Real-rooted atlas

A013613 → x ↦ x+1 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(7\,n + 6\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 6\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
166
28472156
388215126483042
482322116818144518452728
57203024696031752018144038880856830
6605052259308044452803810240163296027993613366548
7494125825412184544546806223392040007520137168641959552202725978
839530064237180384609892416871274880746807040384072192109734912134369283011928816
9311299254213462345664038703681097806348811762210880806551603234566497288465264649069926444049458934
102421216420186779552406403870368012807740736016467095232014114653056080655160320296284262406348948480604661760636269962380
111864336643415980028372061637252292014088514809602113277221440217365657062415526118361607604629401602444345164804655895552039907676169098660462034
121423675254961342322383248575281021392014792940550080253593266572803043119198873626083878847488159697217433606844166461440195547613184033522447974426121388032129035548370664
1310796204016781110466698868852350572946672149573065561920288462340726560395605495853568395605495853568290648935729152155704786997760593161093324801525271382835223770463109121697890222081817250639553518
14813867687418890688113740952466396013524896146581604250681631410343768003204846167324206208553847694194995247472659502428163051813825156096145324467864576049825531839283211647526923468816639324176384109709829734425441508953749252
Technical details
Node id
n-06c3ea6d255a8d34 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-8e9037e0ccb64a62

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:        84,         72;
  3:       882,       1512,        648;
  4:      8232,      21168,      18144,        5184;
  5:     72030,     246960,     317520,      181440,       38880;
  6:    605052,    2593080,    4445280,     3810240,     1632960,     279936;
  7:   4941258,   25412184,   54454680,    62233920,    40007520,   13716864,    1959552;
  8:  39530064,  237180384,  609892416,   871274880,   746807040,  384072192,  109734912,  13436928;
  9: 311299254, 2134623456, 6403870368, 10978063488, 11762210880, 8065516032, 3456649728, 846526464, 90699264;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (7*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[((7*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] = ((7*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.