Real-rooted atlas

A038279 → basis A013611 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(9\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
27232104
39728641922028
411664155526912102435152
5131220233280155520460805120571220
61417176314928027993601244160276480245768911032
71488034839680928440899202612736087091201548288114688135150652
81530550084761711366348948484702924802090188805573836882575365242882007952544
915496819565509980288857108044876187381764232632320150493593633443020842467328235929629366305956
101549681956061987278240110199605760114281072640761873817603386105856010032906240191102976021233664010485760424179974920
1115341851364468186006064013637201212801616260884480125709179904067044895948824831442944063063982080105106636801038090240461373446065773641356
1215062908612327364088654912163646414553602181952194048019395130613760120680812707845363591675904170272751616037838389248056056872960498283315220132659286023698913776
13146863358970127832712478406419146630502771228365378522624028365378522624020170935838310410459003768012839843823878144110677288550402186218045440291495739392235552112648724152321211500426369012
141423444863864248224348102326722193159493953792357403769385062439711529931673603176922394533888188262067824230483672030144102427890676714700868865868431360122428210544641483978309632109924319232375809638416961005969166168
Technical details
Node id
n-d6b3763ad78802c2 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-12e1fe8c90bb4450

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:          4;
  2:         72,         32;
  3:        972,        864,        192;
  4:      11664,      15552,       6912,       1024;
  5:     131220,     233280,     155520,      46080,       5120;
  6:    1417176,    3149280,    2799360,    1244160,     276480,      24576;
  7:   14880348,   39680928,   44089920,   26127360,    8709120,    1548288,    114688;
  8:  153055008,  476171136,  634894848,  470292480,  209018880,   55738368,   8257536,   524288;
  9: 1549681956, 5509980288, 8571080448, 7618738176, 4232632320, 1504935936, 334430208, 42467328, 2359296;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (9*n+4*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((9*n + 4*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] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((9*n + 4*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.