Real-rooted atlas

A038279 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2 + x + n\right)\,P_{n}(x) = \left(27 + 18\,x + 3\,x^{2} + 9\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
112315
2135729216
314581215324273024
41530917496729012968141472
5157464229635131220364504860243559872
6159432328343522066715787320164025174967297464960
715943230334807832976069614467005413343068890561236218798537472
81578379773826375204017693962380855688680203019840464275562020995265611289945088
91549681956426162537951656065203615924564160707758446873096289282088106288207085881968316769286144
1015109399071464904586806392438068551656065200271194342309642465504234365481038263752039858075236196059049216710774784
111464449448424986101693437670925682207031681875354261625379001789882659185303356027211048658390157837977014614627577944681771472786281390080
1214121476824055272018014312897498304817492051108186406328513687815306837027288010739295955082727440242564971896275563135190805261265902550916853144135664401793024
13135566177510885507375961379510280435127908411667477962626289749830481740493624067649571994440677372059833220321161329627118248215448838605246227244121865357910829047961594323454721122861056
1412963515724477956937794554569611565489518896951439260917907176122508518607575175389857604661634553684735469911966644064232031412440668609620492655182490488512214194014038664652875268526784626447829695777633090469888
Technical details
Node id
n-a85f19fae77a4cd8 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-966922252c303ef1

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:         12,          3;
  2:        135,         72,          9;
  3:       1458,       1215,        324,         27;
  4:      15309,      17496,       7290,       1296,         81;
  5:     157464,     229635,     131220,      36450,       4860,       243;
  6:    1594323,    2834352,    2066715,     787320,     164025,     17496,      729;
  7:   15943230,   33480783,   29760696,   14467005,    4133430,    688905,    61236,     2187;
  8:  157837977,  382637520,  401769396,  238085568,   86802030,  19840464,  2755620,   209952,   6561;
  9: 1549681956, 4261625379, 5165606520, 3615924564, 1607077584, 468730962, 89282088, 10628820, 708588, 19683;
%F Formula
The row polynomials satisfy (n+2+x)*P(n,x) = (9*n+27+(3*n+18)*x+3*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[((9*n + 27 + (3*n + 18)*x + 3*x^2)*P[n - 1, x])/(n + 2 + 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*n + 27 + (3*n + 18)*x + 3*x^2)*v[i - 1])/(n + 2 + 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.