Real-rooted atlas

A027467 → basis A013610 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(16\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
133
29618114
32304864813249
44915227648518432482308
59830407372802073602592012151954815
6188743681769472066355201244160116640437444569782
735232153639636172818579456046448640653184048988815309987963501
864424509448455716864475634073614863564802786918403135283219595525248821452921736
9115964116992173946175488114152177664428070666241003290624015049359361410877447558272177147458556202107
102061584302080347892350976026091926323201141521776640321052999680601974374407524679680604661760283435205904909680630933370
1136283883716608680322819686405740223791104028701118955520941755465728021189497978883310859059203547348992024942297601039262401948617202325186507433
126333186975989761306219813797888122458107543552068882685493248025831007059968067806393532416127136987873281702727516160159630704640997691904037413446463772924193649320335884
1310977524091715584246994292063600642547128636905881615919553980661760671606183559168020148185506775044407415579607047083346467225683007966412806917330534403890998425613264767362072619986319281843580279
14189151184349560832461056011852054528518688013333561344356598009166823424167155316796948480564149194189701121410372985474252826444493477642243718756895293443873705099264029052788244481485653944324642668576669615661766225305414796478
Technical details
Node id
n-d48c40c60a8e7740 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-e2c46986e009fe93

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:           96,           18;
  3:         2304,          864,           81;
  4:        49152,        27648,         5184,         324;
  5:       983040,       737280,       207360,       25920,        1215;
  6:     18874368,     17694720,      6635520,     1244160,      116640,       4374;
  7:    352321536,    396361728,    185794560,    46448640,     6531840,     489888,     15309;
  8:   6442450944,   8455716864,   4756340736,  1486356480,   278691840,   31352832,   1959552,   52488;
  9: 115964116992, 173946175488, 114152177664, 42807066624, 10032906240, 1504935936, 141087744, 7558272, 177147;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (16*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[((16*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] = ((16*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.