Real-rooted atlas

A038327 → x ↦ x+1 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
111314
2126169196
313950721972744
4152101487882856138416
5165169021970142805371293537824
6178253543940428415222775848268097529536
719135497689599963577971533378766362748517105413504
8110447321230321999270207924081351506525019881368157307211475789056
911176084184548359868646782918405451956225894661273415764891060449937320661046784
10113076052636405997810935658361013629890752982204036707882445106044993730137858491849289254654976
11114392953625059425130171537366222998575820707010610134595568965583247465515151644341033917921603940374049565169664
121156111544833401413769529406405644599715164969682546440378670689523329898620609098660462034215059247284442329808512248156693912375296
131169131826283422042111547785409182828042441076764551721049845437927758221705169539427528668814139788510734886302875106592253302875106592253793714773254144
14118215379799708285895617433285861449490742721535291034424496393551632123020774474613799635034084965234638342946821201257461457714240251492291542393737638569928911112006825558016
Technical details
Node id
n-c16a78b9139e0c46 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-96a9640a0a811b71

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,  13;
  2: 1,  26,  169;
  3: 1,  39,  507,   2197;
  4: 1,  52, 1014,   8788,   28561;
  5: 1,  65, 1690,  21970,  142805,   371293;
  6: 1,  78, 2535,  43940,  428415,  2227758,   4826809;
  7: 1,  91, 3549,  76895,  999635,  7797153,  33787663,   62748517;
  8: 1, 104, 4732, 123032, 1999270, 20792408, 135150652,  501988136,  815730721;
  9: 1, 117, 6084, 184548, 3598686, 46782918, 405451956, 2258946612, 7341576489, 10604499373;
%F Formula
The row polynomials satisfy P(n,x) = (1+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[(1 + 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] = (1 + 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

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.