Real-rooted atlas

A038327 → x ↦ x+1 → D

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
226228
3507783588
487882028156410976
51428054394050702605192080
622277588568301318201014039063226944
73378766315594306299890530758017745546752706752
8501988136270301304623772247997080615160283927288843308032
973415764894517893224121635586818713167217993430110728842588936913282101504
101060449937307341576489022589466120405451956046782918035986860184548060840117010206610467840
111516443410339116649493103040378670689582828042440111499287901029224196659759102900040836551430113181801204736
12215059247284441819732092406869989695861801615146827580248484127320267598290962058448392113101560435006011154017161248594782035968
133028751065922532795770214697721182825860064423032886820678052492271896356460587310325797962970838228327281837900356283420145002202813737020860878848
144240251492291542424025149229154219570391502884045519854013633961061510387237301469783613097815074703724081159592594166689957274285895610879678818454823661411112006825558016
Technical details
Node id
n-4741af999862dcf7 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-96a9640a0a811b71

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:          0;
  1:          1;
  2:         26,          2;
  3:        507,         78,          3;
  4:       8788,       2028,        156,         4;
  5:     142805,      43940,       5070,       260,        5;
  6:    2227758,     856830,     131820,     10140,      390,       6;
  7:   33787663,   15594306,    2998905,    307580,    17745,     546,     7;
  8:  501988136,  270301304,   62377224,   7997080,   615160,   28392,   728,   8;
  9: 7341576489, 4517893224, 1216355868, 187131672, 17993430, 1107288, 42588, 936, 9;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (13*n+n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((13*n + n*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((13*n + n*x)*v[i - 1])/(n - 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.