Real-rooted atlas

A013613 → basis A013611 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(31 + 24\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1312455
296114885763025
329791691925356813824166375
49235212859936332121617141763317769150625
528629151110822520171596160132848640514252807962624503284375
68875036814122597744797922144082366156804782551040148104806419110297627680640625
7275126141111491006184083462982104964468364006403459378585601606937149444146934579245864714241522435234375

All 15 rows

Technical details
Node id
n-43b97c52965f4781 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-38834d2e4c7d1610

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:             31,              24;
  2:            961,            1488,             576;
  3:          29791,           69192,           53568,            13824;
  4:         923521,         2859936,         3321216,          1714176,           331776;
  5:       28629151,       110822520,       171596160,        132848640,         51425280,         7962624;
  6:      887503681,      4122597744,      7979221440,       8236615680,       4782551040,      1481048064,       191102976;
  7:    27512614111,    149100618408,    346298210496,     446836400640,     345937858560,    160693714944,     41469345792,      4586471424;
  8:   852891037441,   5282421909312,  14313659367168,   22163085471744,   21448147230720,  13284013768704,   5142198878208,   1137444913152,   110075314176;
  9: 26439622160671, 184224464087256, 570501566205696, 1030583474436096, 1196806615474176, 926559960367104, 478224495673344, 158673565384704, 30711012655104, 2641807540224;
%F Formula
The row polynomials satisfy P(n,x) = (31+24*x)*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[(31 + 24*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; for(n = 1, N, my(i = n + 1); v[i] = (31 + 24*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

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.