Real-rooted atlas

A130123 → basis A013610 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(2\,n + 6\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 6\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
166
2247296
3724326481152
419217285184518412288
54805760259205184038880122880
61152172801036803110404665602799361179648
7268848384362880145152032659203919104195955211010048
861441290241161216580608017418240313528323135283213436928100663296
9138243317763483648209018887838208018811699228217548824186470490699264905969664
10307208294409953280696729603135283209405849601881169920241864704018139852806046617608053063680
11675842027520273715202189721601149603840413857382410346434560177367449601995383808013302558720399076761670866960384
12147456486604872990720656916480394149888016554295296496628858881064204697601596307046401596307046409577842278426121388032618475290624
1331948811501568189775872189775872012809871360614873825282152058388485533864427521037599580160138346610688012451194961926791560888321697890222085360119185408
14688128268369924830658565313724416398529331202152058388488608233553922582470066176581055764889696842627481601162111529779295081852436484754092621824109709829734446179488366592
Technical details
Node id
n-0eafffd13de70397 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-f9762973ec1089d7

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:     6;
  2:    24,     72;
  3:    72,    432,     648;
  4:   192,   1728,    5184,     5184;
  5:   480,   5760,   25920,    51840,    38880;
  6:  1152,  17280,  103680,   311040,   466560,    279936;
  7:  2688,  48384,  362880,  1451520,  3265920,   3919104,   1959552;
  8:  6144, 129024, 1161216,  5806080, 17418240,  31352832,  31352832,  13436928;
  9: 13824, 331776, 3483648, 20901888, 78382080, 188116992, 282175488, 241864704, 90699264;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (2*n+6*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 6*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] = 6; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 6*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.