Real-rooted atlas

A038327 → basis A013609 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(13\,n + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
252860
31014312241350
417576811212486427000
5285610175760405604160160506250
6445551634273201054560162240124803849112500
7675753266237722423991240492128056784034944896159468750
8100397627210812052164990177921279532801968512018170889318420482733750000
9146831529781807157289697308469442994106752575789760708664325451264239616460846132031250
10212089987460293663059560180715728960648723129601497053376023031590402362214401557504059904010240768867187500
1130328868206784665979724120323029365516013252486790403567977212806587034854484449164807424102404283136014643202252812686308593750
124301184945688872789283696272559917566894402584234924128079514920742401712629062144263481394176289539993602227230720114216960351436849152207594140625000
13605750213184506111830808587908894626068805153648526189130848016797527006832041347758786048742139260262497864517836894100497920643422208029696409683066881064963373404785156250
148480502984583084169610059691661681565631320230723288317664218143363396833239159360940661512382592192956207668224296855704104963425258124288292757104640180158218247559086081938227222937654493461914062500
Technical details
Node id
n-077b2f078944fa2f · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-dd008bbb0b25b199

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:           2;
  2:          52,           8;
  3:        1014,         312,         24;
  4:       17576,        8112,       1248,         64;
  5:      285610,      175760,      40560,       4160,       160;
  6:     4455516,     3427320,    1054560,     162240,     12480,      384;
  7:    67575326,    62377224,   23991240,    4921280,    567840,    34944,     896;
  8:  1003976272,  1081205216,  499017792,  127953280,  19685120,  1817088,   93184,   2048;
  9: 14683152978, 18071572896, 9730846944, 2994106752, 575789760, 70866432, 5451264, 239616, 4608;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (13*n+2*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((13*n + 2*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] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((13*n + 2*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.