Real-rooted atlas

A008278 → x ↦ x+1 (shift +1) → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(1 + 3\,n + n x\right)\,P_{n-1}(x) + \left(-9 - 6\,x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1415
2199129
310975161201
47426732142711657
558156588281955246115821
6511936979237769101721381811170389
749811879518152179817981734281344114812032785
852963219701019749458231553917785211119278632279126546673
96098781712618683111232590755989228170097153202623361075219185381376085653
107549408481742682619175816455610141174803668745668576286312826906116446656527105315736591885
119983845261254525006522872527446218841876308792235666522185204904155620085074885837811931484312080193614616409
121403297687893917787230284893787084063600435545321727341013675646814831812748027384196738186220012825612414245397354413111625661357673
132087182244308633594682590186827725393407085696342958382311449288214308949167473787922387747089219353291977112797910799904130093110845488230129905322979421
143272531507213510736067827052916021945619638714370355730227886191563883681363779979527351106339172962524389948916363861793191394276730399431459138711393702803014857164251580513190237573
Technical details
Node id
n-4b57838a9b5b136d · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-1a0facbf7a815c78

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        4,         1;
  2:       19,         9,         1;
  3:      109,        75,        16,        1;
  4:      742,       673,       214,       27,        1;
  5:     5815,      6588,      2819,      552,       46,       1;
  6:    51193,     69792,     37769,    10172,     1381,      81,      1;
  7:   498118,    795181,    521798,   179817,    34281,    3441,    148,     1;
  8:  5296321,   9701019,   7494582,  3155391,   778521,  111927,   8632,   279,   1;
  9: 60987817, 126186831, 112325907, 55989228, 17009715, 3202623, 361075, 21918, 538, 1;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+1+n*x)*P(n-1,x) - (9+6*x+x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(3*n + 1 + n*x)*P[n - 1, x] - (9 + 6*x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (3*n + 1 + n*x)*v[i - 1] - (9 + 6*x + x^2)*deriv(v[i - 1], x)); 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

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.