Real-rooted atlas

A038208 → D → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + 4\,x - x^{2} + n - 7\,n x + 3\,n x^{2} + 3\,n^{2} x - 3\,n^{2} x^{2} + n^{3} x^{2}\right)\,P_{n}(x) = \left(2\,n - 2\,n x^{2} + 6\,n^{2} x + 6\,n^{2} x^{2} + 2\,n^{3} x^{2} + 2\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
283240
324192216432
464768172810243584
51602560864010240400025600
6384768034560614404800013824165888
789621504120960286720336000193536439041003520
82048573443870721146880179200015482887024641310725767168
9460814745611612164128768806400092897286322176235929637324831850496
10102403686403317760137625603225600046448640421478402359296074649601024000170393600
112252890112091238404325376011827200020437401623181312017301504082114560225280002725888888143872
1249152216268824330240129761280405504000817496064111270297610380902406569164802703360006542131270778884529848320
131064965111808632586243748659201317888000303641395248217128965398069248426995712023429120008504770561840250881799782422682796032
14229376119275521610219521049624576410009600010627448832192868515842519098982423911759872164003840007937785856257635123250393907244957696111803367424
Technical details
Node id
n-6578845aa2c2a369 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d42e6339eab21b9c

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:    2;
  2:    8,     32;
  3:   24,    192,     216;
  4:   64,    768,    1728,    1024;
  5:  160,   2560,    8640,   10240,    4000;
  6:  384,   7680,   34560,   61440,   48000,   13824;
  7:  896,  21504,  120960,  286720,  336000,  193536,   43904;
  8: 2048,  57344,  387072, 1146880, 1792000, 1548288,  702464,  131072;
  9: 4608, 147456, 1161216, 4128768, 8064000, 9289728, 6322176, 2359296, 373248;
%F Formula
The row polynomials satisfy (n-1+(3*n^2-7*n+4)*x+(n^3-3*n^2+3*n-1)*x^2)*P(n,x) = (2*n+6*n^2*x+(2*n^3+6*n^2-2*n)*x^2+2*n^3*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 6*n^2*x + (2*n^3 + 6*n^2 - 2*n)*x^2 + 2*n^3*x^3)*P[n - 1, x])/(n - 1 + (3*n^2 - 7*n + 4)*x + (n^3 - 3*n^2 + 3*n - 1)*x^2)]];
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] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 6*n^2*x + (2*n^3 + 6*n^2 - 2*n)*x^2 + 2*n^3*x^3)*v[i - 1])/(n - 1 + (3*n^2 - 7*n + 4)*x + (n^3 - 3*n^2 + 3*n - 1)*x^2)); 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.