Real-rooted atlas

A038255 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(5 + 2\,x + 2\,n\right)\,P_{n}(x) = \left(49 + 28\,x + 4\,x^{2} + 14\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
19211
277364117
363746210881215
45145509618482881612393
5408175145025480616072032124659
6319333489804308700101920184801728641240029
7247062944706623428628144060035672051744403212812223143
818941489395300643576529618286016576240011415041379849216256119574225
914412002534094680235577057621459177682287072207446403424512354816207365121162261467
101089547389288240050034094680202371803840107295888032914828869148800978432088704046080102411235194181
11819178222123970042558317064055002500276548013044921120472101907212068770562173248002690688021683201013762048108090316431
12612971290331966027733042876405106962536512440001500165928806261562137618884076288413786419265197440071751680520396822118440961035574967097
134567624776331593725354858255583605295224928844260321648733086000780086282976271334359296701408547841344805862418834816001865543681230028847923281929885033776835
143391115364245127893493737242231215496801223854469827552174501909822249232905281600364040265388810853374371842454929917444183840460852737484804748656642870067210321921638494049035648173
Technical details
Node id
n-cd59b83bc19527ec · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-7c3b314392150c36

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:         9,         2;
  2:        77,        36,         4;
  3:       637,       462,       108,         8;
  4:      5145,      5096,      1848,       288,       16;
  5:     40817,     51450,     25480,      6160,      720,       32;
  6:    319333,    489804,    308700,    101920,    18480,     1728,      64;
  7:   2470629,   4470662,   3428628,   1440600,   356720,    51744,    4032,    128;
  8:  18941489,  39530064,  35765296,  18286016,  5762400,  1141504,  137984,   9216,   256;
  9: 144120025, 340946802, 355770576, 214591776, 82287072, 20744640, 3424512, 354816, 20736, 512;
%F Formula
The row polynomials satisfy (2*n+5+2*x)*P(n,x) = (14*n+49+(4*n+28)*x+4*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[((14*n + 49 + (4*n + 28)*x + 4*x^2)*P[n - 1, x])/(2*n + 5 + 2*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] = ((14*n + 49 + (4*n + 28)*x + 4*x^2)*v[i - 1])/(2*n + 5 + 2*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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.