Real-rooted atlas

A135065 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4\,n^{2} - 2\,n^{2} x + 6\,n^{3} x + n^{4} x^{2}\right)\,P_{n}(x) = \left(8 + 12\,x + 6\,x^{2} + x^{3} + 16\,n + 36\,n x + 22\,n x^{2} + 4\,n x^{3} + 8\,n^{2} + 36\,n^{2} x + 28\,n^{2} x^{2} + 6\,n^{2} x^{3} + 12\,n^{3} x + 14\,n^{3} x^{2} + 4\,n^{3} x^{3} + 2\,n^{4} x^{2} + n^{4} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
181624
23614481261
31287688642562016
440032005400320062512825
511521152025920230409000129671928
631363763210584012544073500211682401369117
781921146883870725734404480001935364390440961772928
82073633177613063682322432226800013063684445288294465618089713
9512009216004147200860160010080000725760032928009216001458001000035429400
10123904247808012545280297369604065600035126784199214407434240176418024200014641150043509
11294912648806436495360973209601520640001532805121043159044866048015396480316800038332820736617888736
1269222416613376102795264304578560535392000616771584489705216274120704108416880297440005398536584064285612484840969
131605632417464322817884169184215041793792000232475443221094993921377632256653837184224224000542622088805888861224384169791268984
1436864001032192007547904002683699200576576000083026944008475667200632586240035026992001441440000436035600943488001384110012348005062537905029325
Technical details
Node id
n-e7912c38ca6c646d · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e1337d43fda52a72

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     1;
  1:     8,     16;
  2:    36,    144,      81;
  3:   128,    768,     864,     256;
  4:   400,   3200,    5400,    3200,      625;
  5:  1152,  11520,   25920,   23040,     9000,    1296;
  6:  3136,  37632,  105840,  125440,    73500,   21168,    2401;
  7:  8192, 114688,  387072,  573440,   448000,  193536,   43904,   4096;
  8: 20736, 331776, 1306368, 2322432,  2268000, 1306368,  444528,  82944,   6561;
  9: 51200, 921600, 4147200, 8601600, 10080000, 7257600, 3292800, 921600, 145800, 10000;
%F Formula
The row polynomials satisfy (4*n^2+(6*n^3-2*n^2)*x+n^4*x^2)*P(n,x) = (8*n^2+16*n+8+(12*n^3+36*n^2+36*n+12)*x+(2*n^4+14*n^3+28*n^2+22*n+6)*x^2+(n^4+4*n^3+6*n^2+4*n+1)*x^3)*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[((8*n^2 + 16*n + 8 + (12*n^3 + 36*n^2 + 36*n + 12)*x + (2*n^4 + 14*n^3 + 28*n^2 + 22*n + 6)*x^2 + (n^4 + 4*n^3 + 6*n^2 + 4*n + 1)*x^3)*P[n - 1, x])/(4*n^2 + (6*n^3 - 2*n^2)*x + n^4*x^2)]];
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] = ((8*n^2 + 16*n + 8 + (12*n^3 + 36*n^2 + 36*n + 12)*x + (2*n^4 + 14*n^3 + 28*n^2 + 22*n + 6)*x^2 + (n^4 + 4*n^3 + 6*n^2 + 4*n + 1)*x^3)*v[i - 1])/(4*n^2 + (6*n^3 - 2*n^2)*x + n^4*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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.