Real-rooted atlas

A027467 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(15 + n x\right)\,P_{n}(x) = \left(225 + 30\,x + x^{2} + 15\,n x + n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
115217
2225603288
33375135013544864
450625270004050240581920
5759375506250101250900037561376256
6113906259112500227812527000016875540723068672
7170859375159468750478406257087500590625283507358385875968
8256289062527337500009568125001701000001771875011340004410096096442450944
9384433593754613203125018452812500382725000047840625038272500198450064800121510107374182400
1057665039062576886718750034599023437582012500000119601562501148175000744187503240000911251500111786706395136
1186497558593751268630859375063431542968751691507812500281917968750315748125002455818750133650000501187512375018151229686813949952
121297463378906252075941406250001141767773437503383015625000063431542968758119237500007367456250048114000002255343757425000163350216013492581209243648
131946195068359375337340478515625020240428710937506596880468750001374350097656251979064140625020523628125001563705000008795840625361968750106177502106002535148162774324609024
142919292602539062554493461914062500354207502441406251259404453125000028861352050781254617816328125005387452382812546911150000003078544218751520268750055743187514742000266175294015135107988821114880
Technical details
Node id
n-d2496557bf6f35d4 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-1697c53452ffcf4a

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:          15,           2;
  2:         225,          60,           3;
  3:        3375,        1350,         135,          4;
  4:       50625,       27000,        4050,        240,         5;
  5:      759375,      506250,      101250,       9000,       375,        6;
  6:    11390625,     9112500,     2278125,     270000,     16875,      540,       7;
  7:   170859375,   159468750,    47840625,    7087500,    590625,    28350,     735,     8;
  8:  2562890625,  2733750000,   956812500,  170100000,  17718750,  1134000,   44100,   960,    9;
  9: 38443359375, 46132031250, 18452812500, 3827250000, 478406250, 38272500, 1984500, 64800, 1215, 10;
%F Formula
The row polynomials satisfy (15+n*x)*P(n,x) = (225+(15*n+30)*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[((225 + (15*n + 30)*x + (n + 1)*x^2)*P[n - 1, x])/(15 + 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] = ((225 + (15*n + 30)*x + (n + 1)*x^2)*v[i - 1])/(15 + 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

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.