Real-rooted atlas

A131431 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(-1 - 6\,x - 9\,x^{2}\right)\,P_{n-2}(x) + \left(2 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
141216
274263112
31090270270640
413156702140410533328
516240144043206480388816384
61934225651026023085277021385177824
7224624158207906237011226611226648114360448
8256006300378001417503402005103004374001640251638400
9287569072635042857688573041714608220449616533725511247340032
10319301255510044052731018983164745790813564091525956101730183051932505856
1134112216830151470908820381704411451132245381403680721036807210220843266022998142606336
12371332219782197801483515712087224923052640878481201647151602196201441976587865326819663317620756992
134015602808030888023166001250964050038560150115680337760280562933800675520560552698640276349320637729202684354560
144318063521742260434864832091889894135041322748712847215369169443073825416461072772704844207952863395978244620566766711542724608
Technical details
Node id
n-8e47c78f771d4cfd · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-f9762973ec1089d7

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:  4,  12;
  2:  7,  42,   63;
  3: 10,  90,  270,   270;
  4: 13, 156,  702,  1404,   1053;
  5: 16, 240, 1440,  4320,   6480,   3888;
  6: 19, 342, 2565, 10260,  23085,  27702,   13851;
  7: 22, 462, 4158, 20790,  62370, 112266,  112266,   48114;
  8: 25, 600, 6300, 37800, 141750, 340200,  510300,  437400,  164025;
  9: 28, 756, 9072, 63504, 285768, 857304, 1714608, 2204496, 1653372, 551124;
%F Formula
The row polynomials satisfy P(n,x) = -(1+6*x+9*x^2)*P(n-2,x) + (2+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 12*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 12*x;
P[n_, x_] := P[n, x] = Expand[-(1 + 6*x + 9*x^2)*P[n - 2, x] + (2 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = -(1 + 6*x + 9*x^2)*v[i - 2] + (2 + 6*x)*v[i - 1]); 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. [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.