Real-rooted atlas

A038327 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(13 + 4\,n x\right)\,P_{n}(x) = \left(169 + 104\,x + 16\,x^{2} + 52\,n x + 16\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
113821
216920848425
32197405618722568381
4285617030448672133121280162129
5371293114244010545604326408320061443090277
6482680917822064205639201124864032448004792322867258214137
762748517270301304374263344255906560984256002180505626091521310721086190605
881573072140159050886487231296532285644825590656007559086081356759041363148858982420106594977
9106044993735873261191210842943737610379570073659882135040221103267845291360256797442048690094082621440369715144373
10137858491849848359949840176197835736019276344422401297446259200574868496384171969208320345558220804485611520340787200115343366759508960329
11179216039403712131547282712279958783447203445646565504026504973580800137010324971524918319357952123537063936021381414912024366284800164941004850331648122975627927389
1223298085122481172047397827552436735702177632597245404687360516846984825600305337295650816127876303306752385435639480328338751815680126704680960012865398374478517370882181038082227673259996145
13302875106592253242300085273802467098485152745281009344733921638497052378261696006450250370623488308729932269158410856437178695682818498113699845353272770560072475077509126634717839363685954355293952409640200934368853509
143937376385699289339220119383323361017660358149970081670006741579438081766353284362867201304383963837194247023605959123353628226736664608768854944427822284819485912884838403297616026664964025062155878433542184632321709933854724026531840723034196402133401
Technical details
Node id
n-34368fd5ab01f26d · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-65ac278c0221e407

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:          13,           8;
  2:         169,         208,           48;
  3:        2197,        4056,         1872,          256;
  4:       28561,       70304,        48672,        13312,        1280;
  5:      371293,     1142440,      1054560,       432640,       83200,        6144;
  6:     4826809,    17822064,     20563920,     11248640,     3244800,      479232,      28672;
  7:    62748517,   270301304,    374263344,    255906560,    98425600,    21805056,    2609152,    131072;
  8:   815730721,  4015905088,   6487231296,   5322856448,  2559065600,   755908608,  135675904,  13631488,   589824;
  9: 10604499373, 58732611912, 108429437376, 103795700736, 59882135040, 22110326784, 5291360256, 797442048, 69009408, 2621440;
%F Formula
The row polynomials satisfy (13+4*n*x)*P(n,x) = (169+(52*n+104)*x+(16*n+16)*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[((169 + (52*n + 104)*x + (16*n + 16)*x^2)*P[n - 1, x])/(13 + 4*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] = ((169 + (52*n + 104)*x + (16*n + 16)*x^2)*v[i - 1])/(13 + 4*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.