Real-rooted atlas

A003506 → (θ+1)² → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(6 - 11\,n + 5\,n^{2}\right)\,P_{n}(x) = \left(8\,n + 12\,n x - 2\,n x^{2} + 6\,n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}(x) + \left(8\,n + 10\,n x + n x^{2} - n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
210818
3547827159
4224456300641044
5800208019808201255805
625928160100806120183021628998
77840288964368034720153303570343134379
8225289497616934416576096320332646328512589032
96220829491260480070156850400022982465016104407292473497
10166400875520202752027148802318400131040049056011736016290100010038330
1143366425062406462720979968096835206519744303072096096019899024310133139621879
1211059206961152197683203345408037509120292723201623283263993601758240320760349801728152818812
13276889618849792584263681090918401367308801212456967802995236736128125611203043040495924488282197578030661
14682393649946624167731200342358016474073600470486016344408064188348160769728962322320050290247403766643027442150210286
Technical details
Node id
n-9f84bb698ebc24d2 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-2f7d4833c4308026

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:     0;
  1:     1;
  2:    10,      8;
  3:    54,     78,     27;
  4:   224,    456,    300,     64;
  5:   800,   2080,   1980,    820,    125;
  6:  2592,   8160,  10080,   6120,   1830,    216;
  7:  7840,  28896,  43680,  34720,  15330,   3570,   343;
  8: 22528,  94976, 169344, 165760,  96320,  33264,  6328,   512;
  9: 62208, 294912, 604800, 701568, 504000, 229824, 65016, 10440, 729;
%F Formula
The row polynomials satisfy (5*n^2-11*n+6)*P(n,x) = (6*n^2+8*n+(2*n^2+12*n)*x+(n^2-2*n)*x^2)*P(n-1,x) + (8*n+10*n*x+n*x^2-n*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((6*n^2 + 8*n + (2*n^2 + 12*n)*x + (n^2 - 2*n)*x^2)*P[n - 1, x] + (8*n + 10*n*x + n*x^2 - n*x^3)*D[P[n - 1, x], x])/(5*n^2 - 11*n + 6)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((6*n^2 + 8*n + (2*n^2 + 12*n)*x + (n^2 - 2*n)*x^2)*v[i - 1] + (8*n + 10*n*x + n*x^2 - n*x^3)*deriv(v[i - 1], x))/(5*n^2 - 11*n + 6)); 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.