Real-rooted atlas

A131431 → D → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(10 - 21\,n + 9\,n^{2}\right)\,P_{n}(x) = \left(2 + 8\,x + 8\,x^{2} + 3\,n + 12\,n x + 12\,n x^{2} - 9\,n^{2} - 36\,n^{2} x - 36\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-10 - 20\,x - 24\,n - 48\,n x + 18\,n^{2} + 36\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
2142842
330120120270
4523126244161404
5806401920256012806480
61141140456091209120364827702
7154184892402464036960295689856112266
8200280016800560001120001344008960025600437400
9252403228224112896282240451584451584258048645121653372
103105580446402083206249601249920166656014284807142401587206101730
1137474806732035904012566403015936502656057446404308480191488038297622084326
124449768976805860802344320656409613128192187545601875456012503040500121690931278653268
13520124801372809152004118400131788803075072052715520658944005857280035143680127795202129920276349320
146021565218782413773766886880247927686611404813222809619834214422038016017630412896165888320552964931584959782446
Technical details
Node id
n-2447747166f6803a · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-11aec07ce60b6d61

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   0;
  1:   4;
  2:  14,   28;
  3:  30,  120,   120;
  4:  52,  312,   624,    416;
  5:  80,  640,  1920,   2560,   1280;
  6: 114, 1140,  4560,   9120,   9120,   3648;
  7: 154, 1848,  9240,  24640,  36960,  29568,   9856;
  8: 200, 2800, 16800,  56000, 112000, 134400,  89600,  25600;
  9: 252, 4032, 28224, 112896, 282240, 451584, 451584, 258048, 64512;
%F Formula
The row polynomials satisfy (9*n^2-21*n+10)*P(n,x) = -(9*n^2-3*n-2+(36*n^2-12*n-8)*x+(36*n^2-12*n-8)*x^2)*P(n-2,x) + (18*n^2-24*n-10+(36*n^2-48*n-20)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[(-(9*n^2 - 3*n - 2 + (36*n^2 - 12*n - 8)*x + (36*n^2 - 12*n - 8)*x^2)*P[n - 2, x] + (18*n^2 - 24*n - 10 + (36*n^2 - 48*n - 20)*x)*P[n - 1, x])/(9*n^2 - 21*n + 10)]];
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] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(9*n^2 - 3*n - 2 + (36*n^2 - 12*n - 8)*x + (36*n^2 - 12*n - 8)*x^2)*v[i - 2] + (18*n^2 - 24*n - 10 + (36*n^2 - 48*n - 20)*x)*v[i - 1])/(9*n^2 - 21*n + 10)); 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.