Real-rooted atlas

A125092 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(8 + 48\,x + 96\,x^{2} + 64\,x^{3}\right)\,P_{n-3}(x) + \left(-12 - 48\,x - 48\,x^{2}\right)\,P_{n-2}(x) + \left(6 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 16\,x\)
\(P_{2}(x) = 18 + 104\,x + 144\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
151621
218104144266
356456120010242736
41601664633610496640025056
5432544026880652807808036864212976
6112016512998403174405606405222402007041718496
72816474883386881326080308224042577923239936104857613343616
8691213107210752004988928143360002614886429589504190054405308416100590336
9166403502083244032173752325935104013418496020093337619228262410675814426214400740710656
10394249113609400320570163202253619206069288961128529920143130624011855462405793382401268776965351256576
119216023203842635776017842176080019456024979046405540806656873725952096023347207007109120305659904060397977638053380096
12212992579993671909376537067520269254656095504302082458543718446298824704633235046406136266752040001077248157538058242835349504267018633216
1348742414270464191692800156506521686687744003441269145610076395929622042116096036032112230443484446720037666527641622181157273679607889920131533373441852441767936
14110592034635776500957184443704934426896629760118082764800387311468544964455038976183264450969626450539315202854978846720223512782438412000071516163955396444166039797760012725469536256
Technical details
Node id
n-431d0fbb7561f8e5 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-1edb041cf004cf36

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:     5,     16;
  2:    18,    104,     144;
  3:    56,    456,    1200,     1024;
  4:   160,   1664,    6336,    10496,     6400;
  5:   432,   5440,   26880,    65280,    78080,     36864;
  6:  1120,  16512,   99840,   317440,   560640,    522240,    200704;
  7:  2816,  47488,  338688,  1326080,  3082240,   4257792,   3239936,   1048576;
  8:  6912, 131072, 1075200,  4988928, 14336000,  26148864,  29589504,  19005440,   5308416;
  9: 16640, 350208, 3244032, 17375232, 59351040, 134184960, 200933376, 192282624, 106758144, 26214400;
%F Formula
The row polynomials satisfy P(n,x) = (8+48*x+96*x^2+64*x^3)*P(n-3,x) - (12+48*x+48*x^2)*P(n-2,x) + (6+12*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 5 + 16*x, P(2,x) = 18 + 104*x + 144*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 16*x;
P[2, x_] := 18 + 104*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(8 + 48*x + 96*x^2 + 64*x^3)*P[n - 3, x] - (12 + 48*x + 48*x^2)*P[n - 2, x] + (6 + 12*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] = 5 + 16*x; v[3] = 18 + 104*x + 144*x^2; for(n = 3, N, my(i = n + 1); v[i] = (8 + 48*x + 96*x^2 + 64*x^3)*v[i - 3] - (12 + 48*x + 48*x^2)*v[i - 2] + (6 + 12*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.