Real-rooted atlas

A038327 → basis A013610 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(13\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
133
2781896
31521702812304
42636418252421232449152
5428415395460136890210601215983040
666832747711470355914082134094770437418874368
71013629891403487548097043524913980431203539803415309352321536
8150596440824327117361684185048647763480149483880206977681592136524886442450944
9220247294674066103901632841608436151576654324372403490807212952931399566141096177147115964116992
10318134981190660741884010609915585240328416084360113682490740262344209404036064760399171240230291105904902061584302080
11454933023101710498454379270109022410861656709071437640270943269597075030443888414428931517019027162440164658136584440070194861736283883716608
1264517774185332163775888316612188972178826860130826893033980603816429387601950791541098445018266333047420593351608562223098065863254603039842526377292633318697598976
1390862531977675925161931932279483193629822173934245663832474918012755622070813054709768149223281268014501713962508160552840836175392589053710296675802568666929410777623482072619910977524091715584
14127207544768746263816226343062387852840057057786908447108175104350762579470240986639010714722539482962329683770445629676080870102837613167842902414216881849874890155832460383698076373668377216821866961566189151184349560832
Technical details
Node id
n-24ec570186f7209b · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-a39bea8eb66cb9fc

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:           3;
  2:          78,          18;
  3:        1521,         702,          81;
  4:       26364,       18252,        4212,         324;
  5:      428415,      395460,      136890,       21060,       1215;
  6:     6683274,     7711470,     3559140,      821340,      94770,      4374;
  7:   101362989,   140348754,    80970435,    24913980,    4312035,    398034,    15309;
  8:  1505964408,  2432711736,  1684185048,   647763480,  149483880,  20697768,  1592136,   52488;
  9: 22024729467, 40661039016, 32841608436, 15157665432, 4372403490, 807212952, 93139956, 6141096, 177147;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (13*n+3*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((13*n + 3*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] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((13*n + 3*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.