Real-rooted atlas

A013610 → basis A013611 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(4\,n + 12\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 12\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
11212
296288384
3576345651849216
43072276488294482944196608
515360184320829440165888012441603932160
6737281105920663552019906560298598401791590475497472
73440646193152464486401857945604180377605016453122508226561409286144

All 15 rows

Technical details
Node id
n-cd9c6d2409b6123d · 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:      12;
  2:      96,       288;
  3:     576,      3456,       5184;
  4:    3072,     27648,      82944,       82944;
  5:   15360,    184320,     829440,     1658880,     1244160;
  6:   73728,   1105920,    6635520,    19906560,    29859840,    17915904;
  7:  344064,   6193152,   46448640,   185794560,   418037760,   501645312,    250822656;
  8: 1572864,  33030144,  297271296,  1486356480,  4459069440,  8026324992,   8026324992,   3439853568;
  9: 7077888, 169869312, 1783627776, 10701766656, 40131624960, 96315899904, 144473849856, 123834728448, 46438023168;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (4*n+12*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 12.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 12;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 12*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] = 12; for(n = 2, N, my(i = n + 1); v[i] = ((4*n + 12*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.