Real-rooted atlas

A027466 → basis A013609 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(10 + 2\,x\right)\,P_{n-1}(x) + \left(2\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
110212
2100444148
3100072814481880
4100001073634724161624640
51000001488327008013600112032333664
6100000019859841278784357760473602880644672832
710000000258318082187494482817281568000152320716812867716096
8100000000329981696357912832176257536447874566182400462336174082561015601920
9100000000041597803525670743040353594624011586892802132229122253619213424644147251215762302464
1010000000000519173642248770996326467916625920279282995206581836800922242048772915203763200972801024253057484800
11100000000000643008370688133177421414412620859412486385426432001874933350403345299865636994805762524262401025024022528020484200319887360
1210000000000007916100448256199308557393922285692348825614017939460096502695198720011109526405121556935311361396204339279185920027258880516096409672030198976512
13100000000000009699320537907229486418124800040557248729088029803675725824012857491866419234494862065664595855002828867440019046450096144384240148480071032832117145681921275221931966464
14100000000000000118391846454886443220849482301447078888159150080617580660523008031675718878003201016036802772992211994924810240294515050086402751492423680172236832768707602022418189516826378241638423288684287377408
Technical details
Node id
n-05ff6d2347b5ee68 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-73b18cc810b5ad72

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          1;
  1:         10,          2;
  2:        100,         44,          4;
  3:       1000,        728,        144,          8;
  4:      10000,      10736,       3472,        416,         16;
  5:     100000,     148832,      70080,      13600,       1120,        32;
  6:    1000000,    1985984,    1278784,     357760,      47360,      2880,       64;
  7:   10000000,   25831808,   21874944,    8281728,    1568000,    152320,     7168,     128;
  8:  100000000,  329981696,  357912832,  176257536,   44787456,   6182400,   462336,   17408,   256;
  9: 1000000000, 4159780352, 5670743040, 3535946240, 1158689280, 213222912, 22536192, 1342464, 41472, 512;
%F Formula
The row polynomials satisfy P(n,x) = (10+2*x)*P(n-1,x) + 2*x*d/dx 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[(10 + 2*x)*P[n - 1, x] + 2*x*D[P[n - 1, x], 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] = (10 + 2*x)*v[i - 1] + 2*x*deriv(v[i - 1], 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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.