Real-rooted atlas

A038327 → basis A013611 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(4 + 13\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
141317
216104169289
364624202821974913
4256332816224351522856183521
51024166401081603515205712203712931419857
6409679872648960281216068546408911032482680924137569
7163843727363634176196851206397664012475444813515065262748517410338673
8655361703936193822721259847685118131201330714112216241043220079525448157307216975757441
926214476677129968025675590860836850544641197642700825948925184361431457922936630595610604499373118587876497
10104857634078720498401280431947776024567029760958114160642594892518404819086105605873261191204241799749201378584918492015993900449
11419430414994636824366284802375712768015442132992070261705113622835054161925300994716160861411641376093319594482406065773641356179216039403734271896307633
121677721665431142411695816704126704680960926527979520481794549350418268043329536508895492751361033693969651201493113511718401455785673925448602369891377623298085122481582622237229761
1367108864283534950455289315328658864340992535327277056031316645707776135705464733696441042760384512107504172843724819410475652339202523361834804096223661617175817612115004263690123028751065922539904578032905937
142684354561221381324825801680486433542184632322997832751513619485912884838494993825313587235283420830760961003372279874764821739732730619904353270656872573444175016853948595233922011938332336169610059691661683937376385699289168377826559400929
Technical details
Node id
n-ccae868e92935003 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-b7535a41ad3fce2e

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:      4,      13;
  2:     16,     104,      169;
  3:     64,     624,     2028,      2197;
  4:    256,    3328,    16224,     35152,      28561;
  5:   1024,   16640,   108160,    351520,     571220,      371293;
  6:   4096,   79872,   648960,   2812160,    6854640,     8911032,     4826809;
  7:  16384,  372736,  3634176,  19685120,   63976640,   124754448,   135150652,    62748517;
  8:  65536, 1703936, 19382272, 125984768,  511813120,  1330714112,  2162410432,  2007952544,   815730721;
  9: 262144, 7667712, 99680256, 755908608, 3685054464, 11976427008, 25948925184, 36143145792, 29366305956, 10604499373;
%F Formula
The row polynomials satisfy P(n,x) = (4+13*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[(4 + 13*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] = (4 + 13*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.