Real-rooted atlas

A038220 → basis A007318 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(2 + 5\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1257
24202549
3860150125343
41616060010006252401
532400200050006250312516807
664960600020000375003750015625117649
71282240168007000017500026250021875078125823543
82565120448002240007000001400000175000012500003906255764801
9512115201152006720002520000630000010500000112500007031250195312540353607
101024256002880001920000840000025200000525000007500000070312500390625009765625282475249
1120485632070400052800002640000092400000231000000412500000515625000429687500214843750488281251977326743
124096122880168960014080000792000003168000009240000001980000000309375000034375000002578125000117187500024414062513841287201
138192266240399360036608000228800000102960000034320000008580000000160875000002234375000022343750000152343750006347656250122070312596889010407
1416384573440931840093184000640640000320320000012012000000343200000007507500000012512500000015640625000014218750000088867187500341796875006103515625678223072849
Technical details
Node id
n-76dd49aa0a9e8b92 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-9629d1e9c04dbfbe

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:   2,     5;
  2:   4,    20,     25;
  3:   8,    60,    150,    125;
  4:  16,   160,    600,   1000,     625;
  5:  32,   400,   2000,   5000,    6250,    3125;
  6:  64,   960,   6000,  20000,   37500,   37500,    15625;
  7: 128,  2240,  16800,  70000,  175000,  262500,   218750,    78125;
  8: 256,  5120,  44800, 224000,  700000, 1400000,  1750000,  1250000,  390625;
  9: 512, 11520, 115200, 672000, 2520000, 6300000, 10500000, 11250000, 7031250, 1953125;
%F Formula
The row polynomials satisfy P(n,x) = (2+5*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2 + 5*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; for(n = 1, N, my(i = n + 1); v[i] = (2 + 5*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

Candidate equalities ?

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.