Real-rooted atlas

A027467 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(1 + 15\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
111516
2130225256
314567533754096
41601350135005062565536
51752250337502531257593751048576
619033756750075937545562501139062516777216
71105472511812517718751594687579734375170859375268435456
811206300189000354375042525000318937500136687500025628906254294967296
9113581002835006378750956812509568125006150937500230660156253844335937568719476736
10115010125405000106312501913625002392031250205031250001153300781253844335937505766503906251099511627776
111165123755568751670625035083125052624687505638359375042287695312521143847656256343154296875864975585937517592186044416
1211801485074250025059375601425000105249375001353206250001268630859375845753906250038058925781250103797070312500129746337890625281474976710656
131195175509652503619687597731562519546312500293194687500329844023437527487001953125164922011718750674680957031250168670239257812519461950683593754503599627370496
14121020475122850050675625152026875034206046875586389375000769636054687576963605468750577227041015625314851113281250011806916748046875272467309570312502919292602539062572057594037927936
Technical details
Node id
n-ff73176658f063e0 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-8deca67d09eebe9e

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: 1,  15;
  2: 1,  30,  225;
  3: 1,  45,  675,   3375;
  4: 1,  60, 1350,  13500,   50625;
  5: 1,  75, 2250,  33750,  253125,   759375;
  6: 1,  90, 3375,  67500,  759375,  4556250,  11390625;
  7: 1, 105, 4725, 118125, 1771875, 15946875,  79734375,  170859375;
  8: 1, 120, 6300, 189000, 3543750, 42525000, 318937500, 1366875000,  2562890625;
  9: 1, 135, 8100, 283500, 6378750, 95681250, 956812500, 6150937500, 23066015625, 38443359375;
%F Formula
The row polynomials satisfy P(n,x) = (1+15*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[(1 + 15*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] = (1 + 15*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

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.