Real-rooted atlas

A125092 → D+1 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(-16 - 32\,x - 24\,x^{2} - 8\,x^{3} - x^{4}\right)\,P_{n-4}(x) + \left(32 + 48\,x + 24\,x^{2} + 4\,x^{3}\right)\,P_{n-3}(x) + \left(-24 - 24\,x - 6\,x^{2}\right)\,P_{n-2}(x) + \left(8 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 9 + 4\,x\)
\(P_{2}(x) = 44 + 44\,x + 9\,x^{2}\)
\(P_{3}(x) = 170 + 264\,x + 123\,x^{2} + 16\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
19413
24444997
317026412316573
45761208888264252961
517924720474022404853614013
6524816608211201372047408044962289
714688542088332868880328308904123964264141
83968016716830105630195218368068880153441808811079649
9104192493056101721611988488870404253761312082476825291004284333
102672641402880326016044121603843840224582488704023208037980342012116592769
116722563874816100108801529088015322560105557765085696170544038709055880449914462965917
121662976104386562966937650462720571507204534041625783296105557763072960615560794645784169234837873
1340550402752921685343232159903744201893120181209600118774656574379522042726452509609409401098247293196862883037
14976486471278592239296512489588736681640960682665984506618112282577152118774656373172808584576139120814814890442253129656049
Technical details
Node id
n-b7bae5ed687a8eab · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-abecc458534a7b8a

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:      9,      4;
  2:     44,     44,       9;
  3:    170,    264,     123,      16;
  4:    576,   1208,     888,     264,     25;
  5:   1792,   4720,    4740,    2240,    485,     36;
  6:   5248,  16608,   21120,   13720,   4740,    804,     49;
  7:  14688,  54208,   83328,   68880,  32830,   8904,   1239,    64;
  8:  39680, 167168,  301056,  301952, 183680,  68880,  15344,  1808,   81;
  9: 104192, 493056, 1017216, 1198848, 887040, 425376, 131208, 24768, 2529, 100;
%F Formula
The row polynomials satisfy P(n,x) = -(16+32*x+24*x^2+8*x^3+x^4)*P(n-4,x) + (32+48*x+24*x^2+4*x^3)*P(n-3,x) - (24+24*x+6*x^2)*P(n-2,x) + (8+4*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 9 + 4*x, P(2,x) = 44 + 44*x + 9*x^2, P(3,x) = 170 + 264*x + 123*x^2 + 16*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 9 + 4*x;
P[2, x_] := 44 + 44*x + 9*x^2;
P[3, x_] := 170 + 264*x + 123*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[-(16 + 32*x + 24*x^2 + 8*x^3 + x^4)*P[n - 4, x] + (32 + 48*x + 24*x^2 + 4*x^3)*P[n - 3, x] - (24 + 24*x + 6*x^2)*P[n - 2, x] + (8 + 4*x)*P[n - 1, x]];
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] = 9 + 4*x; v[3] = 44 + 44*x + 9*x^2; v[4] = 170 + 264*x + 123*x^2 + 16*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(16 + 32*x + 24*x^2 + 8*x^3 + x^4)*v[i - 4] + (32 + 48*x + 24*x^2 + 4*x^3)*v[i - 3] - (24 + 24*x + 6*x^2)*v[i - 2] + (8 + 4*x)*v[i - 1]); 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

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.