Real-rooted atlas

image under θ+1 → (θ+1)² → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(n^{2} - 2\,n^{3} + n^{4}\right)\,P_{n}(x) = \left(-n^{2} - 4\,n^{2} x - 4\,n^{2} x^{2} - 2\,n^{3} - 8\,n^{3} x - 8\,n^{3} x^{2} - n^{4} - 4\,n^{4} x - 4\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(2 + 4\,x - 4\,n^{2} - 8\,n^{2} x + 2\,n^{4} + 4\,n^{4} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8 + 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
181624
227108108243
3643847685121728
4125100030004000200010125
5216216086401728017280691252488
634341162058054880823206585621952250047
7512716843008143360286720344064229376655361119744
87291166481648326592816480130636813063687464961866244782969
91000180001440006720002016000403200053760004608000230400051200019683000
1013312662023958012777604472160107331841788864020444160153331206814720136294478594219
11172838016380160228096091238402554675251093504729907207299072048660480194641923538944306110016
12219752728580008386672017400240556807681299217922227230722784038402474700801484820485399347289989121167575877
132744713448561286278272313913601130088963013570566027141129040711681004523520803618816438337536146112512224788484374822312
1433759450012285009828000540540002162160006486480001482624000259459200034594560003459456000251596800012579840003870720005529600016142520375
Technical details
Node id
n-b0b9f0e48753af20 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-11aec07ce60b6d61

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:    8,    16;
  2:   27,   108,    108;
  3:   64,   384,    768,    512;
  4:  125,  1000,   3000,   4000,    2000;
  5:  216,  2160,   8640,  17280,   17280,    6912;
  6:  343,  4116,  20580,  54880,   82320,   65856,   21952;
  7:  512,  7168,  43008, 143360,  286720,  344064,  229376,   65536;
  8:  729, 11664,  81648, 326592,  816480, 1306368, 1306368,  746496,  186624;
  9: 1000, 18000, 144000, 672000, 2016000, 4032000, 5376000, 4608000, 2304000, 512000;
%F Formula
The row polynomials satisfy (n^4-2*n^3+n^2)*P(n,x) = -(n^4+2*n^3+n^2+(4*n^4+8*n^3+4*n^2)*x+(4*n^4+8*n^3+4*n^2)*x^2)*P(n-2,x) + (2*n^4-4*n^2+2+(4*n^4-8*n^2+4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 8 + 16*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 8 + 16*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^4 + 2*n^3 + n^2 + (4*n^4 + 8*n^3 + 4*n^2)*x + (4*n^4 + 8*n^3 + 4*n^2)*x^2)*P[n - 2, x] + (2*n^4 - 4*n^2 + 2 + (4*n^4 - 8*n^2 + 4)*x)*P[n - 1, x])/(n^4 - 2*n^3 + n^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 8 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^4 + 2*n^3 + n^2 + (4*n^4 + 8*n^3 + 4*n^2)*x + (4*n^4 + 8*n^3 + 4*n^2)*x^2)*v[i - 2] + (2*n^4 - 4*n^2 + 2 + (4*n^4 - 8*n^2 + 4)*x)*v[i - 1])/(n^4 - 2*n^3 + n^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.