Real-rooted atlas

A131431 → θ+1 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(10 - 21\,n + 9\,n^{2}\right)\,P_{n}(x) = \left(2 + 8\,x + 8\,x^{2} + 3\,n + 12\,n x + 12\,n x^{2} - 9\,n^{2} - 36\,n^{2} x - 36\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-10 - 20\,x - 24\,n - 48\,n x + 18\,n^{2} + 36\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8 + 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
181624
2218484189
3402404803201080
4655201560208010405265
596960384076807680307223328
613315967980212803192025536851296957
717624641478449280985601182727884822528384912
8225360025200100800252000403200403200230400576001476225
92805040403201881605644801128960150528012902406451201433605511240
1034168206138032736011457602749824458304052377603928320174592034918420135709
114088976897605385602154240603187212063744172339201723392011489280459571283558472275976
12481115441269848465603809520121904642844441648761856609523205417984032507904118210561970176255623121
135601456017472012812806406400230630406150144012300288018450432020500480016400384089456640298188804587520892820880
14645180602347801878240103303204132128012396384028334592049585536066114048066114048048082944024041472073973760105676803085015005
Technical details
Node id
n-2f7b77f9de6690e6 · 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:  21,   84,    84;
  3:  40,  240,   480,    320;
  4:  65,  520,  1560,   2080,   1040;
  5:  96,  960,  3840,   7680,   7680,    3072;
  6: 133, 1596,  7980,  21280,  31920,   25536,    8512;
  7: 176, 2464, 14784,  49280,  98560,  118272,   78848,   22528;
  8: 225, 3600, 25200, 100800, 252000,  403200,  403200,  230400,  57600;
  9: 280, 5040, 40320, 188160, 564480, 1128960, 1505280, 1290240, 645120, 143360;
%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) = 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[(-(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] = 1; v[2] = 8 + 16*x; 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.