Real-rooted atlas

A027467 → x ↦ x+1 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(15 + x + n\right)\,P_{n}(x) = \left(256 + 32\,x + x^{2} + 16\,n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
117118
2288341323
348648645115780
481920194561728681103173
513762564096004864028808511837462
62306867282575361228800972804320102132656711
738587596816148070428901376286720017024060481191579301656
8644245094430870077446459228167707033657344002723848064136110258466825
91073741824005798205849613891534848193776844817340825610321920408576103681531181369693466
1017867063951361073741824000289910292480463051161604844421120346816512172032005836801296017013201872665419
11296868139499521965377034649659055800320001063004405760127339069440106577264646358302722703360080256015840187156447829212572
12492581209243648356241767399424117922622078976236223201280003189013217280305613766656213154529281089994752405504001070080190082041993884992921357
1381627743246090246403555720167424231557148809625651099802900889676772540416000829143436492866216316108839585841152177124147258572800139110422464221117478667116892830
1413510798882111488011427884054452633644824890041171968108060002777825281788493101531136214963113164800193466801848321324326322176692752220162755264512820019201770496262082381307041919020084047
Technical details
Node id
n-7a54632a32fd58d6 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-2fb4b36f3b7046e1

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:           17,           1;
  2:          288,          34,           1;
  3:         4864,         864,          51,          1;
  4:        81920,       19456,        1728,         68,         1;
  5:      1376256,      409600,       48640,       2880,        85,        1;
  6:     23068672,     8257536,     1228800,      97280,      4320,      102,      1;
  7:    385875968,   161480704,    28901376,    2867200,    170240,     6048,    119,     1;
  8:   6442450944,  3087007744,   645922816,   77070336,   5734400,   272384,   8064,   136,   1;
  9: 107374182400, 57982058496, 13891534848, 1937768448, 173408256, 10321920, 408576, 10368, 153, 1;
%F Formula
The row polynomials satisfy (n+15+x)*P(n,x) = (16*n+256+(n+32)*x+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[((16*n + 256 + (n + 32)*x + x^2)*P[n - 1, x])/(n + 15 + 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] = ((16*n + 256 + (n + 32)*x + x^2)*v[i - 1])/(n + 15 + 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.