - 1])/(6 + 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.